Abstract

For analytic functions $f$ normalized by $f(0)=f'(0)-1=0$ in the open unit disk $U$, a class $P_{\alpha}(\lambda)$ of $f$ defined by $|D^{\alpha}_{z}(\frac{z}{f(z)})|\leq \lambda$, where $D^{\alpha}_{z}$ denotes the fractional derivative of order $\a$, $m \leq \alpha < m+1$, $m \in N_{0} $, is introduced. In this article, we study the problem when $\frac{1}{r} f(rz) \in P_{\alpha}(\lambda)$, $3 \leq \alpha < 4$.

Keywords: Analytic functions, univalent functions, Cauchy-Schwarz inequality, fractional differential operator.

MSC: 30C45

Pages:  55$-$58     

Volume  63 ,  Issue  1 ,  2011