From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture

In this paper we study the joint convexity/concavity of the trace functions Psi(p,q,s)(A, B) = Tr(B-q/2 K* A(p)KB(1/2))(s), p, q, s is an element of R, where A and B are positive definite matrices and K is any fixed invertible matrix. We will give full range of (p, q , s)is an element of R-3 for Psi(p,q,s) to be jointly convex/concave for all K. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of (alpha, z) for alpha-z Renyi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Psi(p)(,0)(,1/p) for 0 < p < 1 which was first proved by Epstein using complex analysis. The key is to reduce the problem to the joint convexity/concavity of the trace functions Psi(p,1-p,1)(A, B) = Tr K* A(p)KB(1-P), - 1 <= p <= 1, using a variational method. (C) 2020 Elsevier Inc. All rights reserved.