On strong $(α,\F)$-convexity

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

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