Spirallikeness of shifted hypergeometric functions

Abstract

In the present paper, we study spirallikenss (including starlikeness) of the shifted hypergeometric function $f(z)=z_2{F}_1(a,b;c;z)$ with complex parameters $a,b,c,$ where ${}_2{F}_1(a,b;c;z)$ stands for the Gaussian hypergeometric function. First, we observe the asymptotic behaviour of ${}_2{F}_1(a,b;c;z)$ around the point $z=1$ to obtain necessary conditions for $f$ to be $\lambda$-spirallike for a given $\lambda$ with $-\pi /2<\lambda <\pi /2.$ We next give sufficient conditions for $f$ to be $\lambda$-spirallike. As special cases, we obtain sufficient conditions of strong starlikeness and examples of spirallike, but not starlike, shifted hypergeometric functions.

16 pages, 1 figure