On the coefficients of powers of a class of Bazilevich
Let S denote the class of functions \(f(z)=z+\sum^{\infty}_{n=2}a_ nz^ n\) analytic and univalent in the open unit disk \(U=\{z:| z| <1\}\). For \(\alpha >0\) and f(z) in S, let \[ \{\frac{f(z)}{z}\}^{1/\alpha} = \sum^{\infty}_{n=0}a_ n(\alpha)z^ n\text{ and } \{\frac{k(z)}{z}\}^{1/\alpha} = \sum^{\infty}_{n=0}b_ n(\alpha)z^ n, \] where \(k(z)=z/(1-z)^ 2\) (z\(\in U)\) is the well-known Koebe function of class S. For \(\beta >0\) the class B(\(\beta)\) of Bazilevich functions of type \(\beta\) consists of functions f(z) analytic in U and satisfying \[ zf'(z)\{f(z)\}^{\beta - 1}=\{g(z)\}^{\beta}p(z), \] where g(z) is a normalized starlike function so that \(zg'(z)/g(z)=q(z)\). Here p(z) and q(z) are functions of positive real part with \(p(0)=q(0)=1\). It is known that functions in B(\(\beta)\) are univalent and that B(1) is the class of normalized close- to-convex functions defined by W. Kaplan. In this paper the Bazilevich class B(1) receives special consideration, and for f(z)\(\in B(1)\) sharp estimates for the coefficients \(a_ 2(\alpha)\) are given. Some sharp coefficient estimates for functions in B(m), m a positive integer, are also included. Particular attention is paid to cases in which the appropriate power of the Koebe function is not extremal and also to the validity of the inequality \[ | a_ n(\alpha)| \leq b_ n(\alpha). \]
scientific article; zbMATH DE number 3981493