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Zbl 1126.30010
Swaminathan, A.
Certain sufficiency conditions on Gaussian hypergeometric functions. (English)
[J] JIPAM, J. Inequal. Pure Appl. Math. 5, No. 4, Paper No. 83, 10 p., electronic only (2004). ISSN 1443-5756/e

Let ${\Cal A}$ 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 ${\Cal A}$ that are univalent in $\Delta$. Let $f\in{\Cal A}$, $k\in [0,\infty)$, $\alpha\in [0,1)$. Then we say that $f\in k\text{-UCV}(\alpha)$ if and only if $$\Re\Biggl\{1+ {zf''(z)\over f'(z)}\Biggr\}\ge k\Biggl\vert{zf''(z)\over f'(z)}\Biggr\vert+ \alpha\quad\text{for }z\in\Delta.$$ 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 $a>-1$, $b>-1$, $c>a+ b+2$, $0\le k<\infty$. If $${(a+1)(b+1)\over c+1}\cdot {\Gamma(c- a- b- 1)\Gamma(c+ 1)\over \Gamma(c- a)\Gamma(c- b)}\le {1\over k+ 2}$$ then $f(z)= zF(a,b;c;z)\in k\text{-UCV}= k\text{-UCV}(0)$.\par Some others similar problems are also investigated.
[Jan Stankiewicz (Rzeszów)]

MSC 2000:: *30C45 Special classes of univalent and multivalent functions
33C45 Orthogonal polynomials and functions of hypergeometric type
30C50 Coefficient problems for univalent and multivalent functions

Keywords: Gaussian hypergeometric functions; $k$-UCV; convex functions

Cited in: Zbl 1160.30312

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