Operator inequalities involving real convex functions

We establish a general convexity operator inequality involving real convex functions. Some special cases, examples and a variety of its consequences are also given. In particular, we prove that the inequality f(A)+f(B)<= f(A+B) holds for a real convex function f and positive operators A, B with A, B<= MI<= A+B for some scalar M>0. Also, we show that if f is a real convex function and A, B, C, D are self-adjoint operators with A<= mI<= C, D<= MI<= B for some scalars m<= M, then f(C)+f(D)<= f(A)+f(B).