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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Bounds for the second Hankel determinant of certain univalent functions

See Keong Lee1*, V Ravichandran2 and Shamani Supramaniam1

Author Affiliations

1 School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia

2 Department of Mathematics, University of Delhi, Delhi, 110007, India

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Journal of Inequalities and Applications 2013, 2013:281  doi:10.1186/1029-242X-2013-281

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/281


Received:11 December 2012
Accepted:11 March 2013
Published:5 June 2013

© 2013 Lee 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

The estimates for the second Hankel determinant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M1">View MathML</a> of the analytic function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M2">View MathML</a> , for which either <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M3">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M4">View MathML</a> is subordinate to a certain analytic function, are investigated. The estimates for the Hankel determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike and lemniscate starlike functions are obtained.

MSC: 30C45, 30C80.

Dedication

Dedicated to Professor Hari M Srivastava

1 Introduction

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M5">View MathML</a> denote the class of all analytic functions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M6">View MathML</a>

(1)

defined on the open unit disk <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M7">View MathML</a>. The Hankel determinants<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M8">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M9">View MathML</a> , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M10">View MathML</a>) of the function f are defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M11">View MathML</a>

Hankel determinants are useful, for example, in showing that a function of bounded characteristic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12">View MathML</a>, i.e., a function which is a ratio of two bounded analytic functions with its Laurent series around the origin having integral coefficients, is rational [1]. For the use of Hankel determinants in the study of meromorphic functions, see [2], and various properties of these determinants can be found in [[3], Chapter 4]. In 1966, Pommerenke [4] investigated the Hankel determinant of areally mean p-valent functions, univalent functions as well as of starlike functions. In [5], he proved that the Hankel determinants of univalent functions satisfy

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M13">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M14">View MathML</a> and K depends only on q. Later, Hayman [6] proved that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M15">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M9">View MathML</a> ; A an absolute constant) for areally mean univalent functions. In [7-9], the estimates for the Hankel determinant of areally mean p-valent functions were investigated. ElHosh obtained bounds for Hankel determinants of univalent functions with a positive Hayman index α[10] and of k-fold symmetric and close-to-convex functions [11]. For bounds on the Hankel determinants of close-to-convex functions, see [12-14]. Noor studied the Hankel determinant of Bazilevic functions in [15] and of functions with bounded boundary rotation in [16-19]. In the recent years, several authors have investigated bounds for the Hankel determinant of functions belonging to various subclasses of univalent and multivalent functions [20-27]. The Hankel determinant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M17">View MathML</a> is the well-known Fekete-Szegö functional. For results related to this functional, see [28,29]. The second Hankel determinant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M18">View MathML</a> is given by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M19">View MathML</a>.

An analytic function f is subordinate to an analytic function g, written <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M20">View MathML</a>, if there is an analytic function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M21">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M22">View MathML</a> satisfying <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M23">View MathML</a>. Ma and Minda [30] unified various subclasses of starlike (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M24">View MathML</a>) and convex functions (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M25">View MathML</a>) by requiring that either of the quantity <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M26">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M27">View MathML</a> is subordinate to a function φ with a positive real part in the unit disk <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M29">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M30">View MathML</a>, φ maps <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12">View MathML</a> onto a region starlike with respect to 1 and symmetric with respect to the real axis. He obtained distortion, growth and covering estimates as well as bounds for the initial coefficients of the unified classes.

The bounds for the second Hankel determinant <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M32">View MathML</a> are obtained for functions belonging to these subclasses of Ma-Minda starlike and convex functions in Section 2. In Section 3, the problem is investigated for two other related classes defined by subordination. In proving our results, we do not assume the univalence or starlikeness of φ as they were required only in obtaining the distortion, growth estimates and the convolution theorems. The classes introduced by subordination naturally include several well-known classes of univalent functions and the results for some of these special classes are indicated as corollaries.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M33">View MathML</a> be the class of functions with positive real part consisting of all analytic functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M34">View MathML</a> satisfying <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M35">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M36">View MathML</a>. We need the following results about the functions belonging to the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M33">View MathML</a>.

Lemma 1[31]

If the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M38">View MathML</a>is given by the series

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M39">View MathML</a>

(2)

then the following sharp estimate holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M40">View MathML</a>

(3)

Lemma 2[32]

If the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M38">View MathML</a>is given by the series (2), then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M42">View MathML</a>

(4)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M43">View MathML</a>

(5)

for somex, zwith<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M44">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M45">View MathML</a>.

2 Second Hankel determinant of Ma-Minda starlike/convex functions

Subclasses of starlike functions are characterized by the quantity <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M3">View MathML</a> lying in some domain in the right half-plane. For example, f is strongly starlike of order β if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M3">View MathML</a> lies in a sector <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M48">View MathML</a>, while it is starlike of order α if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M3">View MathML</a> lies in the half-plane <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M50">View MathML</a>. The various subclasses of starlike functions were unified by subordination in [30]. The following definition of the class of Ma-Minda starlike functions is the same as the one in [30] except for the omission of starlikeness assumption of φ.

Definition 1 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M51">View MathML</a> be analytic, and let the Maclaurin series of φ be given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M52">View MathML</a>

(6)

The class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M53">View MathML</a> of Ma-Minda starlike functions with respect toφ consists of functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M54">View MathML</a> satisfying the subordination

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M55">View MathML</a>

For the function φ given by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M56">View MathML</a> , <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M57">View MathML</a>, the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M58">View MathML</a> is the well-known class of starlike functions of order α. Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M59">View MathML</a>

Then the class

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M60">View MathML</a>

is the parabolic starlike functions introduced by Rønning [33]. For a survey of parabolic starlike functions and the related class of uniformly convex functions, see [34]. For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M61">View MathML</a>, the class

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M62">View MathML</a>

is the familiar class of strongly starlike functions of orderβ. The class

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M63">View MathML</a>

is the class of lemniscate starlike functions studied in [35].

Theorem 1Let the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M64">View MathML</a>be given by (1).

1. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M68">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M69">View MathML</a>

2. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M73">View MathML</a>

or the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M74">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M75">View MathML</a>

3. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M79">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M80">View MathML</a>

Proof Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M81">View MathML</a>, there exists an analytic function w with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M82">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M83">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M85">View MathML</a>

(7)

Define the functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M86">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M87">View MathML</a>

or, equivalently,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M88">View MathML</a>

(8)

Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M86">View MathML</a> is analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M91">View MathML</a> and has a positive real part in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12">View MathML</a>. By using (8) together with (6), it is evident that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M93">View MathML</a>

(9)

Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M94">View MathML</a>

(10)

it follows by (7), (9) and (10) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M95">View MathML</a>

Therefore

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M96">View MathML</a>

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M97">View MathML</a>

(11)

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M98">View MathML</a>

(12)

Since the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M99">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M100">View MathML</a>) is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M33">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M38">View MathML</a>, there is no loss of generality in assuming <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M103">View MathML</a>. Write <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M104">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M105">View MathML</a>. Substituting the values of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M106">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M107">View MathML</a> respectively from (4) and (5) in (12), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M108">View MathML</a>

Replacing <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M109">View MathML</a> by μ and substituting the values of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M110">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M111">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M112">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M113">View MathML</a> from (11) yield

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M114">View MathML</a>

(13)

Note that for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M115">View MathML</a>, differentiating <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M116">View MathML</a> in (13) partially with respect to μ yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M117">View MathML</a>

(14)

Then, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M118">View MathML</a> and for any fixed c with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M119">View MathML</a>, it is clear from (14) that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M120">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121">View MathML</a> is an increasing function of μ. Hence, for fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M122">View MathML</a>, the maximum of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121">View MathML</a> occurs at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M124">View MathML</a>, and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M125">View MathML</a>

Also note that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M126">View MathML</a>

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M127">View MathML</a>

(15)

Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M128">View MathML</a>

(16)

we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M129">View MathML</a>

where P, Q, R are given by (15). □

Remark 1 When <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M130">View MathML</a>, Theorem 1 reduces to [[24], Theorem 3.1].

Corollary 1

1. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M131">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M132">View MathML</a>.

2. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M133">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M134">View MathML</a>.

3. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M135">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M136">View MathML</a>.

4. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M137">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M138">View MathML</a>.

Definition 2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M51">View MathML</a> be analytic, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M140">View MathML</a> be given as in (6). The class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M141">View MathML</a> of Ma-Minda convex functions with respect toφ consists of functions f satisfying the subordination

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M142">View MathML</a>

Theorem 2Let the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M143">View MathML</a>be given by (1).

1. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M147">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M148">View MathML</a>

2. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M152">View MathML</a>

or the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M153">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M154">View MathML</a>

3. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M158">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M159">View MathML</a>

Proof Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M160">View MathML</a>, there exists an analytic function w with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M82">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M83">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M164">View MathML</a>

(17)

Since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M165">View MathML</a>

(18)

equations (9), (17) and (18) yield

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M166">View MathML</a>

Therefore

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M167">View MathML</a>

By writing

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M168">View MathML</a>

(19)

we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M169">View MathML</a>

(20)

Similar as in Theorems 1, it follows from (4) and (5) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M170">View MathML</a>

Replacing <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M109">View MathML</a> by μ and then substituting the values of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M110">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M111">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M112">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M113">View MathML</a> from (19) yield

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M176">View MathML</a>

(21)

Again, differentiating <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121">View MathML</a> in (21) partially with respect to μ yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M178">View MathML</a>

(22)

It is clear from (22) that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M179">View MathML</a>. Thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121">View MathML</a> is an increasing function of μ for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M118">View MathML</a> and for any fixed c with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M119">View MathML</a>. So, the maximum of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121">View MathML</a> occurs at <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M124">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M185">View MathML</a>

Note that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M186">View MathML</a>

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M187">View MathML</a>

(23)

By using (16), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M188">View MathML</a>

where P, Q, R are given in (23). □

Remark 2 For the choice of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M189">View MathML</a>, Theorem 2 reduces to [[24], Theorem 3.2].

3 Further results on the second Hankel determinant

Definition 3 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M51">View MathML</a> be analytic, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M140">View MathML</a> be as given in (6). Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M192">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M193">View MathML</a>. A function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M194">View MathML</a> is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M195">View MathML</a> if it satisfies the following subordination:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M196">View MathML</a>

Theorem 3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M192">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M193">View MathML</a>, and let the functionfas in (1) be in the class<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M199">View MathML</a>. Also, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M200">View MathML</a>

1. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M204">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M205">View MathML</a>

2. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M209">View MathML</a>

or the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M210">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M211">View MathML</a>

3. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M215">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M216">View MathML</a>

Proof For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M217">View MathML</a>, there exists an analytic function w with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M82">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M83">View MathML</a> in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M12">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M221">View MathML</a>

(24)

Since f has the Maclaurin series given by (1), a computation shows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M222">View MathML</a>

(25)

It follows from (24), (9) and (25) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M223">View MathML</a>

Therefore

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M224">View MathML</a>

which yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M225">View MathML</a>

(26)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M226">View MathML</a>

It can be easily verified that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M227">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M192">View MathML</a>.

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M229">View MathML</a>

(27)

Then (26) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M230">View MathML</a>

(28)

It follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M231">View MathML</a>

Application of the triangle inequality, replacement of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M109">View MathML</a> by μ and substituting the values of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M110">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M111">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M112">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M113">View MathML</a> from (27) yield

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M237">View MathML</a>

(29)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M238">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M239">View MathML</a>.

Similarly as in the previous proofs, it can be shown that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M121">View MathML</a> is an increasing function of μ for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M241">View MathML</a>. So, for fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M105">View MathML</a>, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M243">View MathML</a>

which is

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M244">View MathML</a>

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M245">View MathML</a>

(30)

Using (16), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M246">View MathML</a>

where P, Q, R are given in (30). □

Remark 3 For the choice <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M247">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M248">View MathML</a>, Theorem 3 reduces to [[36], Theorem 2.1].

Definition 4 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M51">View MathML</a> be analytic, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M140">View MathML</a> be as given in (6). For a fixed real number α, the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M194">View MathML</a> is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M252">View MathML</a> if it satisfies the following subordination:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M253">View MathML</a>

Al-Amiri and Reade [37] introduced the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M254">View MathML</a> and they showed that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M255">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M256">View MathML</a>. Univalence of the functions in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M257">View MathML</a> was also investigated in [38,39]. Singh et al. also obtained the bound for the second Hankel determinant of functions in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M257">View MathML</a>. The following theorem provides a bound for the second Hankel determinant of the functions in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M252">View MathML</a>.

Theorem 4Let the functionfgiven by (1) be in the class<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M260">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M261">View MathML</a>. Also, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M262">View MathML</a>

1. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M266">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M267">View MathML</a>

2. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M271">View MathML</a>

or

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M272">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M273">View MathML</a>

3. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M66">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M67">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M277">View MathML</a>

then the second Hankel determinant satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M278">View MathML</a>

Proof For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M279">View MathML</a>, a calculation shows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M280">View MathML</a>

(31)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M281">View MathML</a>

It can be easily verified that for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M261">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M283">View MathML</a>. Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M284">View MathML</a>

(32)

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M285">View MathML</a>

(33)

Similarly as in earlier theorems, it follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M286">View MathML</a>

(34)

and for fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M105">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M288">View MathML</a> with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M289">View MathML</a>

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M290">View MathML</a>

(35)

By using (16), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M291">View MathML</a>

where P, Q, R are given in (35). □

Remark 4 For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M292">View MathML</a>, Theorem 4 reduces to Theorem 2. For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M293">View MathML</a>, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M294">View MathML</a>. For this function φ, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/281/mathml/M295">View MathML</a>. In this case, Theorem 4 reduces to [[40], Theorem 3.1].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors jointly worked on the results and they read and approved the final manuscript.

Acknowledgements

The work presented here was supported in part by research grants from Universiti Sains Malaysia (FRGS grants) and University of Delhi as well as MyBrain MyPhD programme of the Ministry of Higher Education, Malaysia.

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