A note on Hardy-type inequalities

We use a theorem of Cartlidge and the technique of Redheffer's ``recurrent inequalities" to give some results on inequalities related to Hardy's inequality.

     

Bilateral Hardy-type inequalities

This paper studies the Hardy-type inequalities on the intervals (may be infinite) with two weights, either vanishing at two endpoints of the interval or having mean zero. For the first type of inequalities, in terms of new isoperimetric constants, the factor of upper and lower bounds becomes smaller than the known ones. The second type of the inequalities is motivated from probability theory and is new in the analytic context. The proofs are now rather elementary. Similar improvements are made for Nash inequality, Sobolev-type inequality, and the logarithmic Sobolev inequality on the intervals.     

Refinement of Hardy's inequalities via superquadratic and subquadratic functions

A new refined weighted Hardy inequality for p?2 is proved and discussed. The inequality is reversed for 1<p?2, which means that for p=2 we have equality. The main tool in the proofs are some new results for superquadratic and subquadratic functions.     

The optimal constant in Hardy-type inequalities

To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined.     

On a result of Cartlidge

We give a simple proof of Cartlidge's result on the lp operator norms of weighted mean matrices.     

Some Hardy-Type Inequalities

In this paper, some new Hardy-type inequalities involving many functions are obtained. These on the one hand generalize and on the other hand improve some existing results by Isumi and Isumi, Levinson, and Pachpatte on this famous type of inequalities.     

On weighted iterated Hardy-type inequalities

In this paper the inequality

$$\begin{aligned} \bigg ( \int _0^{\infty } \bigg ( \int _x^{\infty } \bigg ( \int _t^{\infty } h \bigg )^q w(t)\,dt \bigg )^{r / q} u(x)\,{ ds} \bigg )^{1/r}\le C \,\int _0^{\infty } h v, \quad h \in {\mathfrak {M}}^+(0,\infty ) \end{aligned}$$

is characterized. Here \(0< q ,\, r < \infty \) and \(u,\,v,\,w\) are weight functions on \((0,\infty )\).

     

Multidimensional Hardy-type inequalities with general kernels

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, Hardy-type inequalities via convexity, Math. Inequal. Appl. 8 (3) (2005) 403-417] are generalized and complemented.     

关于Hardy型不等式的讨论

用级数的前n项的某种平均构成新级数的一般项并研究其性质,是一个非常有趣的问题.文章从一道数项级数练习题出发,联系Hardy不等式,将研究对象推广到通过一般函数获得的级数前n项的平均值.其次,类比连续型的Hardy不等式,将所得结论推广到了对[0,+∞)上可积函数在[0,x]上积分均值的研究,进一步推广了原命题.     

Hardy's inequalities in function spaces containing derivatives of noninteger order

A theorem on Hardy's inequality in function spaces containing derivatives of noninteger order is proved. Translated fromMatematichcskie Zametki, Vol. 63, No. 5, pp. 673–678, May, 1998. The author wishes to thank Professor V. A. Kondrat'ev for his attention to this work.     

Conjugate Hardy's inequalities with decreasing weights

We prove that for a decreasing weight on , the conjugate Hardy transform is bounded on () if and only if it is bounded on the cone of all decreasing functions of . This property does not depend on .

     

Hardy-type inequalities on strong and weak Orlicz-Lorentz spaces

In the present paper, the characterization of strong-type modular inequality ∫ 0 ∞φ(Sf (t))w(t)dt≤∫0∞ φ(Cf (t))w(t)dt, f↓ is given, where φ∈Δ’ and S is a Hardy operator. Furthermore, the equivalent conditions of modular inequalities and norm inequalities related to weak Orlicz-Lorentz spaces are researched. We also explore the conditions for Orlicz-Lorentz spaces and weak Orlicz-Lorentz spaces to be normable. Finally, the weak boundedness of certain Hardy-type operators on Orlicz-Lorentz spaces is studied.     

Hardy-type inequalities on planar and spatial open sets

Several Hardy-type inequalities with explicit constants are proved for compactly supported smooth functions on open sets in the Euclidean space ? ⁿ .     

Hardy-type inequalities on the cones of monotone functions

We establish necessary and sufficient conditions for various Hardy-type inequalities on the cones of monotone functions.