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Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli

Mohsan Raza1* and Sarfraz Nawaz Malik2

Author Affiliations

1 Department of Mathematics, G.C. University Faisalabad, Faisalabad, Pakistan

2 Department of Mathematics, COMSATS Institute of Information Technology, Defense Road Lahore, Pakistan

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Journal of Inequalities and Applications 2013, 2013:412  doi:10.1186/1029-242X-2013-412


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/412


Received:15 February 2013
Accepted:8 August 2013
Published:28 August 2013

© 2013 Raza and Malik; 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, the upper bound of the Hankel determinant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M1">View MathML</a> for a subclass of analytic functions associated with right half of the lemniscate of Bernoulli <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M2">View MathML</a> is investigated.

MSC: 30C45, 30C50.

Keywords:
starlike functions; subordination; lemniscate of Bernoulli; Toeplitz determinants; Hankel determinants

1 Introduction and preliminaries

Let A be the class of functions f of the form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M3">View MathML</a>

(1.1)

which are analytic in the open unit disk <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M4">View MathML</a>. A function f is said to be subordinate to a function g, written as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M5">View MathML</a>, if there exists a Schwartz function w with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M6">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M7">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M8">View MathML</a>. In particular, if g is univalent in E, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M9">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M10">View MathML</a>.

Let P denote the class of analytic functions p normalized by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M11">View MathML</a>

(1.2)

such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M12">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M13">View MathML</a> be the class of functions defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M14">View MathML</a>

Thus a function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15">View MathML</a> is such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M16">View MathML</a> lies in the region bounded by the right half of the lemniscate of Bernoulli given by the relation <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M17">View MathML</a>. It can easily be seen that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15">View MathML</a> if it satisfies the condition

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M19">View MathML</a>

(1.3)

This class of functions was introduced by Sokół and Stankiewicz [1] and further investigated by some authors. For details, see [2,3].

Noonan and Thomas [4] have studied the qth Hankel determinant defined as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M20">View MathML</a>

(1.4)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M21">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M22">View MathML</a>. The Hankel determinant plays an important role in the study of singularities; for instance, see [[5], p.329] and Edrei [6]. This is also important in the study of power series with integral coefficients [[5], p.323] and Cantor [7]. For the use of the Hankel determinant in the study of meromorphic functions, see [8], and various properties of these determinants can be found in [[9], Chapter 4]. It is well known that the Fekete-Szegö functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M23">View MathML</a>. This functional is further generalized as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M24">View MathML</a> for some μ (real as well as complex). Fekete and Szegö gave sharp estimates of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M24">View MathML</a> for μ real and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M26">View MathML</a>, the class of univalent functions. It is a very great combination of the two coefficients which describes the area problems posted earlier by Gronwall in 1914-15. Moreover, we also know that the functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M27">View MathML</a> is equivalent to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M28">View MathML</a>. The qth Hankel determinant for some subclasses of analytic functions was recently studied by Arif et al.[10] and Arif et al.[11]. The functional <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M27">View MathML</a> has been studied by many authors, see [12-14]. Babalola [15] studied the Hankel determinant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M30">View MathML</a> for some subclasses of analytic functions. In the present investigation, we determine the upper bounds of the Hankel determinant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M30">View MathML</a> for a subclass of analytic functions related with lemniscate of Bernoulli by using Toeplitz determinants.

We need the following lemmas which will be used in our main results.

Lemma 1.1[16]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M32">View MathML</a>and of the form (1.2). Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M33">View MathML</a>

When<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M34">View MathML</a>or<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M35">View MathML</a>, the equality holds if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M36">View MathML</a>is<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M37">View MathML</a>or one of its rotations. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M38">View MathML</a>, then the equality holds if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M39">View MathML</a>or one of its rotations. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M40">View MathML</a>, the equality holds if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M41">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M42">View MathML</a>) or one of its rotations. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M43">View MathML</a>, the equality holds if and only ifpis the reciprocal of one of the functions such that the equality holds in the case of<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M40">View MathML</a>. Although the above upper bound is sharp, when<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M38">View MathML</a>, it can improved as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M46">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M47">View MathML</a>

Lemma 1.2[16]

If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M48">View MathML</a>is a function with positive real part inE, then forva complex number

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M49">View MathML</a>

This result is sharp for the functions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M50">View MathML</a>

Lemma 1.3[17]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M32">View MathML</a>and of the form (1.2). Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M52">View MathML</a>

for somex, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M53">View MathML</a>, and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M54">View MathML</a>

for somez, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M55">View MathML</a>.

2 Main results

Although we have discussed the Hankel determinant problem in the paper, the first two problems are specifically related with the Fekete-Szegö functional, which is a special case of the Hankel determinant.

Theorem 2.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15">View MathML</a>and of the form (1.1). Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M57">View MathML</a>

Furthermore, for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M58">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M59">View MathML</a>

and for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M60">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M61">View MathML</a>

These results are sharp.

Proof If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15">View MathML</a>, then it follows from (1.3) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M63">View MathML</a>

(2.1)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M64">View MathML</a>. Define a function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M65">View MathML</a>

It is clear that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M32">View MathML</a>. This implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M67">View MathML</a>

From (2.1), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M68">View MathML</a>

with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M69">View MathML</a>

Now

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M70">View MathML</a>

Similarly,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M71">View MathML</a>

Therefore

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M72">View MathML</a>

(2.2)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M73">View MathML</a>

(2.3)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M74">View MathML</a>

(2.4)

This implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M75">View MathML</a>

Now, using Lemma 1.1, we have the required result. □

The results are sharp for the functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M76">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M77">View MathML</a>, such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M78">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M79">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M80">View MathML</a>.

Theorem 2.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15">View MathML</a>and of the form (1.1). Then for a complex numberμ,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M82">View MathML</a>

Proof Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M83">View MathML</a>

therefore, using Lemma 1.2, we get the result. This result is sharp for the functions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M84">View MathML</a>

or

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M85">View MathML</a>

 □

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M86">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M87">View MathML</a>.

Corollary 2.3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15">View MathML</a>and of the form (1.1). Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M89">View MathML</a>

Theorem 2.4Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15">View MathML</a>and of the form (1.1). Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M91">View MathML</a>

Proof From (2.2), (2.3) and (2.4), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M92">View MathML</a>

Putting the values of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M93">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M94">View MathML</a> from Lemma 1.3, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M95">View MathML</a>, and taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M96">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M97">View MathML</a>

After simple calculations, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M98">View MathML</a>

Now, applying the triangle inequality and replacing <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M99">View MathML</a> by ρ, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M100">View MathML</a>

Differentiating with respect to ρ, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M101">View MathML</a>

It is clear that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M102">View MathML</a>, which shows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M103">View MathML</a> is an increasing function on the closed interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M104">View MathML</a>. This implies that maximum occurs at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M105">View MathML</a>. Therefore <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M106">View MathML</a> (say). Now

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M107">View MathML</a>

Therefore

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M108">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M109">View MathML</a>

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M110">View MathML</a>. This shows that maximum of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M111">View MathML</a> occurs at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M110">View MathML</a>. Hence, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M113">View MathML</a>

This result is sharp for the functions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M114">View MathML</a>

or

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M115">View MathML</a>

 □

Theorem 2.5Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15">View MathML</a>and of the form (1.1). Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M117">View MathML</a>

Proof Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M118">View MathML</a>

Therefore, by using Lemma 1.3, we can obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M119">View MathML</a>

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M120">View MathML</a>

(2.5)

We assume that the upper bound occurs at the interior point of the rectangle <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M121">View MathML</a>. Differentiating (2.5) with respect to ρ, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M122">View MathML</a>

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M123">View MathML</a> and fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M124">View MathML</a>, it can easily be seen that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M125">View MathML</a>. This shows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M126">View MathML</a> is a decreasing function of ρ, which contradicts our assumption; therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M127">View MathML</a>. This implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M128">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M129">View MathML</a>

for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M110">View MathML</a>. Therefore <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M110">View MathML</a> is a point of maximum. Hence, we get the required result. □

Lemma 2.6If the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M132">View MathML</a>belongs to the class<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M13">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M134">View MathML</a>

These estimations are sharp. The first three bounds were obtained by Sokół[3]and the bound for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M135">View MathML</a>can be obtained in a similar way.

Theorem 2.7Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M15">View MathML</a>and of the form (1.1). Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M137">View MathML</a>

Proof Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M138">View MathML</a>

Now, using the triangle inequality, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M139">View MathML</a>

Using the fact that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M140">View MathML</a> with the results of Corollary 2.3, Theorem 2.4, Theorem 2.5 and Lemma 2.6, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/412/mathml/M141">View MathML</a>

 □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MR and SNM jointly discussed and presented the ideas of this article. MR made the text file and all the communications regarding the manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for helpful comments and suggestions which improved the presentation of the paper.

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