Geometric properties of the shifted hypergeometric functions

Abstract: We will provide sufficient conditions for the shifted hypergeometric function $z_2F_1(a,b;c;z)$ to be a member of a specific subclass of starlike functions in terms of the complex parameters $a,b$ and $c.$ For example, we study starlikeness of order $\alpha,$ $\lambda$-spirallikeness of order $\alpha$ and strong starlikeness of order $\alpha.$ In particular, those properties lead to univalence of the shifted hypergeometric functions on the unit disk.