Multidimensional Hardy-type inequalities with general kernels

Abstract

Some new multidimensional Hardy-type inequalities involving arithmetic mean operators with general positive kernels are derived. Our approach is mainly to use a convexity argument and the results obtained improve some known results in the literature and, in particular, some recent results in [S. Kaijser, L. Nikolova, L.-E. Persson, A. Wedestig, Hardytype inequalities via convexity, Math. Inequal. Appl. 8 (3) (2005) 403-417] are generalized and complemented.

Refinement multidimensional dynamic inequalities with general kernels and measures

“…This gives us the desired inequality (19). Similarly to the proof of (19), by the same calculations with Ψ subquadratic, we see that only the inequality sign will be reversed.…”

“…In the following, we will establish some particular cases of inequalities (19) and (26) by applying Theorem 2.1 with…”

Some new multidimensional hardy-type inequalities with kernels via convexity

“…The multidimensional version of these results were recently obtained in [22] (see e.g., [11,20] and the references given there). In particular, Oguntuase et al [22] obtained the following new multidimensional Hardy-type inequality…”

“…Furthermore, Oguntuase et al [21] obtained a new class of general multidimensional strengthened Hardy type inequalities with power weights, related to all possible choices of parameter p ∈ R {0} and to arbitrary almost everywhere positive convex (or concave) function φ, such that Ax p φ(x) Bx p holds on (0, ∞) with some positive real constants A B. The multidimensional version of these results were recently obtained in [22] (see e.g., [11,20] and the references given there). In particular, Oguntuase et al [22] obtained the following new multidimensional Hardy-type inequality…”

On the equivalence between some multidimensional Hardy-type inequalities

“…After that a great number of results based on this convexity argument have been presented (see e.g. [10,14,17,18,19,20,21,22,23,24] and the PhD thesis [13]) and the references cited therein.…”