Strengthening of strong and approximate convexity (Q653854)

The motivation of the investigations can be traced back to a basic result of Hyers and Ulam. This result states that, if a function \(f\) is \(\varepsilon\)-convex (that is, it satisfies the inequality of convexity with an additional positive \(\varepsilon\) on the right-hand side), then \(f\) is the sum of a convex and a bounded function. The notion of \(\varepsilon\)-convexity was generalized and studied by several authors. The authors of the present paper introduce the notion of \((E,t)\)-convexity which contains as special cases most of the earlier approximate convexity notions. Let \(X\) be a real linear space and \(D\subseteq X\) be a nonempty convex set. Given an error function \(E \colon [0,1]\times(D-D)\longrightarrow \mathbb R\cup\{+\infty\}\) and an element \(t\in]0,1[\), a function \(f \colon D\longrightarrow\mathbb R\) is called \((E,t)\)-convex if, for all \(x,y\in D\), the following inequality holds: \[ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+E(t,x-y). \] The main result of the paper states that, for all \(a,b\in(\mathbb N\cup\{0\})+\{0,t,1-t\}\) such that \(\{a,b,a+b\} \cap\mathbb N\neq\emptyset\), every \((E,t)\)-convex function is also \(\bigl(F,\frac{a}{a+b}\bigr)\)-convex, where \[ F(s,u):= \frac{(a+b)2s(1-s)}{t(1-t)} E\Bigl(t,\frac{u}{a+b}\Bigr) \qquad\bigl(u\in(D-D),s\in]0,1[\bigl). \] As almost immediate consequences, Kuhn-type results are presented, namely, if \(E\) satisfies a certain limit property, then \((E,t)\)-convexity implies rational convexity. The particular case when \(E\) is the power of the norm reduces to known results, e.g., \((\gamma,p,t)\)-convex functions.