Univalence, strong starlikeness and integral transforms
Volume 86 / 2005
Annales Polonici Mathematici 86 (2005), 1-13
MSC: 30C45, 30C55.
DOI: 10.4064/ap86-1-1
Abstract
Let $\mathcal A$ represent the class of all normalized analytic functions $f$ in the unit disc ${\mit\Delta}$. In the present work, we first obtain a necessary condition for convex functions in ${\mit\Delta}$. Conditions are established for a certain combination of functions to be starlike or convex in ${\mit\Delta}$. Also, using the Hadamard product as a tool, we obtain sufficient conditions for functions to be in the class of functions whose real part is positive. Moreover, we derive conditions on $f$ and $\mu$ so that the non-linear integral transform $\int_0^z ({\zeta}/{f(\zeta)})^{\mu}\,d\zeta$ is univalent in ${\mit\Delta}$. Finally, we give sufficient conditions for functions to be strongly starlike of order $\alpha$.
