Essential Conditions for Subclasses of Spiral-Like Functions and Their Univalent Derivatives associated with Poisson distribution series

In this article, we investigate the necessary and sufficient conditions for specific functions that involve Poisson distribution series to be in few subclasses of analytic functions, where both and are univalent in the open unit disc We also consider an integral operator associated with Poisson distribution series and discuss several mapping properties of integral operator. Furthermore, we point out certain corollaries and consequences of the main results. In addition, we determine the necessary conditions
for specific subclasses of analytic functions linked to Poisson distribution series
to belong to subclasses of spiral-like univalent functions.