Reducible means and reducible inequalities

It is well-known that if a real valued function acting on a convex set satisfies the n-variable Jensen inequality, for some natural number (nge 2), then, for all (kin {1,dots , n}), it fulfills the k-variable Jensen inequality as well. In other words, the arithmetic mean and the Jensen inequality (as a convexity property) are both reducible. Motivated by this phenomenon, we investigate this property concerning more general means and convexity notions. We introduce a wide class of means which generalize the well-known means for arbitrary linear spaces and enjoy a so-called reducibility property. Finally, we give a sufficient condition for the reducibility of the (M, N)-convexity property of functions and also for Hölder–Minkowski type inequalities.