Research Article Applications of Mittag-Leffer Type Poisson Distribution to a Subclass of Analytic Functions Involving Conic-Type Regions Muhammad Ghaffar Khan , 1 Bakhtiar Ahmad , 2 Nazar Khan, 3 Wali Khan Mashwani , 1 Sama Arjika , 4,5 Bilal Khan, 6 and Ronnason Chinram 7 1 Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat, Pakistan 2 Govt: Degree College Mardan, 23200 Mardan, Pakistan 3 Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan 4 Department of Mathematics and Informatics, University of Agadez, Agadez, Niger 5 International Chair of Mathematical Physics and Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, Post Box 072, Cotonou 50, Benin 6 School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China 7 Algebra and Applications Research Unit, Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand Correspondence should be addressed to Sama Arjika; rjksama2008@gmail.com Received 20 April 2021; Revised 2 July 2021; Accepted 10 July 2021; Published 27 July 2021 Academic Editor: Teodor Bulboaca Copyright © 2021 Muhammad Ghaffar Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this article, we introduce a new subclass of analytic functions utilizing the idea of Mittag-Leffler type Poisson distribution associated with the Janowski functions. Further, we discuss some important geometric properties like necessary and sufficient condition, convex combination, growth and distortion bounds, Fekete-Szegö inequality, and partial sums for this newly defined class. 1. Introduction, Definitions, and Motivation Let A represent the collections of holomorphic (analytic) functions f defined in the open unit disc: D = z : z ∈ ℂ and z jj <1 f g, ð1Þ such that the Taylor series expansion of f is given by fz ðÞ = z + 〠 ∞ n=2 a n z n z ∈ D ð Þ: ð2Þ By convention, S stands for a subclass of class A com- prising of univalent functions of the form (2) in the open unit disc D. Let P represent the class of all functions p that are holomorphic in D with the condition R pz ðÞ ð Þ > 0, ð3Þ and has the series representation pz ðÞ =1+ 〠 ∞ n=1 c n z n z ∈ D ð Þ: ð4Þ Next, we recall the definition of subordination, for two functions h 1 , h 2 ∈ A , we say h 1 is subordinated to h 2 and is Hindawi Journal of Function Spaces Volume 2021, Article ID 4343163, 9 pages https://doi.org/10.1155/2021/4343163