CERTAIN CLASSES OF THE MEROMORPHIC HARMONIC FUNCTIONS WITH A POLE AT SOME FIXED POINT OF THE UNIT DISK

Abstract

The class $S_{H}(p)$, $0 \leq p \lt 1$, of complex valued, meromorphic harmonic univalent sense-preserving functions in the unit disk $U \backslash \{p\}$ is studied. The functions belong to $S_{H}(p)$ have the expansion $f(z) = \frac{\alpha}{z-p} + \sum_{n=0}^{\infty} c_{n} z^{n} + \overline{\sum_{n=1}^{\infty} d_{n} z^{n}} + A\log |z-p|$ and $\lim_{z \rightarrow p} f(z) = \infty$. Some coefficient estimates, distortion and area theorems are obtained. Sufficient coefficient conditions for a class of meromorphic harmonic univalent sense-preserving functions that are starlike and convex are given.