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The main goal of this paper is to present some new properties of convex functions, including Mercer-type, superadditive-type, and Jensen-type inequalities. Applications of these new properties will cover the celebrated Hermite–Hadamard inequality, Clarkson inequality, and the operator Mercer inequality. Comparisons with some existing results will be discussed too. [ABSTRACT FROM AUTHOR]

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 fx+y2≤1(y-x)2∫xy∫xyfs+t2dsdt≤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. [ABSTRACT FROM AUTHOR]

The main goal of this article is to present a new approach, made up of integrals, to refining convex functions inequalities, with applications to several known inequalities like Jensen, Hermite–Hadamard and means inequalities. The first main result will be a Hilbert space operator version, which will be utilized to obtain the aforementioned refinements. In the end, we present graphical interpretations of the new results we have. [ABSTRACT FROM AUTHOR]