COEFFICIENT ESTIMATES FOR BOUNDED STARLIKE FUNCTIONS OF COMPLEX ORDER

Let $F(b,M,n)$($b\neq 0$, complex, $M >1/2$, and $n$ is a positive integer) denote the classof functions $f(z)=z+\sum_{k=n+1}^\infty a_kz^k$ analytic in $U=\{z: |z|< 1\}$ which satisfy for fixed $M$, $f (z)/z \neq 0$ in $U$ and \[ \left|\frac{b-1+\frac{zf'(z)}{f(z)}}{b}-M\right|<M, \quad z\in U.\]Also let $F^*(b,M,n)$ ($b\neq 0$, complex, $M >1/2$, and $n$ is a positive integer) denote the class of functions $f(z)=1/z+\sum_{k=n}^\infty a_kz^k$ analytic in the annulus $U^* = \{z : 0 < |z| < 1\}$ which satisfy \[ \left|\frac{b-1+\frac{zf'(z)}{f(z)}}{b}-M\right|<M, \quad z\in U^*.\]In this paper we obtain bounds for the coefficients of functions of the above classes.