On an equality of means and a characterization of the logarithmic mean

Abstract Starting from the famous Hermite-Hadamard inequality for $$\varphi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>φ</mml:mi> </mml:math> -convex functions and a characterization of the logarithmic mean, we present several regularity results and observations concerning the equality of means. Finally, after weakening the conditions, we examine whether the characterization remains valid. We generalize some previous studies and provide a complete result regarding the equality of means.