SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Sum of squared logarithms - an inequality relating positive definite matrices and their matrix logarithm

Mircea Bîrsan12, Patrizio Neff1* and Johannes Lankeit1

Author Affiliations

1 Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Essen, Germany

2 Department of Mathematics, University ‘A.I. Cuza’ of Iaşi, Iaşi, Romania

For all author emails, please log on.

Journal of Inequalities and Applications 2013, 2013:168  doi:10.1186/1029-242X-2013-168

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/168


Received:21 January 2013
Accepted:28 March 2013
Published:12 April 2013

© 2013 Bîrsan et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M1">View MathML</a> be such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M2">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M3">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M4">View MathML</a>

This can also be stated in terms of real positive definite <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M5">View MathML</a>-matrices <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M6">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M7">View MathML</a>: If their determinants are equal, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M8">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M9">View MathML</a>

where log is the principal matrix logarithm and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M10">View MathML</a> denotes the Frobenius matrix norm. Applications in matrix analysis and nonlinear elasticity are indicated.

MSC: 26D05, 26D07.

Keywords:
matrix logarithm; elementary symmetric polynomials; inequality; characteristic polynomial; positive definite matrices; means

1 Introduction

Convexity is a powerful source for obtaining new inequalities; see, e.g., [1,2]. In applications coming from nonlinear elasticity, we are faced, however, with variants of the squared logarithm function; see the last section. The function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M11">View MathML</a> is neither convex nor concave. Nevertheless, the sum of squared logarithms inequality holds. We will proceed as follows: In the first section, we will give several equivalent formulations of the inequality, for example, in terms of the coefficients of the characteristic polynomial (Theorem 1), in terms of elementary symmetric polynomials (Theorem 3), in terms of means (Theorem 5) or in terms of the Frobenius matrix norm (Theorem 7). A proof of the inequality will be given in Section 2, and some counterexamples for slightly changed variants of the inequality are discussed in Section 3. In the last section, an application of the sum of squared logarithms inequality in matrix analysis and in the mathematical theory of nonlinear elasticity is indicated.

2 Formulations of the problem

All theorems in this section are equivalent.

Theorem 1For<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M12">View MathML</a>or<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M13">View MathML</a>let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M14">View MathML</a>be positive definite real matrices. Let the coefficients of the characteristic polynomials of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M6">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M7">View MathML</a>satisfy

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M17">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M18">View MathML</a>

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M13">View MathML</a>, we will now give equivalent formulations of this statement. The case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M12">View MathML</a> can be treated analogously. For its proof, see Remark 15. By orthogonal diagonalization of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M6">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M7">View MathML</a>, the inequalities can be rewritten in terms of the eigenvalues <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M23">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M24">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M25">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M26">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M27">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M28">View MathML</a>, respectively.

Theorem 2Let the real numbers<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M29">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M30">View MathML</a>be such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M31">View MathML</a>

(1)

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M32">View MathML</a>

(2)

The elementary symmetric polynomials, see, e.g., [[3], p.178]

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M33">View MathML</a>

are known to have the Schur-concavity property (i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M34">View MathML</a> is Schur-convex) [1,4]; see (16). It is possible to express the problem in terms of these elementary symmetric polynomials as follows.

Theorem 3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M29">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M30">View MathML</a>satisfy

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M37">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M38">View MathML</a>

Because <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M39">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M40">View MathML</a>

Thus, we obtain the following theorem.

Theorem 4Let the real numbers<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M29">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M30">View MathML</a>be such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M43">View MathML</a>

(3)

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M44">View MathML</a>

(4)

The conditions (3) are also simple expressions in terms of arithmetic, harmonic and geometric and quadratic mean

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M45">View MathML</a>

Theorem 5Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M29">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M30">View MathML</a>. Then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M48">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M49">View MathML</a> (‘reverse!) and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M50">View MathML</a>imply

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M51">View MathML</a>

We denote by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M52">View MathML</a>

and arrive at

Theorem 6Let the real numbers<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M53">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M54">View MathML</a>be such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M55">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M56">View MathML</a>and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M57">View MathML</a>

(5)

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M58">View MathML</a>

(6)

If we again view <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M54">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M53">View MathML</a> as eigenvalues of positive definite matrices, an equivalent formulation of the problem can be given in terms of their Frobenius matrix norms:

Theorem 7For<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M61">View MathML</a>, let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M14">View MathML</a>be positive definite real matrices. Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M63">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M64">View MathML</a>

Let us reconsider the formulation from Theorem 5. If we denote

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M65">View MathML</a>

from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M66">View MathML</a>, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M67">View MathML</a>

Theorem 8Let the real numbers<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M68">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M69">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M70">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M71">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M72">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M73">View MathML</a>be such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M74">View MathML</a>

(7)

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M75">View MathML</a>

(8)

In order to prove Theorem 8, one can assume without loss of generality that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M76">View MathML</a>

(9)

Thus, we have the equivalent formulation

Theorem 9Let the real numbers<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M77">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M78">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M79">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M80">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M81">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M82">View MathML</a>be such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M83">View MathML</a>

(10)

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M84">View MathML</a>

(11)

Let us prove that Theorem 8 can be reformulated as Theorem 9. Indeed, let us assume that Theorem 9 is valid and show that the statement of Theorem 8 also holds true. We denote by s the sum <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M85">View MathML</a> and we designate

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M86">View MathML</a>

Then the real numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M87">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M88">View MathML</a> satisfy the hypotheses of Theorem 9 and we obtain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M89">View MathML</a>. This inequality is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M90">View MathML</a>

which, by virtue of the condition (7)3, reduces to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M91">View MathML</a>

Thus, Theorem 8 is also valid.

By virtue of the logical equivalence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M92">View MathML</a>

for any statements A, B, C, we can formulate the inequality (11) (i.e., Theorem 9) in the following equivalent manner.

Theorem 10Let the real numbers<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M68">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M69">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M70">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M71">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M72">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M73">View MathML</a>be such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M99">View MathML</a>

(12)

Then one of the following inequalities holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M100">View MathML</a>

(13)

We use the statement of Theorem 10 for the proof.

Before continuing, let us show that our new inequality is not a consequence of majorization and Karamata’s inequality [5]. Consider <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M101">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M102">View MathML</a> arranged already in decreasing order <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M103">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M104">View MathML</a>. If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M105">View MathML</a>

(14)

we say that z majorizes c, denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M106">View MathML</a>. The following result is well known [[6], p.89], [4,5]. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M107">View MathML</a> is convex, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M108">View MathML</a>

(15)

A function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M109">View MathML</a> which satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M110">View MathML</a>

(16)

is called Schur-convex. In Theorem 8, the convex function to be considered would be <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M111">View MathML</a>. Do conditions (7) (upon rearrangement of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M112">View MathML</a> if necessary) yield already majorization <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M106">View MathML</a>? This is not the case, as we explain now. Let the real numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M114">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M115">View MathML</a> be such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M116">View MathML</a>

(17)

These conditions do not imply the majorization <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M106">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M118">View MathML</a>

(18)

Therefore, our inequality (i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M119">View MathML</a>) does not follow from majorization in disguise.

Indeed, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M120">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M121">View MathML</a>

Then we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M122">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M123">View MathML</a>, together with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M124">View MathML</a>

but the majorization inequalities (18) are not satisfied, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M125">View MathML</a>.

3 Proof of the inequality

Of course, we may assume without loss of generality that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M115">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M114">View MathML</a> (and the same for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M128">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M53">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M54">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M131">View MathML</a>).

The proof begins with the crucial lemma.

Lemma 11Let the real numbers<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M132">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M133">View MathML</a>be such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M134">View MathML</a>

(19)

Then the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M135">View MathML</a>

(20)

is satisfied if and only if the relation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M136">View MathML</a>

(21)

holds, or equivalently, if and only if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M137">View MathML</a>

(22)

holds.

Proof Let us denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M138">View MathML</a>. Then, from (19), it follows

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M139">View MathML</a>

and we find

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M140">View MathML</a>

(23)

In view of (19) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M141">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M142">View MathML</a>, one can show that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M143">View MathML</a>

(24)

Indeed, let us verify the relations (24). We have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M144">View MathML</a>

which hold true since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M145">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M146">View MathML</a>. Similarly, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M147">View MathML</a>

which holds true since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M145">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M149">View MathML</a>. Also, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M150">View MathML</a>

which hold true since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M149">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M146">View MathML</a>. One can show in the same way that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M153">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M154">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M155">View MathML</a>, so that (24) has been verified.

We prove now that the inequality (21) holds if and only if (22) holds. Indeed, using (23)2,4 and (24) we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M156">View MathML</a>

since the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M157">View MathML</a> is decreasing for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M158">View MathML</a>.

Let us prove next that the inequalities (20) and (21) are equivalent. To accomplish this, we introduce the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M159">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M160">View MathML</a>

(25)

Taking into account (23) and (24)1, the inequality (20) can be written equivalently as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M161">View MathML</a>

(26)

which is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M162">View MathML</a>

since the function f defined by (25) is monotone increasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M163">View MathML</a>, as we show next. To this aim, we denote by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M164">View MathML</a>

Then the function (25) can be written as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M165">View MathML</a>

(27)

We have to show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M166">View MathML</a> is decreasing with respect to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M167">View MathML</a>. We compute the first derivative

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M168">View MathML</a>

(28)

The function (28) has the same sign as the function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M169">View MathML</a>

(29)

i.e., the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M170">View MathML</a> given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M171">View MathML</a>

(30)

In order to show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M172">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M173">View MathML</a>, we remark that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M174">View MathML</a> for fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M175">View MathML</a> and we compute

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M176">View MathML</a>

since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M177">View MathML</a> implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M178">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M179">View MathML</a>.

Consequently, the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M180">View MathML</a> is decreasing with respect to r and for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M181">View MathML</a> we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M182">View MathML</a>

(31)

From (29) and (31), it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M183">View MathML</a> is decreasing with respect to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M184">View MathML</a>. This means that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M185">View MathML</a> is increasing as a function of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M153">View MathML</a>, i.e., the relation (26) is indeed equivalent to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M187">View MathML</a> and the proof is complete. □

Consequence 12Let the real numbers<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M141">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M133">View MathML</a>be such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M190">View MathML</a>

Then one of the following inequalities holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M191">View MathML</a>

(32)

or

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M192">View MathML</a>

(33)

The inequalities (32) and (33) are satisfied simultaneously if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M193">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M194">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M195">View MathML</a>.

Proof According to Lemma 11, the inequality (32) is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M196">View MathML</a>

(34)

while the inequality (33) is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M197">View MathML</a>

(35)

Since one of the relations (34) and (35) must hold, we have proved that one of the inequalities (32) and (33) is satisfied. They are simultaneously satisfied if and only if both (34) and (35) hold true, i.e., <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M193">View MathML</a> (and consequently <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M194">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M195">View MathML</a>). □

Consequence 13Let the real numbers<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M141">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M133">View MathML</a>be such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M203">View MathML</a>

Then we have<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M193">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M194">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M195">View MathML</a>.

Proof Since by hypothesis <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M207">View MathML</a> holds, we can apply Lemma 11 to deduce <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M208">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M209">View MathML</a>.

On the other hand, by virtue of the inverse inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M210">View MathML</a> and Lemma 11, we obtain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M211">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M212">View MathML</a>. In conclusion, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M193">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M195">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M194">View MathML</a>. □

Proof of Theorem 10 In order to prove (13), we define the real numbers

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M216">View MathML</a>

(36)

Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M217">View MathML</a>

(37)

If we apply the Consequence 12 for the numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M115">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M219">View MathML</a>, then we obtain that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M220">View MathML</a>

(38)

In what follows, let us show that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M221">View MathML</a>

(39)

Using the notations <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M222">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M223">View MathML</a>

we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M224">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M225">View MathML</a>. With the help of the function h defined in (27), we can write the inequality (39) in the form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M226">View MathML</a>

(40)

The relation (40) asserts that the function h defined in (27) is increasing with respect to the first variable <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M227">View MathML</a>. To show this, we compute the derivative

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M228">View MathML</a>

(41)

By virtue of the Chebyshev’s sum inequality, we deduce from (41) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M229">View MathML</a>

(42)

Indeed, the Chebyshev’s sum inequality [[6], 2.17] asserts that: if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M230">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M231">View MathML</a> then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M232">View MathML</a>

In our case, we derive the following result: for any real numbers x, y, z such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M233">View MathML</a>, the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M234">View MathML</a>

(43)

holds true, with equality if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M235">View MathML</a>.

Applying the result (43) to the function (41), we deduce the relation (42). This means that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M183">View MathML</a> is an increasing function of r, i.e. the inequality (40) holds, and hence, we have proved (39).

One can show analogously that the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M237">View MathML</a>

(44)

is also valid. From (38), (39) and (44), it follows that the assertion (13) holds true. Thus, the proof of Theorem 10 is complete. □

Since the statements of the Theorems 8 and 10 are equivalent, we have proved also the inequality (8).

Remark 14 The inequality (8) becomes an equality if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M238">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M239">View MathML</a>.

Proof Indeed, assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M240">View MathML</a>. Then we can apply the Consequence 12 and we deduce that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M241">View MathML</a>

(45)

Taking into account (7)1,2 in conjunction with (45), we find

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M242">View MathML</a>

(46)

By virtue of (46), we can apply the Consequence 13 to derive <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M243">View MathML</a>, and consequently <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M244">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M245">View MathML</a>. □

Let us prove the following version of the inequality (6) for two pairs of numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M246">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M247">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M248">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M249">View MathML</a>:

Remark 15 If the real numbers <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M250">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M251">View MathML</a> are such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M252">View MathML</a>

(47)

then the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M253">View MathML</a>

(48)

holds true. Note that the additional condition

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M254">View MathML</a>

is automatically fulfilled.

Proof Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M255">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M250">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M251">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M258">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M259">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M260">View MathML</a>

so that the inequality (48) is equivalent to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M261">View MathML</a>, i.e., we have to show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M262">View MathML</a>.

Indeed, if we insert <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M263">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M264">View MathML</a> into the inequality (47)1 then we find

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M265">View MathML</a>

which means that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M266">View MathML</a> since the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M267">View MathML</a> is increasing for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M268">View MathML</a>. This completes the proof. □

Alternative proof of Remark 15 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M269">View MathML</a>. Then (47) implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M270">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M271">View MathML</a> as well as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M272">View MathML</a>

(49)

because <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M273">View MathML</a>, and Theorem 6 provides the assertion. □

4 Some counterexamples for weakened assumptions

Example 16 Unlike in the 2D case in Remark 15, for two triples of numbers the second condition (18)2 of Theorem 2, namely <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M274">View MathML</a>, cannot be removed. Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M275">View MathML</a>

Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M276">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M277">View MathML</a>

but

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M278">View MathML</a>

Example 17 The condition <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M279">View MathML</a> cannot be weakened to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M280">View MathML</a>. Indeed, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M281">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M282">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M283">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M284">View MathML</a>

But nevertheless

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M285">View MathML</a>

A counterexample for the two variable case can be constructed analogously.

Example 18 Even with an analogous condition, the inequality (4) does not hold for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M286">View MathML</a> numbers (without further assumptions). Indeed, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M287">View MathML</a>

Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M288">View MathML</a>. Also,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M289">View MathML</a>

Furthermore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M290">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M291">View MathML</a>

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M292">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M293">View MathML</a> and, therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M294">View MathML</a>

Nevertheless, for the sum of squared logarithms, the ‘reverse’ inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M295">View MathML</a>

holds true.

Example 19 The inequality (4) does not remain true either, if the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M296">View MathML</a> is replaced by its linearization <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M297">View MathML</a>. Indeed, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M298">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M299">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M300">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M301">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M302">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M303">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M304">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M305">View MathML</a>

But

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M306">View MathML</a>

5 Conjecture for arbitrary n

The structure of the inequality in dimensions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M12">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M13">View MathML</a> and extensive numerical sampling strongly suggest that the inequality holds for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M309">View MathML</a> if the n corresponding conditions are satisfied. More precisely, in terms of the elementary symmetric polynomials, we expect the following:

Conjecture 20Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M309">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M311">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M312">View MathML</a>. If for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M313">View MathML</a>we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M314">View MathML</a>

then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M315">View MathML</a>

6 Applications

The investigation in this paper has been motivated by some recent applications. The new sum of squared logarithms inequality is one of the fundamental tools in deducing a novel optimality result in matrix analysis and the conditions in the form (3) had been deduced in the course of that work. Optimality in the matrix problem suggested the sum of squared logarithms inequality. Indeed, based on the present result in [7], it has been shown that for all invertible <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M316">View MathML</a> and for any definition of the matrix logarithm as possibly multivalued solution <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M317">View MathML</a> of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M318">View MathML</a> it holds

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M319">View MathML</a>

(50)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M320">View MathML</a> is the Hermitian part of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M321">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M322">View MathML</a> is the unitary factor in the polar decomposition of Z into unitary and Hermitian positive definite matrix H

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M323">View MathML</a>

(51)

This result (50) generalizes the fact that for any complex logarithm and for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M324">View MathML</a>

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M325">View MathML</a>

(52)

The optimality result (50) can now also be viewed as another characterization of the unitary factor in the polar decomposition. In addition, in a forthcoming contribution [8], we use (50) to calculate the geodesic distance of the isochoric part of the deformation gradient <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M326">View MathML</a> to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M327">View MathML</a> in the canonical left-invariant Riemannian metric on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M328">View MathML</a>, to the effect that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M329">View MathML</a>

(53)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M330">View MathML</a> is the orthogonal projection of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M331">View MathML</a> to trace free matrices. Thereby, we provide a rigorous geometric justification for the preferred use of the Hencky-strain measure <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/168/mathml/M332">View MathML</a> in nonlinear elasticity and plasticity theory [9].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed fully to all parts of the manuscript. Notably all ideas have emerged by continuous discussions among them.

Acknowledgements

The first author (MB) was supported by the German state grant: ‘Programm des Bundes und der Länder für bessere Studienbedingungen und mehr Qualität in der Lehre’.

References

  1. Guan, K: Schur-convexity of the complete elementary symmetric functions. J. Inequal. Appl.. 2006, Article ID 67624. doi:10.1155/JIA/2006/67624 (2006)

  2. Roventa, I: A note on Schur-concave functions. J. Inequal. Appl.. 159, 1–9 (2012)

  3. Steele, JM: The Cauchy-Schwarz Master Class: an Introduction to the Art of Mathematical Inequalities, Cambridge University Press, Cambridge (2004)

  4. Khan, AR, Latif, N, Pečarić, J: Exponential convexity for majorization. J. Inequal. Appl.. 105, 1–13 (2012)

  5. Karamata, J: Sur une inégalité relative aux fonctions convexes. Publ. Math. Univ. Belgrad. 1, 145–148 (1932)

  6. Hardy, GH, Littlewood, JE, Pólya, G: Inequalities, The University Press, Cambridge (1934)

  7. Neff, P, Nagatsukasa, Y, Fischle, A: The unitary polar factor Q=Up minimizes ∥Log(Q∗Z)∥2 and ∥sym∗Log(Q∗Z)∥2 in the spectral norm in any dimension and the Frobenius matrix norm in three dimensions (2013, submitted)

  8. Neff, P, Eidel, B, Osterbrink, F, Martin, R: The isotropic Hencky strain energy ∥logU∥2 measures the geodesic distance of the deformation gradient F∈GL+(n) to SO(n) in the unique left-invariant Riemannian metric on GL+(n) which is also right O(n)-invariant (2013, in preparation)

  9. Hencky, H: Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Z. Techn. Physik. 9, 215–220 (1928)