On log-subharmonicity of singular values of matrices

Let F be an analytic function from an open subset Ω of the complex plane into the algebra of n×n matrices. Denoting by ${\displaystyle s_1,...,s_n}$ the decreasing sequence of singular values of a matrix, we prove that the functions ${\displaystyle {\log s}_1\left(F(\lambda )\right)+...+{\log s}_k\left(F(\lambda )\right)}$ and ${\displaystyle {\log }^{+}{s}_1\left(F(\lambda )\right)+...+{\log }^{+}{s}_k\left(F(\lambda )\right)}$ are subharmonic on Ω for 1 ≤ k ≤ n.