Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 391-399

INVARIANT GRADIENT IN REFINEMENTS OF SCHWARZ AND HARNACK INEQUALITIES

Petar Melentijevic

University of Belgrade, Faculty of Mathematics
Studentski trg 16; 11000 Beograd, Serbia; petarmel 'at' matf.bg.ac.rs

Abstract. In this paper we prove a refinement of Schwarz's lemma for holomorphic mappings from the unit ball Bⁿ ⊂ Cⁿ to the unit disk D ⊂ C obtained by Kalaj in [3]. We also give some corollaries of this result and a similar result for pluriharmonic functions. In particular, we give an improvement of Schwarz's lemma for non-vanishing holomorphic functions from Bⁿ to D that was obtained in a recent paper by Dyakonov [2]. Finally, we give a new and short proof of Markovic's theorem on contractivity of harmonic mappings from the upper half-plane H to the positive reals. The same result does not hold for higher dimensions, as is shown by given counterexamples.

2010 Mathematics Subject Classification: Primary 30H05; Secondary 31A05.

Key words: M-invariant gradient, Schwarz's lemma, Harnack's inequality, hyperbolic distance, Bergman distance.

Reference to this article: P. Melentijevic: Invariant gradient in refinements of Schwarz and Harnack inequalities. Ann. Acad. Sci. Fenn. Math. 43 (2018), 391-399.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4324