A submajorization of Carlen and Lieb convexity

We show the following function on m-tuples of positive operators in finite von Neumann algebra M:Φp(A1,A2,…,Am)=‖(∑k=1mAkp)1p‖LE(M) is jointly convex for 1≤p≤2 and for any symmetric norm ‖.‖LE(M) associated with M.