Geometric properties of Clausen's Hypergeometric Function $_3F_2(a,b,c;d,e;z)$

Abstract

The Clausen's Hypergeometric Function is given by $${}_3F_2(a,b,c;d,e;z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n(c)_n}{(d)_n(e)_n(1)_n}z^n\, ; \ a,b,c,d,e\in \mathbb{C}$$ provided $d,\, e\, \neq 0,-1,-2,\cdots$ 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.