In this work, an operator superquadratic function (in operator sense) for positive Hilbert space operators is defined. Several examples with some important properties together with some observations which are related to the operator convexity are pointed out. Equivalent statements of a non-commutative version of Jensen's inequality for operator superquadratic function are established. A generalization of the main result to any positive unital linear map is also provided.

In this work, we introduce the class of h-MN-convex functions by generalizing the concept of MN-convexity and combining it with h-convexity. Namely, let M : [0, 1] → [a, b] be a Mean function given by M (t) = M (t; a, b); where by M (t; a, b) we mean one of the following functions: A_t (a, b) := (1 − t) a + tb, G_t (a, b) = a^1−tb^t and H_t (a, b) := $\frac{ab}{ta+\left(1-t\right)b}=\frac{1}{A_t\left(\frac{1}{a},\frac{1}{b}\right)}$ with the property that M (0; a, b) = a and M (1; a, b) = b. Let I, J be two intervals subset of (0, ∞) such that (0, 1) ⊆ J and [a, b] ⊆ I. Consider a non-negative function h : J → (0, ∞), a function f : I → (0, ∞) is said to be h-MN-convex (concave) if the inequality f (M (t; x, y)) ≤ (≥) N (h(t); f (x), f (y)), holds for all x, y ∈ I and t ∈ [0, 1]. In this way, nine classes of h-MN-convex functions are established, and therefore some analytic properties for each class of functions are explored and investigated. Characterizations of each type are given. Various Jensen’s type inequalities and their converses are proved.

In this work, several inequalities of Popoviciu type for h-MN-convex functions are proved, where M and N are denote to Arithmetic, Geometric and Harmonic means and h is a non-negative superadditive or subadditive function.

We show that if $f$ is a non-negative superquadratic function, then $A\mapsto\mathrm{Tr}f(A)$ is a superquadratic function on the matrix algebra. In particular, \begin{align*} \tr f\left( {\frac{{A + B}}{2}} \right) +\tr f\left(\left| {\frac{{A - B}}{2}}\right|\right) \leq \frac{{\tr {f\left( A \right)} + \tr {f\left( B \right)} }}{2} \end{align*} holds for all positive matrices $A,B$. In addition, we present a Klein's inequality for superquadratic functions as $$ \mathrm{Tr}[f(A)-f(B)-(A-B)f'(B)]\geq \mathrm{Tr}[f(|A-B|)] $$ for all positive matrices $A,B$. It gives in particular improvement of Klein's inequality for non-negative convex function. As a consequence, some variants of the Jensen trace inequality for superquadratic functions have been presented.

In this work, in spite of Milne’s recommendation using the three-point Newton–Cotes open formula (Milne’s rule) as a predictor rule and three-point Newton–Cotes closed formula (Simpson’s rule) as a corrector rule for 4-th differentiable functions with bounded derivatives. There is still a great need to introduce such formulas in other Lp spaces. Often, we need to approximate real integrals under the assumptions of the function involved. Because of that, the aim of this work is to introduce several Lp error estimates for the proposed perturbed Milne’s quadrature rule. Numerical experiments showing that our proposed quadrature rule is better than the classical Milne rule for certain types of functions are provided as well.

In this work, new refinements of some numerical radius inequalities are proved. Namely, new improvements and refinements purify the recent inequalities of some famous inequalities concerning the numerical radius of Hilbert space operators. The proven inequalities in this work are not only an improvement over old inequalities, but rather they are stronger than them. Several examples that support the validity of our results are established as well.