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Meromorphic multivalent functions with positive coefficients. II. (English) Zbl 0718.30009

[For part I see the author in ibid. 35, No.1, 1-11 (1990; Zbl 0705.30019).]
Let \(H^*(p;A,B)\) denote the class of functions of the form \[ f(z)=z^{-p}+\sum^{\infty}_{n=1}a_{n+p-1}z^{n+p-1} \] which are regular and p-valent in \(D=\{z:\) \(0<| z| <1\}\) and satisfying the conditions: \[ a_{n+p-1}\geq 0,\quad | z^{p+1}f'(z)+p| \leq | \beta z^{p+1}f'(z)+Ap| \text{ for } z\in D. \] Sharp coefficient estimates, distortion theorems, radius of meromorphic convexity estimates and extreme points are determined for this class \(H^*(p;A,B)\). Some convolution properties and integral transforms of functions in the class \(H^*(p;(2\alpha -1)\beta,\beta)\), \(0\leq \alpha <1\), \(0<\beta \leq 1\), are also studied. In particular it is proved: Theorem. If \(f(z)\in H^*(p;A,B)\), then for \(0<| z| =r<1\) \[ r^{-p}-\frac{B-A}{1+B}r^ p\leq | f(z)| \leq r^ p+\frac{B- A}{1+B}r^ p \] with equality on the right holds for the function \[ f_ p(z)=z^{-p}+\frac{B-A}{1+B}z^ p. \]

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

Citations:

Zbl 0705.30019
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