Convexity of Some Spectral Functions on Hermitian Matrices.

We prove in this note the convexity of the functions $u\circ \lambda $ and more generally $u\circ \lambda_B $ on the space of Hermitian matrices, for $B$ a fixed positive definite hermitian matrix, when $u:\mathbb{R}^m\rightarrow \mathbb{R}\cup {+\infty }$ is a symmetric convex function which is lower semi-continuous on $\mathbb{R}^m$, and finite in at least one point of $\mathbb{R}^m$. This is performed by using some optimisation techniques and a generalized Ky Fan inequality.