NDLI: Bounded harmonic mappings

Content Provider	Springer Nature Link
Author	Szapiel, Wojciech
Copyright Year	2010
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).
Starting Page	47
Ending Page	76
Page Count	30
File Format	PDF
ISSN	00217670
Journal	Journal d'Analyse Mathématique
Volume Number	111
Issue Number	1
e-ISSN	15658538
Language	English
Publisher	The Hebrew University Magnes Press
Publisher Date	2010-12-19
Publisher Place	Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Partial Differential Equations Abstract Harmonic Analysis Dynamical Systems and Ergodic Theory Functional Analysis Analysis
Content Type	Text
Resource Type	Article
Subject	Analysis

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