Convolutory operators preserving univalent functions (Q1073952)

Let \(f(z)=z+a_ 2z^ 2+..\). be regular and univalent in the open disk. Denoting the Hadamard convolution by *, the authors let \[ {\mathcal F}={\mathcal F}(a,b,c)f(z)=zF(a,b,c;z)*f(z) \] for \[ F(a,b,c;z)=1+\sum ((a)_ k(b)_ k/(c)_ k(1)_ k)z^ k,\quad with\quad (a)_ k=\Gamma (a+k)/\Gamma (a). \] For suitable choices of parameters \({\mathcal F}\) is the Biernacki operator \(\int^{z}_{0}(f(t)/t)dt\), the Libera operator \(2z^{-1}\int^{z}_{0}f(t)dt\), or their inverses. The authors show that for some choices of parameters, the operator \({\mathcal F}\) yields a univalent function and provide generalizations of the Biernacki and Libera operators. These results include some conclusions made earlier by \textit{St. Ruscheweyh}, Proc. Am. Math. Soc. 49, 109-115 (1975; Zbl 0303.30006); \textit{A. E. Livingston}, ibid. 17, 352-357 (1966; Zbl 0158.077); \textit{S. D. Bernardi}, Trans. Am. Math. Soc. 135, 429-446 (1969; Zbl 0172.097), and others. The authors then go on to show that \({\mathcal F}\) has a quasiconformal continuation for certain choices of the parameters.