Coefficient inequalities for certain classes of analytic functions of complex order

Abstract

Let ${\mathcal{Q}}_{\mathcal{b}}\mathcal{(}\mathcal{\Phi }\mathcal{,}\mathcal{\Psi }\mathcal{;}\alpha \mathcal{)}$ be the class ofnormalized analytic functions defined in the open unit disk andsatisfying$$\text{RE}\{1+\frac{1}{b}(\frac{f(z)\ast \mathrm{\Phi }(z)}{f(z)\ast \mathrm{\Psi }(z)}-1)\}>\alpha$$for nonzero complex number $b$ and for $0\le \alpha <1$.Sufficient condition, involving coefficient inequalities, for $f(z)$to be in the class ${\mathcal{Q}}_{\mathcal{b}}\mathcal{(}\mathcal{\Phi }\mathcal{,}\mathcal{\Psi }\mathcal{;}\alpha \mathcal{)}$ isobtained. Our main result contains some interesting corollaries asspecial cases.