×

On the coefficients of some classes of starlike functions. (English) Zbl 0544.30009

For any fixed numbers a,b, \(-1\leq b<a\leq 1,\) let \({\mathcal P}(a,b)\) denote the family of functions P, \(P(0)=1\), holomorphic in the unit disc K and such that \(P(z)=(1+a\omega(z))(1+b\omega(z))^{-1}\) for some function \(\omega\) holomorphic in K and satisfying the conditions \(\omega(0)=0,\quad | \omega(z)|<1\) for \(z\in K\). Next, let \(S^*(a,b)\) stand for the family of functions f holomorphic in the disc K, \(f(0)=0,\quad f'(0)=1,\) such that \(zf'(z)/f(z)=P(z),\) for a function \(P\in {\mathcal P}(a,b);\quad \Sigma^*(a,b)\) denotes the class of functions F meromorphic in K of the form \(F(Z)={1\over2}+\sum^{\infty}_{n=0}a_ nz^ n\) and satisfying the condition \(-zF'(z)/F(z)=P(z),\) for some function \(P\in {\mathcal P}(a,b).\)
In this paper the author obtains sharp estimates of the coefficients of functions of the class \(S^*(a,b)\) and \(\Sigma^*(a,b)\). In particular cases of values of a,b, one gets the earlier results.
Reviewer: J.Kaczmarski

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
PDFBibTeX XMLCite
Full Text: DOI