Generalized problem of starlikeness for products of close-to-star functions

Jacek Dziok¹

¹ Institute of Mathematics University of Rzeszów 35-310 Rzeszów, Poland

Annales Polonici Mathematici, Tome 107 (2013) no. 2, pp. 109-121

Résumé

We consider functions of the type $F(z)=z\prod_{j=1}^{n}[ {% f_{j}(z)/{z}}] ^{a_{j}}$, where $a_{j}$ are real numbers and $f_{j}$ are $\beta _{j}$-strongly close-to-starlike functions of order ${\alpha _{j}} $. We look for conditions on the center and radius of the disk $\mathcal{D}(a,r)=\{z:|z-a| r\}$, ${\vert a\vert r\leq 1-|a|}$, ensuring that $F(\mathcal D(a,r))$ is a domain starlike with respect to the origin.

Détail

DOI : 10.4064/ap107-2-1

Keywords: consider functions type prod where real numbers beta strongly close to starlike functions order alpha look conditions center radius disk mathcal z a vert vert leq ensuring mathcal domain starlike respect origin