In this paper we prove a strengthened general inequality of the Hardy-Knopp type and also derive its dual inequality. Furthermore, we apply the obtained results to unify the strengthened classical Hardy and Po´lya-Knopp's inequalities deriving them as special cases of the obtained general relations. We discuss Po´lya-Knopp's inequality, compare it with Levin-CochranLee's inequalities and point out that these results are mutually equivalent. Finally, we also point out a reversed Po´lya-Knopp type inequality. r

Integral means of arbitrary order, with power weights, and their companion means are introduced and related mixed-means inequalities are derived. These results are then used in proving inequalities of Hardy and Levin-Cochran-Lee type. Also, new proofs of Hardy and Carleman inequality for finite and infinite series are given by using discrete mixed-means.Mathematics subject classification (1991): 26D10, 26D15.

Let an almost everywhere positive function be convex for p > 1 and p < 0, concave for p ∈ (0, 1), and such that Ax p ≤ (x) ≤ Bx p holds on R + 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 (, while the corresponding righthand 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.2000 Mathematics subject classification: primary 26D15; secondary 26A51.

Abstract. By using the notion of the subdifferential of a convex function, we state and prove a new general refined weighted Hardy-type inequality for convex functions and the integral operator with a non-negative kernel. We point out that the obtained result generalizes and refines the classical one-dimensional Hardy's, Pólya-Knopp's, and Hardy-Hilbert's inequalities, as well as related dual inequalities. We show that our results may be seen as generalizations of some recent results related to Riemann-Liouville's and Weyl's operator, as well as a generalization and a refinement of the so-called Godunova's inequality.Mathematics subject classification (2010): 26D10, 26D15.

Abstract. We consider integral power means of arbitrary real order, taken over cells in R n , and their dual means. We establish related mixed-means inequalities and then apply obtained results to derive multivariable analogues and some new generalizations of Hardy and Levin-CochranLee type inequalities. Moreover, we prove the constant factors involved in the right-hand sides of these relations to be the best possible, that is, they cannot be replaced with smaller constants. (2000): Primary 26D10, 26D15. Mathematics subject classification

ON STRENGTHENED WEIGHTED CARLEMAN'S INEQUALITYALEKSANDRA CIZMESIJA, JOSIP PECARIC AND LARS-ERIK PERSSON In this paper we prove a new refinement of the weighted arithmetic-geometric mean inequality and apply this result in obtaining a sharpened version of the weighted Carleman's inequality.

Abstract. This paper deals with some new sharp generalizations of inequalities of Hardy and Levin-Cochran-Lee type for n-dimensional balls.Mathematics subject classification (2000): 26D10, 26D15.

Abstract. We consider Hardy's integral inequality and we obtain some new generalizations of Bicheng-Debnath's recent results. We derive two distinguished classes of inequalities covering all admissible choices of parameter k from Hardy's original relation. Moreover, we prove the constant factors involved in the right-hand sides of some particular inequalities from both classes to be the best possible, that is, none of them can be replaced with a smaller constant.2000 Mathematics Subject Classification. 26D15.