Abstract.A recently discovered Hardy-Pólya type inequality described by a convex function is considered and further developed both in weighted and unweighted cases. Also some corresponding multidimensional and reversed inequalities are pointed out. In particular, some new multidimensional Hardy and Pólya-Knopp type inequalities and some new integral inequalities with general integral operators (without additional restrictions on the kernel) are derived.Mathematics subject classification (2000): 26D10, 26D15.

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Abstract. Some new Carleman-Knopp type inequalities are proved as "end point" inequalities of modern forms of Hardy's inequalities. Both finite and infinite intervals are considered and both the cases p q and q < p are investigated. The obtained results are compared with similar results in the literature and the sharpness of the constants is discussed for the power weight case. Moreover, some reversed Carleman-Knopp inequalities are derived and applied.Mathematics subject classification (2000): 26D15, 26D07.

A discrete Hardy-type inequality (For kernels of product type some scales of weight characterizations of the inequality are proved with the corresponding estimates of the best constant C. A sufficient condition for the inequality to hold in the general case is proved and this condition is necessary in special cases. Moreover, some corresponding results for the case when {a n } ∞ n=1 are replaced by the nonincreasing sequences {a * n } ∞ n=1 are proved and discussed in the light of some other recent results of this type.

We consider T f = x1 0 x2 0 f (t 1 ,t 2 )dt 1 dt 2 and a corresponding geometric mean operatorcould be characterized by three independent conditions on the weights. We give a simple proof of the fact that if the weight v is of product type, then in fact only one condition is needed. Moreover, by using this information and by performing a limiting procedure we can derive a weight characterization of the corresponding two-dimensional Pólya-Knopp inequality with the geometric mean operator G involved.