Spirallikeness of shifted hypergeometric functions

In the present paper, we study spirallikenss (including starlikeness) of the shifted hypergeometric function $f(z)=z_2F_1(a,b;c;z)$ with complex parameters $a,b,c,$ where $_2F_1(a,b;c;z)$ stands for the Gaussian hypergeometric function. First, we observe the asymptotic behaviour of $_2F_1(a,b;c;z)$ around the point $z=1$ to obtain necessary conditions for $f$ to be $λ$-spirallike for a given $λ$ with $- π/2< λ<π/2.$ We next give sufficient conditions for $f$ to be $λ$-spirallike. As special cases, we obtain sufficient conditions of strong starlikeness and examples of spirallike, but not starlike, shifted hypergeometric functions.

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