Euclidean maximum moduli of plane domains and their applications. Issue 11 (2nd November 2019)

• Record Type: Journal Article
• Authors: Avkhadiev, F. G.
• Is Part Of: Complex variables and elliptic equations. Volume 64:Issue 11(2019)
• Page Start: 1869
• Page End: 1880
• Publication Date: 2019-11-02
• Subjects: Conformal modulus -- Poincaré metric -- Schwarzian -- Hardy type inequality; Primary 30F45 -- Secondary 30C20; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2018.1561670
• Languages: English
• ISSNs: 1747-6933

Abstract

ABSTRACT: For a plane domain we study correlations of the Euclidean maximum modulus and three hyperbolic domain characteristics connected with the Poincaré metric of the domain and the distance function. We prove that the Laplacian of the hyperbolic radius of every domain of hyperbolic type is a subharmonic function. Also, for any doubly connected domain we prove asymptotically sharp estimates for the hyperbolic characteristics using the Euclidean maximum modulus of the domain. In addition, we obtain applications of these estimates to Hardy and Schwarz–Pick type inequalities.