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On a certain class of meromorphic univalent functions with positive coefficients. (English) Zbl 0745.30011

Let \(\Sigma_ p(\alpha,\beta,A,B,\gamma)\) (\(0\leq\alpha<1\), \(0<\beta\leq1\), \(-1\leq A<B\leq1\), \(0<B\leq1\), \({A\over A- B}\leq\gamma\leq 1\)) denote the class of meromorphic univalent functions \[ f(z)={1\over z}+\sum_{n=1}^ \infty a_ n z^ n \qquad (a_ n\geq0), \] in \(U^*=\{z:\;0<| z|<1\}\) and satisfying the condition \[ \left|{z^ 2f'(z)+1} \over {[(B-A)\gamma+A]z^ 2f'(z)+[(B-A)\gamma\alpha+A]} \right|<\beta. \] In this paper the author determines the coefficient estimates, distortion theorems, radii of starlikeness and convexity for the class \(\Sigma_ p(\alpha,\beta,A,B,\gamma)\). It is further shown that this class is closed under convex linear combinations and convolutions.
Reviewer: R.M.Goel (Patiala)

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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