On a Characterization of the Logarithmic Mean

In the present note we are interested in proving the counterpart of the (right-hand side of the) celebrated Hermite–Hadamard inequality for \(\varphi \) $\phi$ -convex functions. In particular, we prove that the only \(\varphi \) $\phi$ -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 \(\log \) $log$ -convex function.