Singular values of convex functions of operators and the arithmetic–geometric mean inequality

We prove singular value inequalities for convex functions of products and sums of operators that generalize the arithmetic–geometric mean inequality for operators. Among other results, we prove that if AiAi, BiBi, XiXi, YiYi, i=1,…,ni=1,…,n are operators on a complex separable Hilbert space such that |Xi|2+|Yi|22n≤I, i=1,…,ni=1,…,n and if f   is a nonnegative increasing convex function on [0,∞)[0,∞) satisfying f(0)=0f(0)=0, thensj(f(|∑i=1nAiXiYi⁎Bi⁎|))≤12sj(⨁k=1n(Xk⁎(∑i=1nf(|Ai⁎Ak|))Xk+Yk⁎(∑i=1nf(|Bi⁎Bk|))Yk)) for j=1,2,…j=1,2,… .