Abstract
In the present investigation a sufficient condition is obtained for normalized Mittag-Leffler function to be starlike and strongly starlike in the open unit disk. Results obtained are new and their usefulness is depicted by deducing several interesting corollaries and examples.
1. Introduction
Let denote the class of analytic functions in the open unit disk: and denote the subclass of , which are normalized by the condition and have representation of the following form: A function is said to be univalent in a domain if it is one to one in . A function is called starlike, denoted by if is univalent in and is a starlike domain with respect to the origin. For a given , a function is called starlike function of order , class of such function denoted by , if and only if , . Further, a function is called a convex function, if , . It is well known that . Further, let , , be the class of strongly starlike functions of order defined by Note that . For more details about these classes, one can refer to [1].
If , then the function is said to be subordinate to , written as , if there exists a Schwarz function with and () such that In particular, if is univalent in , then we have the following equivalence: The function was introduced by Mittag-Leffler (see [2]) in connection with the summing of divergent series. An important generalization, was introduced by Wiman [3, 4]. The Mittag-Leffler function has a close connection to differential equations of fractional order and integral equations of Abel type, such equations are becoming more and more popular in modelling natural and technical process [5–10]. The Mittag-Leffler function arises naturally in the solution of fractional order differential and integral equations and especially in the investigations of fractional generalization of kinetic equation, random walks, Lévy flights, and superdiffusive transport and in the study of complex systems. These functions interpolate between a purely exponential law and power-law like behavior of phenomena governed by ordinary kinetic equations and their fractional counterparts [6–8, 11]. The most essential properties of these entire functions, investigated by many mathematicians, can be found in [12, 13].
Observe that Mittag-Leffler function does not belong to the family Thus, it is natural to consider the following normalization of Mittag-Leffler function in : Whilst formula (7) holds for complex-valued , , in this paper we shall restrict our attention to the case of real-valued , . It is easy to see that satisfy the following relation:
Several researchers studied families of analytic functions involving special functions to find conditions such that the images of functions under have certain geometric properties like univalency, starlikeness, or convexity. In this context, many results are available in the literature regarding the hypergeometric functions [14], Bessel functions [15–17], Wright function [18], and Mittag-Leffler functions [19]. In this paper, we study some more properties of normalized Mittag-Leffler function .
2. Strongly Starlikeness
To prove results in this section, we shall need the following lemma.
Lemma 1 (Hallenbeck and Ruscheweyh [20]). Let be convex and univalent in with . Let be analytic in , and in . Then, for all , we have
Further, it is easy to see that, under the hypothesis , the inequality, , holds for all , which is equivalent to
Theorem 2. If and , then , where for
Proof. Using inequality (10), we get Note that, under the hypothesis, From (12), we conclude that which gives Using Lemma 1, for and with , we get Consequently, Now from (15) and (17), we conclude that that is, , for given in (11).
Corollary 3. Let and . If and where , then .
Proof. If we put from (19) to (11), we obtain .
Putting in Corollary 3, we get
Corollary 4. If and , where is positive root of , then .
Example 5. If and then From (11), we get and thus, from Theorem 2, we have
3. A Nonlinear Differential Equation
In order to prove results in this section, we required the following known results.
Lemma 6 (see [9]). Let . Suppose that the function satisfies the condition for all , , and . Also, let be an analytic function of the following form: such that for all . Then , where .
Lemma 7 (see [21]). Let be of form (2) which satisfies the inequality . Then is starlike in .
Theorem 8. For all , let the analytic function in satisfy the following inequality: and also let be the (unique) solution of the initial value problem for higher-order linear differential equation given by where (). Then the inequality holds.
Proof. Define by Clearly, function belongs to (23), and then by the related implicit function it immediately follows that and also denotes and by respectively. Then, clearly, Further, for any we have which gives Therefore, in view of Lemma 6, it follows that This completes the proof.
By taking in the above theorem we get the following.
Corollary 9. Let the analytic function in satisfy inequality (24), and also let be the (unique) solution of the initial value problem for third-order linear differential equation given by Then the inequality holds.
Corollary 10. If the analytic function in satisfies the following inequality and the function is the (unique) solution of the initial value problem for third-order linear differential equation given by (34), then .
Proof. By taking , in Corollary 9 (or in Theorem 8 with ), and then using Lemma 7, the proof can easily be obtained.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.