SCHWEITZER 1977 Problem 8

Let $p \geq 1$ be a real number and $\mathbb{R}_+=(0, \infty)$ . For which continuous functions $g : \mathbb{R}_+ \rightarrow \mathbb{R}_+$ are following functions all convex? $$M_n(x)=\left[ \frac{\sum_{i=1}^n g(\frac{x_i}{x_{i+1}}) x_{i+1}^p}{\sum_{i=1}^n g(\frac{x_i}{x_{i+1}})} \right ]^\frac 1p ,$$ $$x=(x_1,\ldots, x_{n+1}) \in \mathbb{R}_+ ^ {n+1} , n=1,2,\ldots$$

L. Losonczi