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Koebe domains for the classes of functions with ranges included in given sets

Koczan L., Zaprawa P.

Journal of Applied Analysis

2008

Vol. 14, nr 1

43-52

In this paper we present a new method of determining Koebe domains. We apple this method by giving a new proof of the well-known theorem of A. W. Goodman concerning the Koebe domain for the class T of typically real functions. We applied also the method to determine Koebe sets for classes of the special type , i.e. for TM,g = {∫ ∈ T : ∫(Δ) ⊂ Mg(Δ)}, g ∈ T ∩ S, M > 1, where Δ = {z ∈ C: IzI < 1} and T, S stand for the classes of tipically real functions and univalent functions respectively. In particular, we find the Koebe domains for the class T (M) of all typically real functions with ranges in a given strip.

On typically real functions which omit two conjugated values

Sobczak-Kneć M.

Demonstratio Mathematica

2008

Vol. 41, nr 4

833-844

In this paper we discuss the class Tp[...] consisting of typically real functions which do not admit values WQ = p[...]. We estimate the second and the third coefficients of a function [...] and we determine the Koebe domain for the class of typically real functions with fixed second coefficient.

Univalence problems for classes of functions preserving sectors

Koczan Leopold

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

2002

z. 26 [199]

89-95

Let A be the set of all functions that are analytic in the disk delta = [z is an element of C : \z\ < 1} and normalized by f(O) = f'(O) - 1 = 0. Abu-Muhanna and MacGregor discussed in the paper [1] different classes of functions which preserved some sectors. For k > 2 they used notation: [...].