CERTAIN CLASSES OF MEROMORPHIC FUNCTIONS WITH POSITIVE COEFFICIENTS

Let $\Sigma_p$ denote the class of functions of the form \[f(z)=\frac{a_{-1}}{z}+\sum_{k=1}^\infty a_kz^k \quad (a_k\ge 0, a_{-1}>0)\]which are analytic in the annulus $D =\{z |0< |z|<1\}$. Let $\Sigma_{p,1}$ and $\Sigma_{p,2}$ denote subclasses of $\Sigma_p$ satisfying $f(z_0)=1/z_0$ and $f'(z_0)=-1/z^2_0$ ($-1<z_0<1$, $z_0\neq 0$), respectively. Properties of certain subclasses of $\Sigma_{p,1}$ and $\Sigma_{p,2}$ are investigated and sharp results are obtained. Also a new characterization for certain subclass of $\Sigma_p$ is proved.