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.

Denote by B(τ) the class of all complex functions of the form f (z) ≡ τ + ∞ n=1 a n (f )z n + b n (f ) z n that are harmonic in the open unit disk D with f (D) ⊂ D.Both B(τ) and some of its closed convex subsets are strongly convex, e.g.,Simple extremal problems in B(τ) may have nontrivial solutions. To find them, we present various methods: the method of subordination, the Poisson integral, and calculus of variations. For instance, by analogy to the known result a n ( (τ)) = {w : |w| ≤ 1 − |τ| 2 }, we find the variability regions a n (B(τ)) = b n (B(τ)) = w :x + cos t √x 2 + 2x cos t + 1 dt .In the case n = 1, |τ| < 2/π (resp., |τ| = 2/π), the extremal functions realizing points of the circle w : |w| = ϕ 1/ϕ −1 (|τ|) are univalent self-mappings of the disk D (resp., univalent mappings of D onto a half unit disk). PreliminariesThe classes of all complex analytic functions and all complex harmonic functions in the disk D = {z ∈ C : |z| < 1} that are bounded there by one have a surprising property. Both of them are strongly convex, and hence their extreme and support points form dense subsets. The geometric nature of the sets considered requires some notions of abstract analysis.Let H be a separated locally convex linear space. Suppose that ∅ = F ⊂ H and write conv F, convF, ext F and spt F to denote the convex hull of F, the closed convex hull of F, the set of all extreme points of F, and the set of all (proper)

We consider a class T O H of typically real harmonic functions on the unit disk that contains the class of normalized analytic and typically real functions. We also obtain some partial results about the region of univalence for this class.

No abstract