On consequences of certain boundary conditions on the unit circle. (English) Zbl 0965.30004

Adamczyk, Gabriela

Math. Bohem. 125, No. 2, 173-186 (2000).

Summary: Let \(\mathcal P\) denote the well-known class of Carathéodory functions of the form \(p(z)=1+c_1 z+\ldots \), \(z\in \Delta =\{z\in \mathbb C\:|z|<1\}\), with positive real part in the unit disc and let \(\mathbf{H}(M)\) stand for the class of holomorphic functions commonly bounded by \(M\) in \(\Delta \). In 1992, J. Fuka and Z. J. Jakubowski began an investigation of families of mappings \(p\in \mathcal P\) fulfilling certain additional boundary conditions on the unit circle \(T\). At first, the authors examine the class \(\mathcal P(B,b;\alpha)\) of functions defined by conditions given by the upper limits for two disjoint open arcs of \(T\). After that, such boundary conditions given, in particular, by the nontangential limits, were assumed for different subsets of the unit circle. In parallel, G. Adamczyk started to search for properties of families, contained in \(\mathbf{H}(M)\) and satisfying certain similar conditions on \(T\). The present article belongs to the above series of papers. In the first section we will consider subclasses of \(\mathcal P\) of functions satisfying some inequalities on several arcs of \(T\), whereas in Sections 2 and 3 families of mappings \(f\in \mathbf{H}(M)\) with conditions given for measurable subsets of the unit circle \(T\).

MSC:

30C45

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

Keywords:

bounded holomorphic functions; Carathéodory functions; harmonic measure; extreme points