ON SOME SUBCLASSES OF CLOSE-TO-CONVEX FUNCTIONS OF ORDER $\alpha$ TYPE $\delta$

Abstract

Let $Q_\lambda^*(\alpha, \delta)$ denote the class of analytic functions $f$ in the unit disc $E$, with $f(0)=0$, $f'(0) =1$ and satisfying the condition

\[Re \left\{(1-\lambda)\frac{zf'(z)}{g(z)}+\lambda\frac{(zf'(z))'}{g'(z)}\right\}>\alpha,\]

for $z\in E$, $g$ starlike function of order $\delta$ ($0 \le \delta \le 1$), $0\le \alpha \le 1$ and $\lambda$ complex with Re$\lambda\ge 0$. It is shown that $Q_\lambda^*(\alpha, \delta)$ with $\lambda\ge 0$ arc close-to-convex and hence univalent in $E$. Coeffiicient results, an integral representation for $Q_\lambda^*(\alpha, \delta)$ and some other propertie of $Q_\lambda^*(\alpha, \delta)$ are discussed. The class $Q_\lambda^*(\alpha, 1)$ is also investigated in some detail.