Open Access Research

Applications of Kato’s inequality for n-tuples of operators in Hilbert spaces, (I)

Sever S Dragomir12, Yeol Je Cho3* and Young-Ho Kim4*

Author Affiliations

1 School of Computer Science and Mathematics, Victoria University of Technology, P.O. Box 14428, MCMC, Melbourne, VIC, 8001, Australia

2 School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg, 2050, South Africa

3 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, 660-701, Republic of Korea

4 Department of Mathematics, Changwon National University, Changwon, 641-773, Republic of Korea

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Journal of Inequalities and Applications 2013, 2013:21  doi:10.1186/1029-242X-2013-21


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/21


Received:21 August 2012
Accepted:28 December 2012
Published:16 January 2013

© 2013 Dragomir 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

In this paper, by the use of the famous Kato’s inequality for bounded linear operators, we establish some inequalities for n-tuples of operators and apply them for functions of normal operators defined by power series as well as for some norms and numerical radii that arise in multivariate operator theory.

MSC: 47A63, 47A99.

Keywords:
bounded linear operators; functions of normal operators; inequalities for operators; norm and numerical radius inequalities; Kato’s inequality

1 Introduction

The ‘square root’ of a positive bounded self-adjoint operator on H can be defined as follows (see, for instance, [[1], p.240]).

If the operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M1">View MathML</a> is self-adjoint and positive, then there exists a unique positive self-adjoint operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M2">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M3">View MathML</a>. If A is invertible, then so is B.

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M1">View MathML</a>, then the operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M5">View MathML</a> is self-adjoint and positive. Define the ‘absolute value’ operator by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M6">View MathML</a>.

In 1952, Kato [2] proved the following generalization of Schwarz inequality:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M7">View MathML</a>

(1.1)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a> and T is a bounded linear operator on H.

Utilizing the modulus notation introduced before, we can write (1.1) as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M10">View MathML</a>

(1.2)

For results related to the Kato’s inequality, see [2-18] and [19].

In the recent paper [20], by employing Kato’s inequality (1.2), Dragomir established the following results for sequences of bonded linear operators on complex Hilbert spaces.

Theorem 1.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M11">View MathML</a>be ann-tuple of bounded linear operators on the Hilbert space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M12">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M13">View MathML</a>be ann-tuple of nonnegative weights not all of them equal to zero. Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M14">View MathML</a>

(1.3)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a>.

He also obtained the following result.

Theorem 1.2With the assumptions in Theorem 1.1, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M18">View MathML</a>

(1.4)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>.

For various related results, see the papers [21-31].

Motivated by the above results, we establish in this paper other similar inequalities for n-tuples of bounded linear operators that can be obtained from Kato’s result (1.2) and apply them to functions of normal operators defined by power series as well as to some norms and numerical radii that can be associated with these n-tuples of bonded linear operators on Hilbert spaces.

2 Some inequalities for an n-tuple of linear operators

Employing Kato’s inequality (1.2), we can state the following new result.

Theorem 2.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M20">View MathML</a>be ann-tuple of bounded linear operators on the Hilbert space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M12">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M13">View MathML</a>be ann-tuple of nonnegative weights, not all of them equal to zero. Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M23">View MathML</a>

(2.1)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a>and, in particular, for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M26">View MathML</a>

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M27">View MathML</a>

(2.2)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>.

Proof Utilizing Kato’s inequality, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M29">View MathML</a>

and by replacing α with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M30">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M31">View MathML</a>

which by summation gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M32">View MathML</a>

(2.3)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>. By the elementary inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M35">View MathML</a>

(2.4)

we have

which by (2.3) produces

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M37">View MathML</a>

(2.5)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>. Multiplying the inequalities (2.5) with the positive weights <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M40">View MathML</a>, summing over j from 1 to n and utilizing the weighted Cauchy-Buniakowski-Schwarz inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M41">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M42">View MathML</a>, we have

(2.6)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>, and the inequality in (2.1) is proved. □

Remark 2.1 In order to provide some applications for functions of normal operators defined by power series, we need to state the inequality (2.1) for normal operators <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M45">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33">View MathML</a>, namely,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M47">View MathML</a>

(2.7)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a> and for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>.

From a different perspective that involves quadratics, we can state the following result as well.

Theorem 2.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M20">View MathML</a>be ann-tuple of bounded linear operators on the Hilbert space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M12">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M13">View MathML</a>be ann-tuple of nonnegative weights, not all of them equal to zero. Then we have

(2.8)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a>.

Proof We must prove the inequalities only in the case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M57">View MathML</a>, since the case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M58">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M59">View MathML</a> follows directly from the corresponding case of Kato’s inequality.

Utilizing Kato’s inequality for the operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M60">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M61">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M62">View MathML</a>

(2.9)

and, by replacing α with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M30">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M64">View MathML</a>

(2.10)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>.

By the Hölder-McCarthy inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M66">View MathML</a> that holds for the positive operator P, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M67">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M68">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M69">View MathML</a>, we also have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M70">View MathML</a>

(2.11)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M71">View MathML</a>

(2.12)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M75">View MathML</a>.

If we add (2.9) with (2.10) and make use of (2.11) and (2.12), we deduce

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M76">View MathML</a>

(2.13)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M75">View MathML</a>.

Now, if we multiply (2.13) with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M81">View MathML</a>, sum over j from 1 to n, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M82">View MathML</a>

(2.14)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M75">View MathML</a>.

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M86">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M87">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M33">View MathML</a>, then we get from (2.14) the first inequality in (2.8).

Now, on making use of the weighted Hölder discrete inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M89">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M42">View MathML</a>, we also have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M91">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M92">View MathML</a>

Summing these two inequalities, we deduce the second inequality in (2.8).

Finally, on utilizing the Hölder inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M93">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M94">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M95">View MathML</a>, we have

and the proof is concluded. □

Remark 2.2 For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M26">View MathML</a>, we get from (2.8) that

(2.15)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a>.

3 Inequalities for functions of normal operators

Now, by the help of power series <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M101">View MathML</a>, we can naturally construct another power series which will have as coefficients the absolute values of the coefficient of the original series, namely, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M102">View MathML</a>. It is obvious that this new power series will have the same radius of convergence as the original series. We also notice that if all coefficients <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M103">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M104">View MathML</a>.

As some natural examples that are useful for applications, we can point out that if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M105">View MathML</a>

(3.1)

then the corresponding functions constructed by the use of the absolute values of the coefficients are as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M106">View MathML</a>

(3.2)

The following result is a functional inequality for normal operators that can be obtained from (2.1).

Theorem 3.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M107">View MathML</a>be a function defined by power series with complex coefficients and convergent on the open disk<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M108">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M109">View MathML</a>. IfNis a normal operator on the Hilbert spaceH, for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M57">View MathML</a>, we have that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M111">View MathML</a>, then we have the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M112">View MathML</a>

(3.3)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>. In particular, if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M114">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M115">View MathML</a>

(3.4)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>.

Proof If N is a normal operator, then for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M117">View MathML</a>, we have that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M118">View MathML</a>

Now, utilizing the inequality (2.9), we can write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M119">View MathML</a>

(3.5)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M121">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M111">View MathML</a>, then it follows that the series <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M123">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M124">View MathML</a> are absolute convergent in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M125">View MathML</a>, and by taking the limit over <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M126">View MathML</a> in (3.5), we deduce the desired result (3.3). □

Remark 3.1 With the assumptions in Theorem 3.1, if we take the supremum over <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M127">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M128">View MathML</a>, then we get the vector inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M129">View MathML</a>

(3.6)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M68">View MathML</a>, which in its turn produces the norm inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M131">View MathML</a>

(3.7)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a>. Making use of the examples in (3.1) and (3.2), we can state the vector inequalities

(3.8)

and

(3.9)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M136">View MathML</a>. We also have the inequalities

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M137">View MathML</a>

(3.10)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M138">View MathML</a>

(3.11)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> and N a normal operator.

If we utilize the following function as power series representations with nonnegative coefficients:

(3.12)

where Γ is the gamma function, then we can state the following vector inequalities:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M141">View MathML</a>

(3.13)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> and N a normal operator. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M136">View MathML</a>, then we also have the inequalities

(3.14)

(3.15)

and

(3.16)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>. From a different perspective, we also have

Theorem 3.2With the assumption of Theorem 3.1 and ifNis a normal operator on the Hilbert spaceHand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M148">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M149">View MathML</a>, then we have the inequalities

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M150">View MathML</a>

(3.17)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a>. In particular, for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M26">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M155">View MathML</a>

(3.18)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a>.

Proof If we use the second and third inequality from (2.8) for powers of operators, we have

(3.19)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a>. Since N is a normal operator on the Hilbert space H, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M162">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M163">View MathML</a>

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M164">View MathML</a> and for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a>. Then from (3.19), we have

(3.20)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a>. By the weighted Cauchy-Buniakowski-Schwarz inequality, we also have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M171">View MathML</a>

(3.21)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a>.

Now, since the series <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M174">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M175">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M176">View MathML</a> are convergent, then by (3.20) and (3.21), on letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M126">View MathML</a>, we deduce the desired result (3.17). □

Similar inequalities for some particular functions of interest can be stated. However, the details are left to the interested reader.

4 Applications for the Euclidean norm

In [29], the author has introduced the following norm on the Cartesian product <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M178">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M125">View MathML</a> denotes the Banach algebra of all bounded linear operators defined on the complex Hilbert space H:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M180">View MathML</a>

(4.1)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M181">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M182">View MathML</a> is the Euclidean closed ball in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M183">View MathML</a>.

It is clear that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M184">View MathML</a> is a norm on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M185">View MathML</a> and, for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M186">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M187">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M188">View MathML</a> is the adjoint operator of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M60">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M61">View MathML</a>. We call this the Euclidean norm of an n-tuple of operators <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M191">View MathML</a>.

It has been shown in [29] that the following basic inequality for theEuclidean norm holds true:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M192">View MathML</a>

(4.2)

for any n-tuple <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M191">View MathML</a> and the constants <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M194">View MathML</a> and 1 are best possible.

In the same paper [29], the author has introduced the Euclidean operator radius of an n-tuple of operators <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M195">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M196">View MathML</a>

(4.3)

and proved that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M197">View MathML</a> is a norm on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M185">View MathML</a> and satisfies the double inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M199">View MathML</a>

(4.4)

for each n-tuple <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M191">View MathML</a>.

As pointed out in [29], the Euclidean numerical radius also satisfies the double inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M201">View MathML</a>

(4.5)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M202">View MathML</a> and the constants <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M203">View MathML</a> and 1 are best possible.

In [30], by utilizing the concept of hypo-Euclidean norm on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M204">View MathML</a>, we obtained the following representation for the Euclidean norm.

Proposition 4.1For any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M205">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M206">View MathML</a>

(4.6)

We can state now the following result.

Theorem 4.1For any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M205">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M208">View MathML</a>

(4.7)

and

(4.8)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a>.

Proof We have from the second inequality in (2.8)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M211">View MathML</a>

(4.9)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M16">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a>. Taking the supremum over <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M215">View MathML</a>, we have

which proves the first part of (4.7). The second part follows by the elementary inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M217">View MathML</a>

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M218">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M9">View MathML</a>. The inequality (4.8) follows from (4.9) by taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M220">View MathML</a> and then the supremum over <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M69">View MathML</a>. □

5 Applications for s-1-norm and s-1-numerical radius

Following [20], we consider the s-p-norm of the n-tuple of operators <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M191">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M223">View MathML</a>

(5.1)

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M224">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M225">View MathML</a>

We are interested in this section in the case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M226">View MathML</a>, namely, on the s-1-norm defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M227">View MathML</a>

Since for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a> we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M229">View MathML</a>, then by the properties of the supremum, we get the basic inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M230">View MathML</a>

(5.2)

Similarly, we can also consider the s-p-numerical radius of the n-tuple of operators <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M205">View MathML</a> by [20]

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M232">View MathML</a>

(5.3)

which for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M224">View MathML</a> reduces to the Euclidean operator radius introduced previously.

We observe that the s-p-numerical radius is also a norm on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M234">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M235">View MathML</a>, and for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M226">View MathML</a> it satisfies the basic inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M237">View MathML</a>

(5.4)

We can state the following result.

Theorem 5.1For any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M205">View MathML</a>, we have

(5.5)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M240">View MathML</a>

(5.6)

Proof From (2.1) we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M241">View MathML</a>

(5.7)

for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M8">View MathML</a>.

Taking the supremum over <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M128">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M69">View MathML</a> in (5.7), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M245">View MathML</a>

and the first inequality in (5.5) is proved. The second part follows by the arithmetic mean-geometric mean inequality.

Now, if we take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M220">View MathML</a> in (5.7), then we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M247">View MathML</a>

Taking the supremum over <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M69">View MathML</a>, we deduce the desired result (5.6). □

Remark 5.1 If we take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M26">View MathML</a> in the first inequality in (5.5), then we deduce

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M250">View MathML</a>

(5.8)

and then we get the following refinement of the generalized triangle inequality:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M251">View MathML</a>

From (5.6) we also have, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M26">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/21/mathml/M253">View MathML</a>

(5.9)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The authors wish to thank the anonymous referees for their valuable comments. Also, this research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0008474).

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