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 φ φ -convex functions. In particular, we prove that the only φ φ -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.