Generalized problem of starlikeness for products of close-to-star functions
Volume 107 / 2013
Annales Polonici Mathematici 107 (2013), 109-121
MSC: 30C45, 30C50, 30C55.
DOI: 10.4064/ap107-2-1
Abstract
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.
