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:

$\sum _{k=1}^{n}|f(\sigma _{k}(X))-f(\sigma _{k}(Y))|\leq \sum _{k=1}^{n}f(\sigma _{k}(X-Y))$

where σ₁(M) ≥ σ₂(M) ≥ ... ≥ σ_n(M) denotes the singular values of the matrix M arranged in non-increasing order.