User:Saung Tadashi/Miao's inequality

Miao's inequalities are a set of inequalities relating the image of the singular values of two matrices by a concave function. It was conjectured by W. Miao and proved in 2016.

Statement
Let f be a concave function f:R+→R+ with f(0)=0 and let X and Y be n×n complex matrices. The following inequality is valid: $$ \sum_{k=1}^{n} | f(\sigma_{k}(X)) - f(\sigma_{k}(Y)) | \le \sum_{k=1}^{n} f(\sigma_{k}(X-Y)) $$ where σ1(M) ≥ σ2(M) ≥ ... ≥ σn(M) denotes the singular values of the matrix M arranged in non-increasing order.