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

Let A₁ and A₂ be two circular annuli and let be a radial metric defined in the annuli A₂. We study the existence and uniqueness of the extremal problem for weighted combined energy between A₁ and A₂, and obtain that the extremal mapping is a certain radial mapping. In fact, this extremal mapping generalizes the -harmonic mapping and satisfies equation (2. 7) obtained by mean of variation for weighted combined energy. Meanwhile, we get a -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 =1 and =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) ).