Abstract
We introduce a new subclass of harmonic functions by using a certain linear operator. For this class we derive coefficient bounds, extreme points, and inclusion results and also show that this class is closed under an integral operator.
1. Introduction
Harmonic functions have long been used in the representation of minimal surface; for example, Heinz [1] in 1952 used such mappings in the study of the Gaussian curvature of nonparametric minimal surface over the unit disc. Such mappings have vast application in the field of engineering, physics, electronics, medicine, operations research, aerodynamics, and other branches of applied mathematics (see [2]).
is said to be complex valued harmonic function if both and are continuous and real harmonic; that is, and . In simply connected domain [3], we can write , where and are analytic in We call the analytic part and the coanalytic part of . The necessary and sufficient condition for a function to be univalent and sense preserving in is (see [3]). A function is in class if it is harmonic, univalent, and sense preserving in , where and have the following series:
Hence
Note that if the coanalytic part is identically zero, then reduces to well-known class of normalized univalent analytic functions For this class, the function may be expressed as
A function given by (2) is said to be harmonic starlike of order for , if or equivalently
The class of all harmonic starlike functions of order is denoted by This class was studied by Jahangiri [4]. The cases and were studied by Silverman and Silvia [5] and Silverman [6].
For and given by (2), the convolution is denoted by and defined as
Let and be positive real parameter such that
The Wright generalized hypergeometric function [7] is defined as
which is defined by
If and , we have the relationship
is the generalized hypergeometric function [8], where denotes the set of all positive integers and is the Pochhammer symbol and
Dziok and Srivastava [8] introduced a linear operator which is generalization of Dziok-Srivastava operator [8–10], Carlson-Shaffer operator [11], and the generalized Bernardi’s integral operator [12].
Dziok and Raina [13] considered the linear operator defined by where is given by
For , we havewhere is given by
For our convenience we write
Motivated by the work of [4, 14–21], we extend the work of Chandrashekar et al. [22] by introducing some new subclasses of using the generalized hypergeometric function.
Definition 1. A function is in class if
Definition 2. Let denote the class of functions of the formand .
Throughout this paper, we shall assume , , , and as given in (11) and (15), respectively, unless otherwise mentioned.
2. Main Results
In Theorem 3, we shall present a sufficient condition for to be in class
Theorem 3. Let be given by (2). Ifthen
Proof. When inequality (18) holds for the coefficients of given in (2), we have to show that inequality (16) is satisfied. Arranging the left side inequality (16), we have As we know if and only if , it is sufficient to show thatSubstituting the values and in left side of (20), we obtain by hypothesis in (18), which implies that
Now we obtain the necessary and sufficient condition for the function given by (16) to be in
Theorem 4. Let be given by (16). Then if and only if
Proof. Since , we only have to prove the necessary part of theorem. Assume that , and then by virtue of (16), we obtain This is equivalent toThis condition must hold for all values of and for real , so that, by taking and , the above inequality reduces toLetting through real values, we obtain condition (22). This completes the proof.
We determine the extreme points of closed convex hulls of denoted by
Theorem 5. A function if and only if where , , and In particular, the extreme points of are and
Proof. First, we consider where , and
Using (22) for the coefficients in (29), we have and hence
Conversely, suppose that , and set where Then which is the required result.
Next we show that is close under convex combinations of its members.
Theorem 6. The family is closed under convex combination.
Proof. For , suppose that , whereFor coefficients in (22) the relation given in (33) takes the formFor , , the convex combination of may be written asFrom inequality (22) for (35), and therefore
Theorem 7. For , let and Then
Proof. Let and thenWe note that and . Therefore by using (22), since and . This proves that .
Now for the class the closure property under the generalized Bernardi-Livingston integral operator is examined which is defined by
Theorem 8. Let and then
Proof. Consider the generalized Bernardi-Livingston integral operator given in (41): where Therefore Since , therefore, by Theorem 4,
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work is supported by AP-2013-009 and GP-K006392, Universiti Kebangsaan Malaysia.