Uniformly convex spiral functions and uniformly spirallike function associated with Pascal distribution series

TL;DR

This paper investigates conditions under which Pascal distribution series belong to classes of uniformly spirallike and convex functions, providing new inclusion relations and properties of related special functions.

Contribution

It establishes necessary and sufficient conditions for Pascal distribution series to be in specific function classes and explores properties of related special functions.

Findings

Derived conditions for Pascal series to be uniformly spirallike and convex.

Established inclusion relations between Pascal series and function classes.

Analyzed properties of a special function related to Pascal distribution series.

Abstract

The aim of this paper is to find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in the classes SP_{p}({\alpha},\b{eta}) and UCV_{p}({\alpha},\b{eta}) of uniformly spirallike functions. Further, we consider properties of a special function related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.