The extremal problem for weighted combined energy and $\rho-$Nitsche type inequality

TL;DR

This paper investigates the extremal problem for weighted combined energy between two circular annuli under a radial metric, generalizing previous results and establishing a $ ho$-Nitsche type inequality with unique radial extremal mappings.

Contribution

It extends existing extremal energy and distortion results to a broader class involving a radial metric and weighted energy, introducing new inequalities and characterizations.

Findings

Existence and uniqueness of the extremal mapping as a radial map.

Generalization of $ ho$-harmonic mappings satisfying a specific equation.

Derivation of a $ ho$-Nitsche type inequality.

Abstract

Let $A_1$ and $A_2$ be two circular annuli and let $\rho$ be a radial metric defined in the annuli $A_2$. We study the existence and uniqueness of the extremal problem for weighted combined energy between $A_1$ and $A_2$, and obtain that the extremal mapping is a certain radial mapping. In fact, this extremal mapping generalizes the $\rho-$harmonic mapping and satisfies equation (2.7) obtained by mean of variation for weighted combined energy. Meanwhile, we get a $\rho-$Nitsche type inequality. This extends the results of Kalaj (J. Differential Equations, 268(2020)) and YTF (Arch. Math., 122(2024)), where they considered the case $\rho=1$ and $\rho=\frac{1}{|h|^{2}}$, respectively. Moreover, in the course of proving the extremal problem for weighted combined energy we also investigate the extremal problem for the weighted combined distortion (see Theorem 4.1). This extends the result obtained by Kalaj (J. London Math. Soc., 93(2016)).