Approximate convexity with respect to a subfield
This paper is concerned with approximate convexity in the following sense. \smallskip 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 \(\alpha:D^*\longrightarrow\mathbb{R}\) be a nonnegative even function. The function \(f:D\longrightarrow\mathbb{R}\) is called \((\alpha,\mathbb{F})\)-convex, if it satisfies the inequality \[ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+t\alpha((1-t)(x-y))+(1-t)\alpha(t(y-x)) \] for all \(x,y\in D\) and for all \(t\in \mathbb{F}\cap[0,1]\). Combining the approach of approximate convexity from \textit{J. Makó} and \textit{Z. Páles} [Publ. Math. 80, No. 1--2, 107--126 (2012; Zbl 1313.26016)] with the concept of restricted strong convexity by \textit{J. Makó et al. [Math. Inequal. Appl. 15, No. 2, 289--299 (2012; Zbl 1246.26011)], the authors obtain characterizations of \((\alpha,\mathbb{F})\)-convex functions by comparison of modified difference ratios and support properties. Suitable differentiability properties are investigated under some additional assumptions on the function \(\alpha\).}
scientific article; zbMATH DE number 6806139
scientific article