Abstract

Let $\Cal{Q}_{b}(\Phi ,\Psi ;\alpha )$ be the class of normalized analytic functions defined in the open unit disk and satisfying $$ \RE\left\{ 1+\frac{1}{b}\left( \frac{f(z)\ast \Phi (z)}{f(z)\ast \Psi (z)}-1\right) \right\} >\alpha $$ for nonzero complex number $b$ and for $0\leq \alpha <1$. Sufficient condition, involving coefficient inequalities, for $f(z)$ to be in the class $\Cal{Q}_{b}(\Phi ,\Psi ;\alpha )$ is obtained. Our main result contains some interesting corollaries as special cases.

Keywords: Analytic functions; Starlike and convex functions of complex order; Hadamard product; Coefficient inequalities.

MSC: 30C45

Pages:  73$-$78     

Volume  63 ,  Issue  1 ,  2011