The main objective of this paper is a study of some new refinements and converses of multidimensional Hilbert-type inequalities with nonconjugate exponents. Such extensions are deduced with the help of some remarkable improvements of the well-known Hölder inequality. First, we obtain refinements and converses of the general multidimensional Hilbert-type inequality in both quotient and difference form. We then apply the results to homogeneous kernels with negative degree of homogeneity. Finally, we consider some particular settings with homogeneous kernels and weighted functions, and compare our results with those in the literature.

General three-point quadrature formulas for the approximate evaluation of an integral of a function f over [0,1], through the values f(x), f(1/2), f(1−x), f′(0) and f′(1), are derived via the extended Euler formula. Such quadratures are sometimes called “corrected” or “quadratures with end corrections” and have a higher accuracy than the adjoint classical formulas, which only include the values f(x), f(1/2)and f(1−x) . The Gauss three-point, corrected Simpson, corrected dual Simpson, corrected Maclaurin and corrected Gauss two-point formulas are recaptured as special cases. Finally, sharp estimates of error are given for this type of quadrature formula.

Some inequalities of Jensen type for Q-class functions are proved. More precisely, a refinement of the inequality f((1/P)∑ ni=1pixi)≤P∑ ni=1(f(xi)/pi) is given in which p1,…,pn are positive numbers, P=∑ ni=1pi and f is a Q-class function. The notion of the jointly Q-class function is introduced and some Jensen type inequalities for these functions are proved. Some Ostrowski and Hermite–Hadamard type inequalities related to Q-class functions are presented as well.

Bounds for Hardy differences, that is, improvements and reverses of the well-known Hardy inequality, are obtained.

We find new properties for the space R(X), introduced by Soria in the study of the best constant for the Hardy operator minus the identity. In particular, we characterize when R(X) coincides with the minimal Lorentz space Λ(X). The condition that R(X) ≠ {0} is also described in terms of the embedding (L1, ∞ ∩ L∞) ⊂ X. Finally, we also show the existence of a minimal rearrangement-invariant Banach function space (RIBFS) X among those for which R(X) ≠ {0} (which is the RIBFS envelope of the quasi-Banach space L1, ∞ ∩ L∞).

The superadditivity and subadditivity of some functionals associated with the Riemann–Stieltjes integral are established. Applications in connection to Ostrowski’s and the generalized trapezoidal inequalities and for special means are provided.

Some inequalities in terms of the Gâteaux derivatives related to Jensen’s inequality for convex functions defined on linear spaces are given. Applications for norms, mean f-deviations and f-divergence measures are provided as well.

A new sharp general Ostrowski type inequality in L∞ norm is established. Some special cases are discussed.

We show that the conjecture of Kannan, Lovász, and Simonovits on isoperimetric properties of convex bodies and log-concave measures is true for log-concave measures of the form ρ(∣x∣B) dx on ℝn and ρ(t,∣x∣B) dx on ℝ1+n, where ∣x∣B is the norm associated to any convex body B already satisfying the conjecture. In particular, the conjecture holds for convex bodies of revolution.

The quasilinearity of certain composite functionals defined on convex cones in linear spaces is investigated. Applications in refining the Jensen, Hölder, Minkowski and Schwarz inequalities are given.

A concept of synchronicity associated with convex functions in linear spaces and a Chebyshev type inequality are given. Applications for norms, semi-inner products and convex functions of several real variables are also given.

We consider families of general two-point quadrature formulae, using the extension of Montgomery’s identity via Taylor’s formula. The formulae obtained are used to present a number of inequalities for functions whose derivatives are from Lp spaces and Bullen-type inequalities.

Some new results related to Jensen’s celebrated inequality for convex functions defined on convex sets in linear spaces are given. Applications for norm inequalities in normed linear spaces and f-divergences in information theory are provided as well.

The inverse stochastic dominance of degree r is a stochastic order of interest in several branches of economics. We discuss it in depth, the central point being the characterization in terms of the weak r-majorization of the vectors of expected order statistics. The weak r-majorization (a notion introduced in the paper) is a natural extension of the classical (reverse) weak majorization of Hardy, Littlewood and Pòlya. This work also shows the equivalence between the continuous majorization (of higher order) and the discrete r-majorization. In particular, our results make it clear that the cases r = 1, 2 differ substantially from those with r ≥ 3, a fact observed earlier by Muliere and Scarsini (1989), among other authors. Motivated by this fact, we introduce new stochastic orderings, as well as new social inequality indices to compare the distribution of the wealth in two populations, which could be considered as natural extensions of the first two dominance rules and the S-Gini indices, respectively.

We give a characterization of pairs of weights for the validity of weighted inequalities involving certain generalized geometric mean operators generated by some Volterra integral operators, which include the Hardy averaging operator and the Riemann–Liouville integral operators. The estimations of the constants are also discussed. Our results generalize the work done by J. A. Cochran, C.-S. Lee, H. P. Heinig, B. Opic, P. Gurka, and L. Pick.

In two recent papers a global upper bound is derived for Jensen’s inequality for weighted finite sums. In this paper we generalize this result on positive normalized functionals.

Sharp bounds for the deviation of a real-valued function f defined on a compact interval [a,b] to the chord generated by its end points (a,f(a)) and (b,f(b)) under various assumptions for f and f′, including absolute continuity, convexity, bounded variation, and monotonicity, are given. Some applications for weighted means and f-divergence measures in information theory are also provided.

Some new Gronwall–Ou-Iang type integral inequalities in two independent variables are established. We also present some of its application to the study of certain classes of integral and differential equations.

A new sharp L2 inequality of Ostrowski type is established, which provides some other interesting results as special cases. Applications in numerical integration are also given.

We give here some extensions of inequalities of Popoviciu and Rado. The idea is to use an inequality [C. P. Niculescu and L. E. Persson, Convex functions. Basic theory and applications (Universitaria Press, Craiova, 2003), Page 4] which gives an approximation of the arithmetic mean of n values of a given convex function in terms of the value at the arithmetic mean of the arguments. We also give more general forms of this inequality by replacing the arithmetic mean with others. Finally we use these inequalities to establish similar inequalities of Popoviciu and Rado type.