On strong $(\alpha,\F)$-convexity

Abstract

In this paper, strongly $(\alpha,T)$-convex functions, i.e., functions $f:D\to \R$ satisfying the functional inequality $$ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-t\alpha\big((1-t)(x-y)\big)-(1-t)\alpha\big(t(y-x)\big)$$ 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 $T\subseteq\R$ is a nonempty set. The main results provide various characterizations of strong $(\alpha,T)$-convexity in the case when $T$ is a subfield of $\R$.