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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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