The objective of the current paper is to find the necessary and sufficient condition for the function to belong to the subclass of $C\left(d,\delta ,\mu ,\beta \right)$ of analytic functions involving the Poisson distribution defined by the $\left(p,q\right)$ derivative operator. Furthermore, distortion bounds, covering theorems, a radius of starlikeness, and convexity for functions belonging to this class are obtained.

This paper establishes the upper bounds for the second and third coefficients of holomorphic and bi-univalent functions in a family which involves Bazilevic functions and $\mu$-pseudo-starlike functions under a new operator, joining the neutrosophic Poisson distribution with the modified Caputo’s derivative operator. We also discuss Fekete–Szego’s function problem in this family. Examples are given to support our case for the neutrosophic Poisson distribution. The fields of physics, mechanics, engineering, and biology all make extensive use of fractional derivatives.