On a Characterization of the Logarithmic Mean

Abstract

In the present note we are interested in proving the counterpart of the (right-hand side of the) celebrated Hermite–Hadamard inequality for \(\varphi \)-convex functions. In particular, we prove that the only \(\varphi \)-convex function for which the Hermite–Hadamard inequality holds with the Lagrangian mean on the right-hand side is (up to an affine transformation) the \(\log \)-convex function.