On homogeneous Páles means

Abstract. Let \(f,g:I\to \mathbb{R}\) be continuous functions such that \(g\) is positive, \(f/g\) is strictly monotonic on the open interval \(I\) and let \(\mu\) be a probability measure on the Borel sets of \([0,1]\). The two-variable mean \(M_{f,g;\mu}:I^2\to I\) defined by $$ M_{f,g;\mu}(x,y):=\left(\frac{f}{g}\right)^{-1}\left(\frac{\int\limits_{[0,1]} f(tx+(1-t)y)d\mu(t)}{\int\limits_{[0,1]} g(tx+(1-t)y)d\mu(t)}\right). $$ was proposed by Zsolt Páles in 2005. It is a common generalization of quasi-arithmetic, Bajraktarević, Lagrangian and Cauchy means. Extending earlier results we solve the homogeneity equation $$ M_{f,g;\mu}(tx,ty)=tM_{f,g,\mu}(x,y)\quad(x,y\in I,t\in \mathbb{R} \hbox{ such that }tx, ty\in I) $$ if the centralized moments \(c_k\) of the measure satisfy the condition $$ 5c_2c_4^2+c_4c_6-6c_2^2c_6\ne 0. $$