Auxiliary Monge-Amp\`ere Equations in Orbifold Setting -- a Mean-Value Inequality

In this note, we generalize a mean-value inequality of Guo-Phong-Sturm to the setting of a compact K\"ahler orbifold. This shows that their reasoning is insensitive to quotient singularities. As we aim for a self-contained exposition, we generalize some fundamental results, namely the $\alpha$-invariant estimate by H\"ormander and Tian, an approximation result for the psh-envelope of a $(1,1)$-form in a K\"ahler class by Berman and an $L^\infty$ estimate for this envelope by Guo-Phong-Tong-Wang.