Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings

Abstract: We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasi-re-gu-lar, shortly HQR) mappings with the strip codomain. Along the way using the principle of subordination and the corresponding conformal mapping, depicted on the Figure 1, we get a simple proof of a new version of the Schwarz lemma for real valued harmonic functions (see Theorems 4 and 5) and Theorem 6 related to holomorphic mappings. Using the Schwarz-Pick lemma related to distortion for harmonic mappings and the elementary properties of the hyperbolic geometry of the strip we prove Lemma 4, which is a key ingredient in the proof of Theorem 7 which yields optimal estimates for modulus of HQR mappings.