Abstract

Functions that are analytic in the unit disk and satisfy the differential equation are considered, where is subordinated to a normalized convex univalent function . These functions are given by a double integral operator of the form with subordinated to . The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function .

1. Introduction

Let denote the class of all analytic functions defined in the open unit disk and normalized by the conditions , . Further, let be the subclass of consisting of univalent functions, and let be its subclass of starlike functions. A starlike function is characterized analytically by the condition in , that is, the domain is starlike with respect to origin. For two functions and in , the Hadamard product (or convolution) of and is the function defined by

For and in , a function is subordinate to , written as , if there is an analytic function satisfying and , such that , . When is univalent in , then is subordinated to which is equivalent to and .

In a recent paper, Miller and Mocanu [1] investigated starlikeness properties of functions defined by double integral operators of the form

In this paper, conditions on a different kernel are investigated from the perspective of starlikeness. Specifically, we consider functions given by the double integral operator of the form In this case, it follows that

where . Further, the function satisfies a third-order differential equation of the form

for appropriate parameters and . The investigation of such functions can be seen as an extension to the study of the class

The class or its variations for an appropriate function have been investigated in several works; see, for example, [2–10] and more recently [11, 12].

2. Results on Differential Subordination

We first recall the definition of best dominant solution of a differential subordination.

Definition 2.1 ((dominant and best dominant) [13]). Let , and let be univalent in . If is analytic in and satisfies the differential subordination then is called a solution of the differential subordination. A univalent function is called a dominant if for all satisfying (2.1). A dominant that satisfies for all dominants of (2.1) is said to be the best dominant of (2.1).

In the following sequel, we will assume that is an analytic convex function in with . For , consider the third-order differential equation We will denote the class consisting of all solutions as , that is,

Let The discriminant is denoted by . Note that and .

We will rewrite the solution of

in its equivalent integral form

It follows from relations (2.4) that

Thus,

Making the substitution in the above integral and integrating again, a change of variables yields

We will use the notation for From [14] it is known that is convex in provided .

Theorem 2.2. Let and be given by (2.4), and Then the function is convex. If , then and is the best dominant.

Proof. It follows from (2.10) that Thus, Since the convolution of two convex functions is convex [15], the function is convex. Let Then, It is known from [16] that Similarly, implies
The differential chain shows that . Since , the function is a solution of the differential subordination , and thus for all dominants . Hence, is the best dominant.

Remark 2.3. (1) When , then and , and the above subordination reduces to the result of [16], that is,
(2) The above proof also reveals that that is, .

Theorem 2.4. Let , , and be as given in Theorem 2.2. If , then

Proof. Let . Then With given by (2.10), this subordination implies

In this paper, starlikeness properties will be investigated for functions given by a double integral operator of the form (1.3).

3. Applications

First, we consider a class of convex univalent functions so that is symmetric with respect to the real axis. Denote by the class

where , and let . When and , let . The class of is of particular significance, and we will simply denote it by

Equivalently,

The following result is an immediate consequence of Theorems 2.2 and 2.4.

Theorem 3.1. Under the assumptions of Theorem 2.2, if then where is the best dominant. Further, if , and if .

4. Starlikeness Property

Starlikeness properties of functions given by a double integral operator are investigated in this section. The following result will be required.

Lemma 4.1 (see [5]). If satisfies for , then . This result is sharp.

Theorem 4.2. Let and be given by (2.4) with and . If where , and satisfies then .

Proof. The function satisfies and thus Now, also implies that , and so It follows from the proof of Theorem 2.2 that where Since an application of Lemma 4.1 yields the result.

Corollary 4.3. Let and If satisfies then .

Proof. In this case, , , and the result now follows from Theorem 4.2.

Example 4.4. If and satisfies then .

Theorem 4.5. Let and let and be given by (2.4) with . If satisfies then .

Proof. Clearly, Since , substituting and in (3.7) gives Hence, it follows that

Acknowledgments

The authors are greatly thankful to the referee for the many suggestions that helped improve the presentation of this paper. The work presented here was supported in part by a Research University grant from Universiti Sains Malaysia.