Let the quadruple (X,%', p, h ) be an invertible abstract dynamical system (see def. 1.1-1.2) and let %'Ih1 = { A E 3 : h -' ( A ) = A ] . In this paper the convex subsets (1.2)-(1.4) of the set M of all finite measures on 37 are examined. We show that if there exists a measurable generator for ( X , < H , p , h ) (see def. 2.2), then the sets (1.2)-(1.4)have extreme points and we determine them (see th. 8.1). As an application we solve some extremal linear problems in the sets (1.2)-(1.4) (see section 9). A curious topological property of the sets of extreme points is given by th. 11.2.

LetIn 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.

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Abstract. In this paper we present a new method of determining Koebe domains. We apply 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, where ∆ = {z ∈ C : |z| < 1} and T , S stand for the classes of typically real functions and univalent functions respectively. In particular, we find the Koebe domains for the class TM of all typically real and bounded functions, and for the class T (M ) of all typically real functions with ranges in a given strip.

Let F denote the class of all functions univalent in the unit disk ≡ {ζ ∈ C : |ζ | < 1} and convex in the direction of the real axis. The paper deals with the subclass F (n) of these functions f which satisfy the property f (εz) = ε f (z) for all z ∈ , where ε = e 2πi/n . The functions of this subclass are called n-fold symmetric. For F (n) , where n is odd positive integer, the following sets, f ∈F (n) f ( )-the Koebe set and f ∈F (n) f ( )-the covering set, are discussed. As corollaries, we derive the Koebe and the covering constants for F (n) .