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1

Koebe domains for the class of typically real functions that are convex in two directions

Gregorczyk M., Koczan L.

Journal of Applied Analysis

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2018

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Vol. 24, nr 2

167--173

EN

In this paper, we discuss the class of typically real functions that are convex in two directions. We determine the Koebe domain for this class and its different representations.

2

On functions convex in the direction of the imaginary axis with real coefficients

Koczan L., Sobczak-Kneć M., Zaprawa P.

Demonstratio Mathematica

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2010

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Vol. 43, nr 3

545-559

EN

Let Υ be a subclass of the class of all analytic functions in the unit disk Δ having the normalization f(0) = f′(0) − 1 = 0. If there exists an analytic, univalent function m satisfying the following conditions: m′ (0) > 0, ∧f ∈Υ m ‹ f and for every analytic function k, k(0) = 0, there is (∧f∈Υ k ‹ f) ⇒ k ‹ m, then this function is called the minorant of Υ. Similarly, if there exists an analytic, univalent function M such that M′ (0) > 0, ∧f ∈Υ f ‹ M and for every analytic function k, k(0) = 0, there is (∧f ∈Υ f ‹ k) ⇒ M ‹ k, then this function is called the majorant of Υ. It is possible to give a number of examples of classes of analytic functions for which the majorant or minorant does not exist. However, if these functions exist then m(Δ) and M(Δ) coincide with the Koebe domain and the covering domain for Υ, respectively. In this paper we determine the Koebe domain and the covering domain as well as the minorant and the majorant for the class consisting of functions convex in the direction of the imaginary axis with real coefficients.

3

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

Koczan L., Zaprawa P.

Journal of Applied Analysis

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2008

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Vol. 14, nr 1

43-52

EN

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.

4

On typically real functions which omit two conjugated values

Sobczak-Kneć M.

Demonstratio Mathematica

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2008

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Vol. 41, nr 4

833-844

EN

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.