Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 457-464

HEINZ-SCHWARZ INEQUALITIES FOR HARMONIC MAPPINGS IN THE UNIT BALL

David Kalaj

University of Montenegro, Faculty of Natural Sciences and Mathematics
Cetinjski put b.b. 81000 Podgorica, Montenegro; davidkalaj 'at' gmail.com

Abstract. We first prove the following generalization of Schwarz lemma for harmonic mappings. If u is a harmonic mapping of the unit ball onto itself then ||u(x) - (1 - ||x||²)/(1 + ||x||²)^n/2u(0)|| ≤ U(|x|N). By using this result we obtain certain sharp estimate of the gradient of a harmonic mapping. Those two results extend some known result from harmonic mapping theory [1]. By using the Schwarz lemma for harmonic mappings we derive Heinz inequality on the boundary of the unit ball by providing a sharp constant C_n in the inequality: ||∂_ru(rη)||_r=1 ≥ C_n, ||η|| = 1, for every harmonic mapping of the unit ball into itself satisfying the condition u(0) = 0, ||u(η)|| = 1.

2010 Mathematics Subject Classification: Primary 31A05; Secondary 42B30.

Key words: Harmonic mappings, Heinz inequality, Schwarz inequality.

Reference to this article: D. Kalaj: Heinz-Schwarz inequalities for harmonic mappings in the unit ball. Ann. Acad. Sci. Fenn. Math. 41 (2016), 457-464.

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doi:10.5186/aasfm.2016.4126