Geometric properties of Clausen's Hypergeometric Function ${}_3{F}_2(a,b,c;d,e;z)$

Abstract

The Clausen's Hypergeometric Function is given by $${}_3{F}_2(a,b,c;d,e;z)=\sum _{n=0}^{\infty }\frac{(a{)}_n(b{)}_n(c{)}_n}{(d{)}_n(e{)}_n(1{)}_n}{z}^n;a,b,c,d,e\in \mathbb{C}$$ provided $d,e\ne 0,-1,-2,\cdots$ in unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. In this paper, an operator ${\mathcal{I}}_{a,b,c}(f)(z)$ involving Clausen's Hypergeometric Function by means of Hadamard Product is introduced. Geometric properties of ${\mathcal{I}}_{a,b,c}(f)(z)$ are obtained based on its Taylor's coefficient.

12 Pages