Convolutions with hypergeometric functions. (Q5952804)

This paper deals with the operator \(T(f)\) defined by the convolution/Hadamard product \((T(f))(z)=zF(a,b;c;z)*f(z)\) where \(F(a,b;c;z)\) denotes the classical hypergeometric function and \(f\in {\mathcal A}\), the class of all normalized analytic functions in the unit disc \(\Delta =\{z:\, | z| <1\}\). Let \(S=\{f\in {\mathcal A}:\, f ~\text{ is ~univalent ~in} ~\Delta\}\), \(R(\beta) = \{f\in {\mathcal A}: \, \text{ Re}\, f'(z) > \beta, ~z\in \Delta\}\) and \(S^{*}(A,B) =\{f:\, zf'(z)/f(z)\prec (1+Az)/(1+Bz)\), for some \(-1\leq B<A\leq 1\). The integral representation for \(T(f)\) was recently given by \textit{Y. C. Kim} and \textit{F. Rønning} [J. Math. Anal. Appl. 258, 466--489 (2001; Zbl 0982.44001)] and by \textit{R. Balasubramanian, S. Ponnusamy} and \textit{M. Vuorinen} [J. Comput. Appl. Math. 139, 299--322 (2002; Zbl 1172.33302)] where one can find a large number of results concerning operators of similar type. In the present paper the authors obtained conditions on the parameters \(a,b,c\) such that \(T(f)\in S^{*}(A,B)\) if \(f\in R(\beta)\) or \(S\).