Sufficient conditions for starlike and convex functions

S. Ponnusamy¹ ; P. Vasundhra¹

¹ Department of Mathematics Indian Institute of Technology Madras Chennai 600 036, India

Annales Polonici Mathematici, Tome 90 (2007) no. 3, pp. 277-288

Résumé

For $n\geq 1$, let ${\mathcal A}$ denote the class of all analytic functions $f$ in the unit disk $\Delta$ of the form $f(z)=z+\sum_{k=2}^\infty a_kz^k$. For $\mathop{\rm Re} \alpha2$ and $\gamma>0$ given, let ${\mathcal P} (\gamma,\alpha)$ denote the class of all functions $f\in{\mathcal A}$ satisfying the condition $$\bigg|f'(z)-\alpha\, \frac{f(z)}{z}+\alpha-1\bigg| \leq \gamma, \quad\ z\in\Delta. $$ We find sufficient conditions for functions in ${\mathcal P} (\gamma,\alpha)$ to be starlike of order $\beta$. A generalization of this result along with some convolution results is also obtained.

Détail

DOI : 10.4064/ap90-3-7

Keywords: geq mathcal denote class analytic functions unit disk delta form sum infty mathop alpha gamma given mathcal gamma alpha denote class functions mathcal satisfying condition bigg alpha frac alpha bigg leq gamma quad delta sufficient conditions functions mathcal gamma alpha starlike order beta generalization result along convolution results obtained