Sur une famille de fonctions univalentes et bornées. (On a family of univalent and bounded functions). (French) Zbl 0712.30020

Śladkowska, Janina

Demonstr. Math. 22, No. 4, 973-982 (1989).

Let \(S_ 1(a)\) denote the space of all functions f, holomorphic and univalent in the unit disk U, of the form \[ f(z)=b_ 1z+b_ 2z^ 2+...,\quad b_ i\in {\mathbb{C}}, \] such that \(a\not\in f(U)\), \(| a| <1\) fixed, and f(U)\(\subset U\). Let \(\psi\) (f) be a complex and continuous functional defined on \(S_ 1(a)\). Using variational methods, the author investigates the functional Re \(\psi\) (f) defined on \(S_ 1(a)\) and proves some properties of extremal functions. An example is given.

Reviewer: O.Fekete

MSC:

30C70	Extremal problems for conformal and quasiconformal mappings, variational methods
30H05	Spaces of bounded analytic functions of one complex variable

Keywords:

extremal functions