Research Article The Sufficient and Necessary Conditions for the Poisson Distribution Series to Be in Some Subclasses of Analytic Functions Abdel Moneim Y. Lashin , Abeer O. Badghaish , and Amani Z. Bajamal Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, Saudi Arabia Correspondence should be addressed to Amani Z. Bajamal; azbajamal@kau.edu.sa Received 5 March 2022; Revised 23 March 2022; Accepted 6 April 2022; Published 5 May 2022 Academic Editor: John R. Akeroyd Copyright © 2022 Abdel Moneim Y. Lashin 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 paper, we introduce new subclasses of analytic functions in the open unit disc. Furthermore, the necessary and sufficient conditions for the Poisson distribution series to be in these new subclasses are found. 1. Introduction Let A be the class of all analytic functions f in the open unit disc U = fz ∈ ℂ : jzj <1g and normalized by f ð0Þ =0 and f ′ ð0Þ =1. A function f ∈ A has the Taylor series expansion of the form fz ðÞ = z + 〠 ∞ k=2 a k z k : ð1Þ We denote by S the subclass of A consisting of normal- ized functions of the form (1) which are univalent in U . Fur- ther, we denote by T the subclass of S consisting of functions with negative coefficients of the form fz ðÞ = z − 〠 ∞ k=2 a k z k , a k ≥ 0: ð2Þ If f , g ∈ A such that f is given by (1) and g is given by gðzÞ = z + ∑ ∞ k=2 b k z k , then, the Hadamard product ð f ∗ gÞðz Þ is defined by f ∗ g ð Þ z ðÞ = z + 〠 ∞ k=2 a k b k z k : ð3Þ In 1837, the French mathematician Siméon Denis Pois- son created the Poisson distribution which is a popular dis- tribution expresses the probability of a given number of events occurring in a fixed interval of time or space. In [1], Porwal introduced a power series such that its coefficients are probabilities of the Poisson distribution Nm, z ð Þ = z + 〠 ∞ k=2 m k−1 k − 1 ð Þ! e −m z k , m > 0, z ∈ U : ð4Þ In addition, he introduced the series Rm, z ð Þ =2z − Nm, z ð Þ = z − 〠 ∞ k=2 m k−1 k − 1 ð Þ! e −m z k , m > 0, z ∈ U : ð5Þ In [2], Porwal and Kumar introduced a new linear oper- ator defined by Nm, z ð Þ ∗ fz ðÞ = z + 〠 ∞ k=2 m k−1 k − 1 ð Þ! e −m a k z k , m > 0, z ∈ U : ð6Þ In [3, 4], El-Ashwah and Kota presented the functions Hindawi Journal of Function Spaces Volume 2022, Article ID 2419196, 6 pages https://doi.org/10.1155/2022/2419196