A Brunn-Minkowski-type inequality involving \(\gamma\)mean variance and its applications

By means of the algebra, functional analysis, and inequality theories, we establish a BrunnMinkowski type inequality involving \(\gamma\)mean variance: \[\overline{var}^{[\gamma]} (f + g) \leq \overline{var}^{[\gamma]} f + \overline{var}^{[\gamma]} g; \quad \gamma \in [1; 2],\] where \(\overline{var}^{[\gamma]} \varphi\) is the \(\gamma\)mean variance of the function \(\varphi: \Omega\rightarrow (0,\infty)\) We also demonstrate the applications of this inequality to the performance appraisal of education and business.