# Wydawnictwa / Czasopisma IMPAN / Annales Polonici Mathematici / Wszystkie zeszyty

## Sufficient conditions for starlike and convex functions

### Tom 90 / 2007

Annales Polonici Mathematici 90 (2007), 277-288 MSC: 30C45, 30C55. DOI: 10.4064/ap90-3-7

#### Streszczenie

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.

#### Autorzy

• S. PonnusamyDepartment of Mathematics
Chennai 600 036, India
e-mail