A Generalization of Lieb concavity theorem

Lieb concavity theorem, successfully solved the Wigner-Yanase-Dyson conjecture, which is a very important theorem, and there are many proofs of it. Generalization of the Lieb concavity theorem has been obtained by Huang, which implies that it is jointly concave for any ( [ s ]) nonnegative matrix monotone function f(x) over Tr perpendicular to k(Aq2 s K*BspKA 2 sq )1 ( [ s ]) we obtained Tr perpendicular to k(f(Aq2 s)K* f(Bsp)Kf(A2 sq ))1 monotone function f(x) by using Epstein's theorem, and some more general results were obtained. 1 k 1 k . In this manuscript, was jointly concave for any nonnegative matrix