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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.

Analytic, symmetric functions that omit zero

Śladkowska Janina

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

2004

z. 27 [212]

69-81

Let Fo] (b), b is an element of R denote the class of functions of the form f{z) = b+b1z+..., analytic in the unit disk U and such that 0 [...] and let Fo(b) be the set of functions of the class Fo(b) such that f(z) €is an element of R iff z (-1,1). The functions from the class ^{b) are closely related to the typically-real functions, namely for each f is an element of Fo(b) there exist two typically-real functions T1 and T2 such that f = bT1/T2. So it is possible to treat Fo (b) as a subclass of the class[...] of all the functions of the form bT1/T2, where T1,T2 are arbitrary typically-real functions. It is easy to obtain the variational formulas in the class [...] (b) by utilizing the known formulas obtained in the class of typically-real functions [1]. Certain extremal problems in the class FoR(b) and also in the class Fo(b), for example estimation from below of the second coefficient, were solved with the aid of these formulas.

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: [...].

On domains of univalence for the class of typically real odd functions

Koczan L., Zaprawa P.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

2001

z. 25 [190]

81-89

The univalence problems for the class T consisting of typically real functions are connected with the name of Goluzin. It was Goluzin, who derived the radius of local univalence in the class T and described the univalence domain H in T [1]. This domain H and the domain of local univalence are equal. Moreover, H is (in some sense) the largest set in which all typically real functions are univalent, but for these functions the domain of local univalence is larger than H. This observation suggests the possibility of existence of other domains of univalence.