On uniformly convex functions. (English) Zbl 0744.30010

Goodman, A. W.

Ann. Pol. Math. 56, No. 1, 87-92 (1991).

The author introduces the class \(UCV\) of uniformly convex functions which consists of those convex functions \(f\) transforming any circular arc in the unit disk \(E\) with center \(\zeta\in E\) into a convex arc. He shows that \(f\in UCV\) if and only if \[ 1+\hbox{Re}\left({f''(z) \over f'(z)}(z- \zeta)\right)\geq 0, \] for every pair \((z,\zeta)\in E\times E\), gives some examples of uniformly convex functions and proves some coefficient inequalities.

Reviewer: K.J.Wirths (Braunschweig)

MSC:

30C45

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

Keywords:

circular arc; convex arc; uniformly convex functions