Properties of close-to-convexity preserved by some integral operators
The authors study operators of the form \[ f(z)=(1/\alpha)z^{1- (1/\alpha)}\int^{z}_{0}z^{(1/\alpha)-2}F(z)dz \] for \(F(z)=z+..\). and z in the open unit disk. They show, for example, that f(z) satisfies \[ Re\{zf'(z)/g(z)\}>\gamma \quad when\quad Re\{zF'(z)/G(z)\}>\beta,\quad Re\{zG'(z)/G(z)\}>\lambda \] and g(z) is the integral of G(z), above. Their work makes use of the methods of \textit{S. S. Miller} [Bull. Am. Math. Soc. 81, 79-81 (1975; Zbl 0302.30003)], and \textit{G. S. Sălăgean} [Lect. Notes Math. 1013, 362-372 (1983; Zbl 0531.30009)].
scientific article; zbMATH DE number 3929371