Mathematics and Statistics Vol. 11(1), pp. 22 - 27
DOI: 10.13189/ms.2023.110103
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Inclusion Results of a Generalized Mittag-Leffler-Type Poisson Distribution in the k-Uniformly Janowski Starlike and the k-Janowski Convex Functions


Jamal Salah 1,*, Hameed Ur Rehman 2, Iman Al Buwaiqi 1
1 Department of Basic Science, College of Applied and Health Science, A'Sharqiyah University, Oman
2 Department of Mathematics, Center for Language and Foundation Studies, A'Sharqiyah University, Oman

ABSTRACT

Due to the Mittag-Leffler function's crucial contribution to solving the fractional integral and differential equations, academics have begun to pay more attention to this function. The Mittag-Leffler function naturally appears in the solutions of fractional-order differential and integral equations, particularly in the studies of fractional generalization of kinetic equations, random walks, Levy flights, super-diffusive transport, and complex systems. As an example, it is possible to find certain properties of the Mittag-Leffler functions and generalized Mittag-Leffler functions [4,5]. We consider an additional generalization in this study, , given by Prabhakar [6,7]. We normalize the later to deduce in order to explore the inclusion results in a well-known class of analytic functions, namely and , -uniformly Janowski starlike and k-Janowski convex functions, respectively. Recently, researches on the theory of univalent functions emphasize the crucial role of implementing distributions of random variables such as the negative binomial distribution, the geometric distribution, the hypergeometric distribution, and in this study, the focus is on the Poisson distribution associated with the convolution (Hadamard product) that is applied to define and explore the inclusion results of the followings: and the integral operator . Furthermore, some results of special cases will be also investigated.

KEYWORDS
-Uniformly Janowski Star-like, -Janowski Convex Functions, Mittag-Leffler Function

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Jamal Salah , Hameed Ur Rehman , Iman Al Buwaiqi , "Inclusion Results of a Generalized Mittag-Leffler-Type Poisson Distribution in the k-Uniformly Janowski Starlike and the k-Janowski Convex Functions," Mathematics and Statistics, Vol. 11, No. 1, pp. 22 - 27, 2023. DOI: 10.13189/ms.2023.110103.

(b). APA Format:
Jamal Salah , Hameed Ur Rehman , Iman Al Buwaiqi (2023). Inclusion Results of a Generalized Mittag-Leffler-Type Poisson Distribution in the k-Uniformly Janowski Starlike and the k-Janowski Convex Functions. Mathematics and Statistics, 11(1), 22 - 27. DOI: 10.13189/ms.2023.110103.