Some subclasses of close-to-convex functions. (English) Zbl 0673.30009

In the paper the class \(S^*_ s\) of functions starlike with respect to symmetric points and some related classes are investigated. The class \(S^*_ s\) is defined as follows: \[ f\in S^*_ s\quad \Leftrightarrow \quad f(z)=z+a_ 2z^ 2+...\quad and\quad Re(zf'(z))/(f(z)-f(-z))>0\quad for\quad | z| <1. \] This class was investigated among others by Sakaguchi, M. S. Robertson and also by the reviewer in the paper [Ann. Univ. Marie Curie-Skłodowska, Sect. A 19, 53-59 (1965; Zbl 0201.409)]. The authors concentrate their attention on the problem of coefficients. For the class \(S^*_ s\) the following results are presented: If \(f(z)=z+a_ 2z^ 2+...\in S^*_ s\) and \(n\geq 2\), then \[ | | a_{n+1}| -| a_ n| | \leq Cn^{-1/2}, \] where C is constant. The growth rate \(n^{-1/2}\) is best possible. Unfortunately the estimation is false as the function \[ f_ 2(z)=z/(1-z^ 2)=z+z^ 3+z^ 5+...\in S^*_ s \] shows. For this function we have \(| | a_{n+1}| -| a_ n| | \equiv 1\). There are some mistakes (in printing, but not only) in the proofs of this paper. The analyse of the proof of the mentioned result suggests that the growth rate will be \(n^{1/2}\) but the extremal function given in this proof \[ f_ 0(z)=(k(z^ 4))^{1/4}=z/(1-z^ 4)^{1/2} \] is not extremal in this case.
Reviewer: J.Stankiewicz


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


Zbl 0201.409