Bounded harmonic mappings

Abstract

Denote by B(τ) the class of all complex functions of the form

$$ f(z) \equiv \tau + \sum\limits_{n = 1}^\infty {(a_n (f)z^n + \overline {b_n (f)} \bar z^n )} $$

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.,

$$ \Lambda (\tau ) = \{ f \in {\rm B}(\tau ):b_n (f) = 0 for all n \geqslant 1\} . $$

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

$$ a_n (\Lambda (\tau )) = \{ w:|w| \leqslant 1 - |\tau |^2 \} , $$

we find the variability regions

$$ a_n (B(\tau )) = b_n (B(\tau )) = \{ w:|w| \leqslant \phi (1/\phi ^{ - 1} (|\tau |))\} , $$

where

$$ \phi (x) \equiv \frac{1} {\pi }\int_0^\pi {\frac{{x + \cos t}} {{\sqrt {x^2 + 2x\cos t + 1} }}} dt. $$

. 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).