×

A remark on certain close-to-convex functions. (English) Zbl 0712.30007

Let \(R_ n(\alpha)\) denote the class of functions \(f(z)=z+\sum^{\infty}_{k=n+1}a_ kz^ k\), \(n\in N\), analytic in the unit disk U, which satisfy the condition \(| f'(z)-1| <1-\alpha\) for some \(\alpha\), \(0\leq \alpha <1\), \(z\in U\). Using a well known result of S. S. Miller and P. T. Mocanu [J. Math. Anal. Appl. 65, 289-305 (1978; Zbl 0367.34005)] (there is a misprint in the references; this paper is omitted), the authors prove that for \(f\in R_ n(\alpha)\), \[ Re (e^{i\beta}f(z)/z)>n \cos \beta /(n+2),\quad z\in U, \] where \(| \beta | \leq \pi /2-\sin^{-1}(1-\alpha)\).
Reviewer: O.Fekete

MSC:

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

Citations:

Zbl 0367.34005
PDFBibTeX XMLCite