Sufficient conditions for starlike and convex functions
Volume 90 / 2007
Annales Polonici Mathematici 90 (2007), 277-288
MSC: 30C45, 30C55.
DOI: 10.4064/ap90-3-7
Abstract
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 $\mathop{\rm Re} \alpha<2$ 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.
