Matrix inequalities from a two variables functional

We introduce a two variables norm functional and establish its joint log-convexity. This entails and improves many remarkable matrix inequalities, most of them related to the log-majorization theorem of Araki. In particular: if A is a positive semidefinite matrix and N is a normal matrix, p >= 1 and Phi is a subunital positive linear map, then vertical bar A Phi (N) A vertical bar(p) is weakly log-majorized by A(p)Phi(vertical bar N vertical bar(p)) A(p). This far extension of Araki's theorem (when F is the identity and N is positive) complements some recent results of Hiai and contains several special interesting cases, such as a triangle inequality for normal operators and some extensions of the Golden-Thompson trace inequality. Some applications to Schur products are also obtained.