On strong $(α,\F)$-convexity

Abstract

In this paper, strongly $(\alpha ,T)$-convex functions, i.e., functions $f:D\to \mathbb{R}$ satisfying the functional inequality $$f(tx+(1-t)y)\le tf(x)+(1-t)f(y)-t\alpha ((1-t)(x-y))-(1-t)\alpha (t(y-x))$$ for $x,y\in D$ and $t\in T\cap [0,1]$ are investigated. Here $D$ is a convex set in a linear space, $\alpha$ is a nonnegative function on $D-D$, and $T\subseteq \mathbb{R}$ is a nonempty set. The main results provide various characterizations of strong $(\alpha ,T)$-convexity in the case when $T$ is a subfield of $\mathbb{R}$.