Functional Inequalities Involving Numerical Differentiation Formulas of Order Two

We write expressions connected with numerical differentiation formulas of order 2 in the form of Stieltjes integral, then we use Ohlin lemma and Levin–Stechkin theorem to study inequalities connected with these expressions. In particular, we present a new proof of the inequality f(x+y2)≤1(y-x)2∫xy∫xyf(s+t2)dsdt≤1y-x∫xyf(t)dtsatisfied by every convex function f:R→R and we obtain extensions of this inequality. Then we deal with non-symmetric inequalities of a similar form

Convexity properties of functions defined on metric Abelian groups

The notions of quasiconvexity, Wright convexity and convexity for functions defined on a metric Abelian group are introduced. Various characterizations of such functions, the structural properties of the functions classes so obtained are established and several well-known results are extended to this new setting