## Close-to-convexity properties of Gaussian hypergeometric functions.(English)Zbl 0901.30007

Let $$F(a, b;c;z)$$ be the classical hypergeometric function. The sufficient conditions on $$a$$, $$b$$, $$c$$ under which $$zF(a,b; c,z)$$ or $${c\over ab} [F(a, b; c;z)- 1]$$ is closed-to-convex of order $$\beta$$ in $$| z|< 1$$ are given.
Example: If $$a\in (0,\infty)$$, $$b\in\left(0,{1\over a}\right]$$ and if for some real $$\eta$$, $$|\eta|< {\pi\over 2}$$, $\beta\leq 1-{1\over\cos\eta} (1-\Gamma(a+ b)/\Gamma(a)\Gamma(b))$ then the function $$f(z)= zF(a,b; a+b;z)$$ satisfy: $$\text{Re}[e^{i\eta}(1- z)f'(z)- \beta)]>0$$, $$(| z|< 1)$$.

### MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 33C20 Generalized hypergeometric series, $${}_pF_q$$
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### References:

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