In this paper we establish operator quasilinearity properties of some functionals associated with Davis–Choi–Jensen’s inequality for positive maps and operator convex or concave functions. Applications for the power function and the logarithm are provided.

Some inequalities of Jensen type for Arg-square-convex functions of unitary operators in Hilbert spaces are given.

Two new reverses of the celebrated Jensen’s inequality for convex functions in the general setting of the Lebesgue integral, with applications to means, Hölder’s inequality and $f$-divergence measures in information theory, are given.

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.

Some bounds in terms of Gâteaux lateral derivatives for the weighted f-Gini mean difference generated by convex and symmetric functions in linear spaces are established. Applications for norms and semi-inner products are also 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.

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.

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.

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.

Some inequalities for superadditive functionals defined on convex cones in linear spaces are given, with applications for various mappings associated with the Jensen, Hölder, Minkowski and Schwarz inequalities.

Let an almost everywhere positive function Φ be convex for p>1 and p<0, concave for p∈(0,1), and such that Axp≤Φ(x)≤Bxp holds on for some positive constants A≤B. In this paper we derive a class of general integral multidimensional Hardy-type inequalities with power weights, whose left-hand sides involve instead of , while the corresponding right-hand sides remain as in the classical Hardy’s inequality and have explicit constants in front of integrals. We also prove the related dual inequalities. The relations obtained are new even for the one-dimensional case and they unify and extend several inequalities of Hardy type known in the literature.

It is proved that a distribution F is strongly unimodal iff any two quantiles of the convolution of F with any other distribution are further apart than the corresponding quantiles of F itself. Some related characterizations of strong unimodality are also given.