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Certain sufficiency conditions on Gaussian hypergeometric functions. (English) Zbl 1126.30010

Let \({\mathcal A}\) denote the class of functions of the form \[ f(z)= z+\sum^\infty_{n=2} a_nz^n \] analytic in \(\Delta= \{z: |z|< 1\}\) and let \(S\) denote a subclass of \({\mathcal A}\) that are univalent in \(\Delta\). Let \(f\in{\mathcal A}\), \(k\in [0,\infty)\), \(\alpha\in [0,1)\). Then we say that \(f\in k\text{-UCV}(\alpha)\) if and only if
\[ \operatorname{Re}\Biggl\{1+ {zf''(z)\over f'(z)}\Biggr\}\geq k\Biggl|{zf''(z)\over f'(z)}\Biggr|+ \alpha\quad\text{for }z\in\Delta. \]
We put \(k\text{-UCV}:= k\text{-UCV}(0)\).
The Gaussian hypergeometric function \(f(z)\) is given by the series \[ f(z)= zF(a,b; c;z)= z \sum^\infty_{n=0} {(a,n)(b,n)\over (c,n)(1,n)} z^n, \] where \((a,n)\) is a Pochhammer symbol.
In this paper some properties of the Gaussian functions are derived. For example it is proved.
Theorem 1. Let \(a\), \(b\), \(c\), \(k\) be a fixed and such that \(a>-1\), \(b>-1\), \(c>a+ b+2\), \(0\leq k<\infty\). If \[ {(a+1)(b+1)\over c+1}\cdot {\Gamma(c- a- b- 1)\Gamma(c+ 1)\over \Gamma(c- a)\Gamma(c- b)}\leq {1\over k+ 2} \] then \(f(z)= zF(a,b;c;z)\in k\text{-UCV}= k\text{-UCV}(0)\).
Some others similar problems are also investigated.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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