Hyperbolic Derivatives for HQC Mappings and Applications

Abstract

We analyse the properties of hyperbolic metrics on various domains to obtain the Schwarz-Pick type inequalities for HQC mappings. We also present several variants of the classical Koebe theorem, along with precise hyperbolic distortion estimates, for a notable class of real-valued harmonic functions defined on the unit disc whose range is contained in the interval (−1, 1). Furthermore, we prove a significant result for the class of real harmonic functions defined on a Riemann surface equipped with a complete conformal metric whose Gaussian curvature is bounded below by a negative constant. This result provides a natural generalization of classical hyperbolic estimates and reveals deeper geometric structure in the behavior of such mappings.