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 \[ f_ p(z)=z^{-p}+\frac{B-A}{1+B}z^ p. \]


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


Zbl 0705.30019