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Inclusion theorems of convolution operators associated with normalized hypergeometric functions. (English) Zbl 1104.30008

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 \(\beta\), \(\gamma\), \(0\leq\gamma< 1\), \(\beta< 1\), \[ P_\gamma(\beta)= \{f\in A:\text{Re}[e^{i\varphi}((1- \gamma)z^{-1} f(z)+\gamma f'(z)- \beta)]\geq 0,\;\varphi\in\mathbb{R},\,z\in D\}. \] 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 \(zF(a,b,c;z)* f(z)\in{\mathcal H}^\infty\). Results are given in quite complicated forms. Proofs are highly computational.

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.)
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