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Integral operators on new families of meromorphic functions of complex order

Aabed Mohammed and Maslina Darus*

Author Affiliations

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia

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Journal of Inequalities and Applications 2011, 2011:121  doi:10.1186/1029-242X-2011-121

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2011/1/121


Received:21 August 2011
Accepted:25 November 2011
Published:25 November 2011

© 2011 Mohammed and Darus; 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 article, we define and investigate new families of certain subclasses of meromorphic functions of complex order. Considering the new subclasses, several properties for certain integral operators are derived.

2010 Mathematics Subject Classification: 30C45.

Keywords:
Analytic function; meromorphic function; integral operator

1 Introduction

Let Σ denote the class of functions of the form:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M1">View MathML</a>

(1.1)

which are analytic in the punctured open unit disk

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M2">View MathML</a>

(1.2)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M3">View MathML</a> is the open unit disk <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M4">View MathML</a>.

We say that a function f ∈ Σ is meromorphic starlike of order δ (0 ≤ δ < 1), and belongs to the class Σ*(δ), if it satisfies the inequality:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M5">View MathML</a>

(1.3)

A function f ∈ Σ is a meromorphic convex function of order δ (0 ≤ δ < 1), if f satisfies the following inequality:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M7">View MathML</a>

(1.4)

and we denote this class by Σk(δ).

For f ∈ Σ, Wang et al. [1] and Nehari and Netanyahu [2] introduced and studied the subclass ΣN(β) of Σ consisting of functions f(z) satisfying

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M10">View MathML</a>

(1.5)

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M11">View MathML</a> denote the class of functions f of the form <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M12">View MathML</a>, which are analytic in the open unit disk <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M13">View MathML</a>.

Analogous to several subclasses [3-10] of analytic functions of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M14">View MathML</a>, we define the following subclasses of Σ.

Definition 1.1 Let a function f ∈ Σ be analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M15">View MathML</a>. Then f is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M16">View MathML</a> if, and only if, f satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M17">View MathML</a>

(1.6)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M18">View MathML</a>, 0 ≤ δ < 1.

Definition 1.2 Let a function f ∈ Σ be analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M19">View MathML</a>. Then f is in the class ΣKb(δ) if, and only if, f satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M21">View MathML</a>

(1.7)

where b C\{0}, 0 ≤ δ < 1. We note that f ∈ Σ Kb(δ) if, and only if, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M24">View MathML</a>.

Furthermore, the classes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M25">View MathML</a>

are the classes of meromorphic starlike functions of order δ and meromorphic convex functions of order δ in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M26">View MathML</a>, respectively. Moreover, the classes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M27">View MathML</a>

are familiar classes of starlike and convex functions in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M28">View MathML</a>, respectively.

Definition 1.3 Let a function f ∈ Σ be analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M29">View MathML</a>. Then f is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M30">View MathML</a> if, and only if, f satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M31">View MathML</a>

(1.8)

where α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M32">View MathML</a>.

Definition 1.4 Let a function f ∈ Σ be analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M33">View MathML</a>. Then f is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M34">View MathML</a> if, and only if, f satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M35">View MathML</a>

(1.9)

where α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M36">View MathML</a>.

We note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M37">View MathML</a> if, and only if, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M38">View MathML</a>.

For α = 0, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M39">View MathML</a>

Definition 1.5 Let a function f ∈ Σ be analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M40">View MathML</a>. Then f is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M41">View MathML</a> if, and only if, f satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M42">View MathML</a>

(1.10)

where α > 0, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M43">View MathML</a>.

Definition 1.6 Let a function f ∈ Σ be analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M44">View MathML</a>. Then f is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M45">View MathML</a> if, and only if, f satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M46">View MathML</a>

(1.11)

where α > 0, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M47">View MathML</a>.

We note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M48">View MathML</a> if, and only if, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M49">View MathML</a>.

Let us also introduce the following families of new subclasses <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M50">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M51">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M52">View MathML</a> as follows.

Definition 1.7 Let a function f ∈ Σ be analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M53">View MathML</a>. Then f is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M54">View MathML</a>, if, and only if, f satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M55">View MathML</a>

(1.12)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M56">View MathML</a>, 0 ≤ δ < 1.

We note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M57">View MathML</a> if, and only if, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M58">View MathML</a>.

Definition 1.8 Let a function f ∈ Σ be analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M59">View MathML</a>. Then f is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M60">View MathML</a> if, and only if, f satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M61">View MathML</a>

(1.13)

where α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M62">View MathML</a>.

We note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M63">View MathML</a> if, and only if, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M64">View MathML</a>.

Definition 1.9 Let a function f ∈ Σ be analytic in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M65">View MathML</a>. Then f is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M66">View MathML</a> if, and only if, f satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M67">View MathML</a>

(1.14)

where α > 0, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M68">View MathML</a>.

We note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M69">View MathML</a> if, and only if, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M70">View MathML</a>.

Recently, many authors introduced and studied various integral operators of analytic and univalent functions in the open unit disk <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M71">View MathML</a>[11-21].

Most recently, Mohammed and Darus [22,23] introduced the following two general integral operators of meromorphic functions Σ:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M72">View MathML</a>

(1.15)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M73">View MathML</a>

(1.16)

Goyal and Prajapat [24] obtained the following results for f ∈ Σ to be in the class Σ*(δ), 0 ≤ δ < 1.

Corollary 1.1 If f ∈ Σ satisfies the following inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M75">View MathML</a>

(1.17)

then f ∈ Σ*(δ).

Corollary 1.2 If f ∈ Σ satisfies the following inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M77">View MathML</a>

(1.18)

then f ∈ Σ*.

Corollary 1.3 If f ∈ Σ satisfies the following inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M79">View MathML</a>

(1.19)

then f ∈ Σ*.

In this article, we derive several properties for the integral operators <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M81">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M82">View MathML</a> of the subclasses given by (1.5) and Definitions 1.1 to 1.6.

2 Some properties for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M83">View MathML</a>

In this section, we investigate some properties for the integral operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M84">View MathML</a> defined by (1.15) of the subclasses given by (1.5), Definitions 1.1, 1.3, and 1.5.

Theorem 2.1 For i ∈ {1,..., n}, let γi > 0, fi ∈ Σ and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M87">View MathML</a>

(2.1)

If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M88">View MathML</a>

(2.2)

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M89">View MathML</a>, where β > 1.

Proof On successive differentiation of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M90">View MathML</a>, which is defined in (1.5), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M91">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M92">View MathML</a>

(2.4)

Then from (2.3) and (2.4), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M93">View MathML</a>

(2.5)

By multiplying (2.5) with z yield

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M94">View MathML</a>

(2.6)

This is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M95">View MathML</a>

(2.7)

Therefore, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M96">View MathML</a>

(2.8)

Then, we easily get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M97">View MathML</a>

(2.9)

Taking real parts of both sides of (2.9), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M98">View MathML</a>

(2.10)

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M99">View MathML</a>

(2.11)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M100">View MathML</a>, applying Corollary 1.1, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M101">View MathML</a>

(2.12)

Then, by the hypothesis (2.2), we have β > 1. Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M102">View MathML</a>, where β > 1. This completes the proof.   □

Letting δ = 0 in Theorem 2.1, we have

Corollary 2.2 For i ∈ {1,..., n}, let γi > 0, fi ∈ Σ and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M106">View MathML</a>

(2.13)

If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M107">View MathML</a>

(2.14)

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M108">View MathML</a>, where β > 1.

Making use of (2.11), Corollary 1.2 and Corollary 1.3, one can prove the following results.

Theorem 2.3 For i ∈ {1,..., n}, let γi > 0, fi ∈ Σ and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M111">View MathML</a>

(2.15)

If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M112">View MathML</a>

(2.16)

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M113">View MathML</a>, where β > 1.

Theorem 2.4 For i ∈ {1,..., n}, let γi > 0, fi ∈ Σ and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M116">View MathML</a>

(2.17)

If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M117">View MathML</a>

(2.18)

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M118">View MathML</a>, where β > 1.

Theorem 2.5 For i ∈ {1,..., n}, let γi > 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M120">View MathML</a> (0 ≤ δ < 1 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M121">View MathML</a>).

If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M122">View MathML</a>

(2.19)

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M123">View MathML</a>is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M124">View MathML</a>.

Proof From (2.7), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M125">View MathML</a>

(2.20)

Equivalently, (2.20) can be written as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M126">View MathML</a>

(2.21)

Taking the real part of both terms of the last expression, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M127">View MathML</a>

(2.22)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M128">View MathML</a>, hence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M129">View MathML</a>

(2.23)

so that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M130">View MathML</a>

(2.24)

Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M131">View MathML</a>.

Now, adopting the techniques used by Breaz et al. [11] and Bulut [21], we prove the following two theorems.

Theorem 2.6 For i ∈ {1,..., n}, let γi > 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M133">View MathML</a> (α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M134">View MathML</a>). If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M135">View MathML</a>

(2.25)

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M136">View MathML</a>is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M137">View MathML</a>.

Proof Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M138">View MathML</a>, it follows from Definition 1.3 that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M139">View MathML</a>

(2.26)

Considering Definition 1.8 and with the help of (2.21), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M140">View MathML</a>

(2.27)

This completes the proof.   □

Theorem 2.7 For i ∈ {1,..., n}, let γi > 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M143">View MathML</a> (α > 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M144">View MathML</a>). If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M145">View MathML</a>

(2.28)

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M146">View MathML</a>is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M147">View MathML</a>.

Proof Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M148">View MathML</a>, it follows from Definition 1.5 that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M149">View MathML</a>

(2.29)

Considering this inequality and (2.21), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M150">View MathML</a>

This completes the proof.   □

3 Some properties for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M152">View MathML</a>

In this section, we investigate some properties for the integral operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M153">View MathML</a> defined by (1.16) of subclasses given by (1.5), Definitions 1.2, 1.4, and 1.6.

Theorem 3.1 For i ∈ {1,..., n}, let γi > 0, fi ∈ Σ and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M156">View MathML</a>

(3.1)

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M157">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M158">View MathML</a>, β > 1.

Proof On successive differentiation of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M159">View MathML</a>, which is defined in (1.16), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M160">View MathML</a>

(3.2)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M161">View MathML</a>

(3.3)

Then from (3.2) and (3.3), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M162">View MathML</a>

(3.4)

By multiplying (3.4) with z yields,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M163">View MathML</a>

(3.5)

that is equivalent to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M164">View MathML</a>

(3.6)

Therefore, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M165">View MathML</a>

(3.7)

so that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M166">View MathML</a>

(3.8)

Taking the real parts of both terms of the last expression, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M167">View MathML</a>

(3.9)

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M168">View MathML</a>

(3.10)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M169">View MathML</a>, fi ∈ Σk(δ), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M171">View MathML</a>

(3.11)

which, in light of the hypothesis (3.1), yields β > 1.

Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M172">View MathML</a>, β > 1. This completes the proof.   □

Theorem 3.2 For i ∈ {1,..., n}, let γi > 0, fi ∈ Σ and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M176">View MathML</a>

(3.12)

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M177">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M178">View MathML</a>, β > 1.

Proof It follows from (3.7) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M179">View MathML</a>

(3.13)

Taking the real parts of both terms of the last expression, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M180">View MathML</a>

(3.14)

Thus, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M181">View MathML</a>

(3.15)

Let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M182">View MathML</a>

(3.16)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M183">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M184">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M185">View MathML</a>

(3.17)

Then, by the hypothesis (3.12), we see that β > 1. Therefore, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M186">View MathML</a>, β > 1. This completes the proof.   □

Now, using the method given in the proofs of Theorems 2.5, 2.6, and 2.7, one can prove the following results:

Theorem 3.3 For i ∈ {1,..., n}, let γi > 0 fi ∈ ΣKb(δi) (0 ≤ δ < 1 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M190">View MathML</a>)). If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M191">View MathML</a>

(3.18)

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M192">View MathML</a>is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M193">View MathML</a>.

Theorem 3.4 For i ∈ {1,..., n}, let γi > 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M195">View MathML</a> (α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M196">View MathML</a>)). If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M197">View MathML</a>

(3.19)

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M198">View MathML</a>is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M199">View MathML</a>.

Theorem 3.5 For i ∈ {1,..., n}, let γi > 0 and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M201">View MathML</a> (α ≥ 0, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M202">View MathML</a>)). If

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M203">View MathML</a>

(3.20)

then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M204">View MathML</a>is in the class <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/121/mathml/M205">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

AM is currently a PhD student under supervision of MD and jointly worked on deriving the results. All the authors read and approved the final manuscript.

Acknowledgements

The above study was supported by MOHE grant: UKM-ST-06-FRGS0244-2010.

References

  1. Wang, Z-G, Sun, Y, Zhang, Z-H: Certain classes of meromorphic multivalent functions. Comput Math Appl. 58, 1408–1417 (2009). Publisher Full Text OpenURL

  2. Nehari, Z, Netanyahu, E: On the coefficients of meromorphic Schlicht functions. Proc Am Math Soc. 8, 15–23 (1957). Publisher Full Text OpenURL

  3. Goodman, AW: Univalent Functions. Mariner, Tampa, FL (1983)

  4. Frasin, BA: Family of analytic functions of complex order. Acta Math Acad Paed Ny. 22(2), 179–191 (2006)

  5. Ronning, F: Uniformly convex functions. Ann Polon Math. 57, 165–175 (1992)

  6. Murugusundaramoorthy, G, Maghesh, N: A new subclass of uniforlmly convex functions and a corresponding subclass of starlike functions with fixed second coefficient. J Inequal Pure Appl Math. 5(4), 1–10 (2005)

  7. Stankiewicz, J, Wisniowska, A: Starlike functions associated with some hyperbola. Folia Scientiarum Universitatis Tehnicae Resoviensis 147, Matematyka. 19, 117–126 (1996)

  8. Acu, M: Subclasses of convex functions associated with some hyperbola. Acta Universitatis Apulensis. 12, 3–12 (2006)

  9. Acu, M, Owa, S: Convex functions associated with some hyperbola. J Approx Theory Appl. 1, 37–40 (2005)

  10. Nasr, MA, Aouf, MK: Starlike function of complex order. J Nat Sci Math. 25(91), 1–12 (1985)

  11. Baricz, Á, Frasin, BA: Univalence of integral operators involving Bessel functions. Appl Math Lett. 23, 371–376 (2010). Publisher Full Text OpenURL

  12. Mohammed, A, Darus, M: New properties for certain integral operators. Int J Math Anal. 4(42), 2101–2109 (2010)

  13. Frasin, BA: Univalence of two general integral operators. Filomat. 23, 223–229 (2009). Publisher Full Text OpenURL

  14. Breaz, D, Breaz, N, Srivastava, HM: An extension of the univalent condition for a family of integral operators. Appl Math Lett. 22, 41–44 (2009). Publisher Full Text OpenURL

  15. Breaz, D, Breaz, N: Two integral operators. Studia Universitatis Babes-Bolyai, Mathematica, Cluj-Napoca. 3, 13–21 (2002)

  16. Breaz, D, Owa, S, Breaz, N: A new integral univalent operator. Acta Univ Apulensis Math Inform. 16, 11–16 (2008)

  17. Breaz, D, Güney, HÖ, Salagean, GS: A new general integral operator. Tamsui Oxford, J Math Sci. 25(4), 407–414 (2009)

  18. Darus, M, Faisal, I: A study on Beckers univalence criteria. Abs Appl Anal. 2011, 13 (2011) Article, ID 759175

  19. Breaz, N, Braez, D, Darus, M: Convexity properties for some general integral operators on uniformly analytic functions classes. Comput Math Appl. 60, 3105–3107 (2010). Publisher Full Text OpenURL

  20. Latha, S, Dileep, L: On a generalized integral operator. Int J Math Anal. 3(30), 1487–1491 (2009)

  21. Bulut, S: A new general integral operator defined by Al-Oboudi differential operator. J Inequal Appl. 2009, 13 (2009) Article, ID 158408

  22. Mohammed, A, Darus, M: A new integral operator for meromorphic functions. Acta Universitatis Apulensis. 24, 231–238 (2010)

  23. Mohammed, A, Darus, M: Starlikeness properties of a new integral operator for meromorphic functions. J Appl Math. 2011, 8 (2011) Article, ID 804150

  24. Goyal, SP, Prajapat, JK: A new class of meromorphic multivalent functions involving certain linear operator. Tamsui Oxford, J Math Sci. 25(2), 167–176 (2009)