Inclusion properties of hypergeometric type functions and related integral transforms

Abstract

In this work, conditions on the parameters $a, b$ and $c$ are given so that the normalized Gaussian hypergeometric function $zF(a,b;c;z)$, where \begin{align*} F(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_n(1)_n}z^n, \quad |z|<1, \end{align*} is in certain class of analytic functions. Using Taylor coefficients of functions in certain classes, inclusion properties of the Hohlov integral transform involving $zF(a,b;c;z)$ are obtained. Similar inclusion results of the Komatu integral operator related to the generalized polylogarithm are also obtained. Various results for the particular values of these parameters are deduced and compared with the existing literature.