Estimates for approximately Jensen convex functions

TL;DR

This paper investigates functions that nearly satisfy Jensen's inequality with an error term, providing bounds and explicit formulas for such functions involving an infinite series, extending the understanding of approximate Jensen convexity.

Contribution

It introduces a new class of approximately Jensen convex functions with explicit bounds and finite sum evaluations for rational convex combinations.

Findings

Derived bounds for $ ext{ extit{ extbf{f}}}$ involving an infinite series.

Proved convergence of the series and finite sum evaluation for rational $ ext{ extit{ extbf{ extlambda}}}$.

Extended Jensen convexity analysis to functions with error functions $ ext{ extit{ extphi}}$.

Abstract

In this paper functions $f:D\to\mathbb{R}$ satisfying the inequality \[ f\Big(\frac{x+y}{2}\Big)\leq\frac12f(x)+\frac12f(y) +\varphi\Big(\frac{x-y}{2}\Big) \qquad(x,y\in D) \] are studied, where $D$ is a nonempty convex subset of a real linear space $X$ and $\varphi:\{\frac12(x-y) : x,y \in D\}\to\mathbb{R}$ is a so-called error function. In this situation $f$ is said to be $\varphi$-Jensen convex. The main results show that for all $\varphi$-Jensen convex function $f:D\to\mathbb{R}$, for all rational $\lambda\in[0,1]$ and $x,y\in D$, the following inequality holds \[ f(\lambda x+(1-\lambda)y) \leq \lambda f(x)+(1-\lambda)f(y)+\sum_{k=0}^\infty \frac{1}{2^k}\varphi\big(\mbox{dist}(2^k\lambda,\mathbb{Z})\cdot(x-y)\big). \] The infinite series on the right hand side is always convergent, moreover, for all rational $\lambda\in[0,1]$, it can be evaluated as a finite sum.