Univalence, strong starlikeness and integral transforms

M. Obradović¹ ; S. Ponnusamy² ; P. Vasundhra²

¹ Department of Mathematics Faculty of Technology and Metallurgy 4 Karnegijeva St. 11000 Belgrad Serbia and Montenegro
² Department of Mathematics Indian Institute of Technology Madras Chennai-600 036, India

Annales Polonici Mathematici, Tome 86 (2005) no. 1, pp. 1-13.

Résumé

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 $\alpha$.

Détail

DOI : 10.4064/ap86-1-1

Keywords: mathcal represent class normalized analytic functions unit disc mit delta present work first obtain necessary condition convex functions mit delta conditions established certain combination functions starlike convex mit delta using hadamard product tool obtain sufficient conditions functions class functions whose real part positive moreover derive conditions non linear integral transform int zeta zeta zeta univalent mit delta finally sufficient conditions functions strongly starlike order alpha