The extremal problem for weighted combined energy.

We study the extremal problem for weighted combined energy between two concentric annuli and obtain that the extremal mapping is a certain radial mapping. This extends the result obtained by Kalaj (J. Differential Equations, 268(2020)) to a non-Euclidean version. Meanwhile, we get a 1 | w | 2 -Nitsche type inequality, which generalizes the result in Arch. Math., 107(2016). Furthermore, based on the relationship between weighted combined energy and weighted combined distortion, we also consider the extremal problem for weighted combined distortion on annuli. [ABSTRACT FROM AUTHOR]