Necessary and sufficient conditions for functions defined by binomial distribution to be in a general class of analytic functions

In this paper, a comprehensive class (E l , E 2 , E 3 ) of univalent functions is examined in light of the recent findings on binomial distribution.Additionally, we provide some requirements for the functions B(, , ), B(, , )b and the operator L(, , ) defined by the binomial distribution to be in these subclasses.This study will motivate the authors to discover new sufficient requirements, not only for binomial distributions but also for numerous other special functions.Additionally, we summarize many previous studies.