Bounded harmonic mappings

Abstract

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)

Some properties of a class of elliptic partial differential operators

“…For α = 0, there are numerous discussions on coefficient estimates of harmonic mappings in the literature, see for example [8,9,10,12,15,21,29]. We investigate the problem of coefficient estimates on the solutions to (1.1) as follows.…”

Schwarz–Pick type estimates of pluriharmonic mappings in the unit polydisk

“…We refer to [5,6,7,8,9,10,13,20,23,28] for further discussion on this topic. In this paper, we generalize Theorem A to higher dimensional case, and give the estimate for the partial derivatives of arbitrary order.…”

Landau's theorems for biharmonic mappings (II)

“…If D is simply connected, then a harmonic mapping f ( z ) can be written as with f (0) = h (0), where h ( z ) and g ( z ) are analytic in D (for details, see ). Thus, we have Suppose that f ( z ) = u ( x , y ) + i v ( x , y ), z = x + i y is a four times continuously differentiable complex‐valued function in a domain .…”