Geometric Properties of a Certain Class of Mittag–Leffler-Type Functions

Abstract

The main objective of this paper is to establish some sufficient conditions so that a class of normalized Mittag–Leffler-type functions satisfies several geometric properties such as starlikeness, convexity, close-to-convexity, and uniform convexity inside the unit disk. Moreover, pre-starlikeness and k-uniform convexity are discussed for these functions. Some sufficient conditions are also derived so that these functions belong to the Hardy spaces Hp and H∞. Furthermore, the inclusion properties of some modified Mittag–Leffler-type functions are discussed. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section.

More on the Unified Mittag–Leffler Function

“…The extensions of Mittag-Leffler function is an interesting topic for researchers in which the classical notions linked with predefined Mittag-Leffler functions are investigated in more general prospect, see [9][10][11]. The Wright function is the generalization of hypergeometric function and several other special functions based on the gamma function, see [12][13][14][15]. The extensions of Mittag-Leffler function which are due to the gamma function can be obtained from the Wright function.…”

“…For a detailed study on hypergeometric functions and their applications, we refer the readers to [6,15,24,25]. In the upcoming section, we obtain relationship of the unified Mittag-Leffler function (5) with some known special functions.…”

“…Proof. In (13) writing the Wright generalized function in terms of Mellin-Barnes integral by using (11), one can have (15).…”

The Grüss-Type and Some Other Related Inequalities via Fractional Integral with Respect to Multivariate Mittag-Leffler Function

“…The M-L functions with different parameters that have been extensively studied by [29][30][31] and the references cited therein. Definition 4 ([1,2]).…”

Properties of a Class of Analytic Functions Influenced by Multiplicative Calculus

“…We referred the survey-cum-expository papers of Srivastava [18][19][20][21] for this study and found that they supplied enough material to analyze this duality theory. However, see Srivastava et al [22][23][24][25][26] for thorough study on the Mittag-Leffler function. Srivastava et al [24] introduced the following multi-index Mittag-Leffler functions as a kernel of specific fractional-calculus operators and as given below:…”