Jensen convex functions and doubly stochastic matrices

TL;DR

The paper investigates when Jensen convexity of a function $f$ on a nonempty interval $I$ can be characterized by a convexity-type inequality involving a doubly stochastic matrix $P$. The authors prove a main theorem: if there exists a linear-algebraic condition on $P$ and the DS-weighted inequality $f\left(\frac{x_1+\cdots+x_n}{n}\right)\le\frac{1}{n}\sum_{i=1}^n f(p_{i1}x_1+\cdots+p_{in}x_n)$ holds for all $x_i\in I$, then $f$ is Jensen convex on $I$; conversely, if $f$ is Jensen convex, the inequality holds for all orders $n$ and all DS matrices $P$. The circulant matrix case is developed by constructing left-circulant $P$ from weights $\lambda_j$ and requiring a spectral nonvanishing condition $\sum_{j=1}^n\lambda_j\omega_n^{k(j-1)}\neq0$, yielding analogous equivalences and several explicit corollaries for small $n$. Additionally, the work connects to Schur convexity and majorization concepts, highlighting how matrix-weighted convexity interacts with classical convexity notions. Overall, the results extend Jensen convexity to DS- and circulant-weighted settings, with concrete implications for matrix-inequality frameworks.

Abstract

Given an nxn doubly stochastic matrix P satisfying an appropriate condition of linear algebraic-type, and a function f defined on a nonempty interval, we show that the validity of a convexity-type functional inequality for f in terms P implies that f is Jensen convex. We also prove that if f is convex, then the functional inequality in question holds for all doubly stochastic matrices of any order. The particular case when the doubly stochastic matrix is a circulant one is also considered.