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We consider the existence and uniqueness of a minimizer of the extremalproblem for weighted combined energy between two concentric annuli and obtainthat the extremal mapping is a certain radial mapping. Meanwhile, this in turnimplies a Nitsche type phenomenon and we get a$\frac{1}{|w|^{\lambda}}-$Nitsche type inequality ($\lambda\neq1$). As anapplication, on the basis of the relationship between weighted combined energyand weighted combined distortion, we also investigate the extremal problem forweighted combined distortion on annuli. This extends the result obtained byKalaj in \cite{Ka1}.

The extremal problem for weighted combined energy and the generalization of Nitsche inequality