ON SUBCLASSES OF STARLIKE FUNCTIONS OF ORDER $\alpha$ AND TYPE $\beta$

Abstract

Let $S^*(\alpha, \beta, A, B) (0\le \alpha<1, 0<\beta\le 1, -1\le A <S\le 1, 0<B\le 1)$, denote the class of functions $f(z) = z+ \sum_{n=2}^\infty a_nz^n$ analytic in $U = \{z : |z|< 1\}$ which satisfy for $z=re^{i\theta}\in U$, \[ \left|\frac{z\frac{f'(z)}{f(z)}-1}{(B-A)\beta\left(z\frac{f'(z)}{f(z)}-\alpha\right)+A\left(z\frac{f'(z)}{f(z)}-1\right)}\right|<1.\] It is the purpose of this paper to show a representation formula, a distortion theo- rem, a sufficient condition for this class $S^*(\alpha, \beta, A, B)$. Furthermore, we maximize $|a_3-\mu a_2^2|$ over the class $S^*(\alpha, \beta, A, B)$ and we give the radii of convexity for functions in the class $S^*(\alpha, \beta, A, B)$.