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Zbl 1104.30008
Swaminathan, A.
Inclusion theorems of convolution operators associated with normalized hypergeometric functions.
(English)
[J] J. Comput. Appl. Math. 197, No. 1, 15-28 (2006). ISSN 0377-0427

Assume the notations: $F(a,b,c; z)$ the Gauss hypergeometric function, $A$ -- the class of functions analytic on the unit disk $D$ and normalized by the conditions $f(0)= f'(0)- 1= 0$, $f*g$ -- the Hadamard convolution. For given $\beta$, $\gamma$, $0\le\gamma< 1$, $\beta< 1$, $$P_\gamma(\beta)= \{f\in A:\text{Re}[e^{i\varphi}((1- \gamma)z^{-1} f(z)+\gamma f'(z)- \beta)]\ge 0,\ \varphi\in\bbfR,\,z\in D\}.$$ The author asked and partially answered the following question. Under what conditions on $\gamma$, $f\in P_\gamma(\beta_1)$ and $f\in P_\gamma(\beta_2)$ implies $f*g\in P_\gamma(\alpha)$, for some $\alpha=\alpha(\beta_1,\beta_2)$? He also, among other results, gave conditions for $a$, $b$, $c$, $\beta$ and $\gamma$ under which $zF(a,b,c; z)$ is in $P_\gamma(\beta)$ or $zF(a,b,c;z)* f(z)\in{\Cal H}^\infty$. Results are given in quite complicated forms. Proofs are highly computational.
[Eligiusz Złotkiewicz (Lublin)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
33C45 Orthogonal polynomials and functions of hypergeometric type

Keywords: convex; starlike; hypergeometric functions; integral transforms

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