Approximate convexity with respect to a subfield

Abstract

Let ${\mathbb{F}}$ be a subfield of ${\mathbb{R}}$ and X be a linear space over ${\mathbb{F}}$. Let ${ D\subseteq X }$ be a nonempty ${\mathbb{F}}$-convex set, ${ D^*:=D-D:=\{x-y : x,y\in D\} }$, and ${\alpha \colon {D^* \rightarrow \mathbb{R}}}$ be a nonnegative even function. The function ${ f \colon {D\rightarrow \mathbb{R}}}$ is called ${(\alpha,\mathbb{F})}$-convex, if it satisfies the inequality

$$\begin{aligned}{f(tx+(1-t)y)\leq tf(x)+(1-t)f(y) + t\alpha((1-t)(x-y))+(1-t)\alpha(t(y-x))}\end{aligned}$$

for all ${x,y\in D}$ and for all ${t\in \mathbb{F}\cap [0,1]}$. In this paper we characterize ${(\alpha,\mathbb{F})}$-convex functions by comparison of modified difference ratios and support properties. If ${\alpha}$ satisfies some additional conditions, we obtain the differentiability of ${(\alpha,\mathbb{F})}$-convex functions in the appropriate sense.