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 $n\geq 2$, then, for all $k\in\{1,\dots, n\}$, 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 H\"older--Minkowski type inequalities

Asymptotic stability of the Cauchy and Jensen functional equations

The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations

A composite functional equation from algebraic aspect

In this paper we discuss the composite functional equation f(x+2f(y))=f(x)+y+f(y) on an Abelian group. This equation originates from Problem 10854 of the American Mathematical Monthly. We give an algebraic description of the solutions on uniquely 3-divisible Abelian groups, and then we construct all solutions f of this equation on finite Abelian groups without elements of order 3 and on divisible Abelian groups without elements of order 3 including the additive group of real numbers