- Open Access
On a new class of Hardy-type inequalities
© Adeleke et al.; licensee Springer 2012
- Received: 11 May 2012
- Accepted: 19 October 2012
- Published: 5 November 2012
In this paper, we generalize a Hardy-type inequality to the class of arbitrary non-negative functions bounded from below and above with a convex function multiplied with positive real constants. This enables us to obtain new generalizations of the classical integral Hardy, Hardy-Hilbert, Hardy-Littlewood-Pólya, and Pólya-Knopp inequalities as well as of Godunova’s and of some recently obtained inequalities in multidimensional settings. Finally, we apply a similar idea to functions bounded from below and above with a superquadratic function.
MSC: 26D10, 26D15.
- Hardy inequality
- power weights
- convex functions
- superquadratic functions
Observe that (1.7) follows also from (1.4) by rewriting it with , instead of with a positive function f, and by taking limit as . Moreover, the constants , , , and e, respectively appearing on the right-hand sides of (1.4)-(1.7), are the best possible, that is, neither of them can be replaced with any smaller constant.
and the n-boxes , , and are defined similarly. Finally, the integral is interpreted as .
holds for all convex functions and non-negative measurable functions f on such that the function is integrable on .
was obtained, which holds for , , , and all non-negative integrable functions , as well as for , , and all a.e. positive integrable functions f on .
Motivated by the idea from , in this paper we generalize the modular Hardy-type inequality (1.2) to the class of arbitrary non-negative modular functions Φ equivalent to a non-negative convex function, that is, such that holds for some real constants and a non-negative convex function Ψ. Applying the result obtained to some particular one-dimensional settings, we get new generalizations of the classical inequalities (1.4)-(1.7). Moreover, our result provides a new generalization of Godunova’s inequality (1.8) and improves inequalities (1.10) and (1.11) by relaxing the conditions on the function Φ and by replacing the constant with the smaller constant . Finally, we show that a similar idea can be applied to the function Φ bounded with a superquadratic function Ψ in the same way. Such an approach enables us to get a new generalization of the refined Hardy-type inequality from .
It is equivalent to the power function , where , since it fulfills , .
Now, we state and prove the central theorem in this section.
holds for all measurable functions with values in I and for defined on by (1.3).
so the proof is completed. □
where denotes the class of all convex functions Ψ on I such that for a.e. .
Remark 2.3 For , inequality (2.2) reduces to (1.2), so Theorem 2.1 can be regarded as a generalization of the corresponding result from . In that case, the function Φ has to be convex.
The following two sections are dedicated to some applications and analogues of Theorem 2.1. Namely, by choosing some standard measure spaces, kernels, and weight functions, we get generalizations of one-dimensional and multidimensional Hardy-Knopp-type inequalities from the papers [1, 4, 7–9], and .
Applying Theorem 2.1 to , , , and to , , instead of k, , , we get the following corollary.
, , , and for k, , and respectively replaced with , , and , Theorem 2.1 provides the results dual to Corollary 3.1.
Remark 3.1 By setting in Corollary 3.1 and Corollary 3.2, we obtain a generalization of [, Theorem 3.1] and [, Theorem 4.3]. Observe that the functions Φ and Ψ need not be non-negative in that case.
As a consequence of Corollary 3.1, we get an inequality related to the so-called Riemann-Liouville operator.
so we obtained a new generalization of (3.10). Notice that for , , , and , , inequalities (3.11) reduce to the classical Hardy inequality (1.4).
Observe that inequality (1.7) follows from (3.13) by taking , and , .
In the sequel, we state and prove inequalities dual to (3.8)-(3.13), related to the so-called Weyl operator.
and for , we define .
For , , , and , relation (3.16) reduces to the so-called classical dual Hardy inequality.
Since for , , and , , relation (3.17) reduces to the so-called classical dual Pólya-Knopp inequality, our result can be regarded as its generalization.
Remark 3.2 It is important to notice that due to variable changes applied, none of the inequalities from Example 3.1 and Example 3.2 can be derived directly from Theorem 2.1.
Our analysis continues by considering . We still assume that and . In the following example, we apply Theorem 2.1 to provide a new generalization of the classical Hardy-Hilbert inequality (1.5).
Similarly, in the next example we generalize the classical Hardy-Littlewood-Pólya inequality (1.6).
(see  for more details). For , , and , we have , so it is not hard to see that our result generalizes the classical Hardy-Littlewood-Pólya inequality (1.6).
where , and either , and or and (for more information regarding the unified Riemann-Zeta function, see, e.g., ).
Observe that for relation (3.20) reduces to a usual Hardy-Hilbert-type inequality, so our result can be seen as a generalization in that direction.
In this section, we give a multidimensional result related to Godunova’s inequality (1.8). Namely, we improve and generalize inequalities (1.10) and (1.11) by considering an arbitrary function Φ, not necessarily convex, such that (2.1) holds.
Suppose that , , , and that the kernel is of the form , where is a non-negative measurable function. Applying Theorem 2.1 to this setting and to and , respectively replaced with and , we get the following generalization of Godunova’s inequality (1.8) and a generalization of [, Theorem 6.1].
The above result can be reformulated with particular convex functions, for example, with power and exponential functions. This leads to multidimensional analogues of corollaries and examples from the previous section. Due to the lack of space, we only give a result regarding the n-dimensional Riemann-Liouville operator. The corresponding result for the n-dimensional Weyl operator, which provides a generalization and a refinement of (1.11), can be obtained by a similar method as in the one-dimensional case.
Following the idea that we have used to get inequality (3.11), we obtain the next result.
Notice that (4.7) generalizes and refines inequality (1.10). Namely, for and , inequality (4.7) reduces to (1.10), only with a smaller constant on its right-hand side (since ).
It holds for all probability measure spaces and all non-negative μ-integrable functions f on Ω if and only if is a superquadratic function (see [, Theorem 2.3]).
The following theorem provides the main result of this section.
holds for all measurable functions with values in I and for defined on by (1.3).
so (5.2) is proved. □
As in previous sections, the above result can be specified for some usual measure spaces. Namely, suppose , , , and is defined by (4.2). Applying Theorem 5.1 to this setting and to , , and respectively replaced with , , and , we immediately obtain the following corollary.
Remark 5.1 Observe that for , , and a superquadratic function Φ (that is, for ), inequality (5.3) reduces to [, Proposition 2.1], so our result can be regarded as its generalization.
Analogously, applying Theorem 5.1 to , , and respectively replaced with , , and , where and , we get the following result dual to Corollary 5.1.
Remark 5.2 For , , and a superquadratic function Φ (that is, for ), relation (5.4) reduces to [, Proposition 2.2]. Hence, our results can be seen as its generalization.
Finally, we apply Corollary 5.1 to the superquadratic function , , where , and to some particular weights and kernels deducing a new class of multidimensional Hardy-type inequalities.
A result dual to (5.7) generalizes the corresponding inequality from [, Theorem 3.2]. It is omitted since it can be deduced similarly, starting from Corollary 5.2.
The third author expresses his gratitude to the Abdus Salam International Centre for Theoretical physics, Trieste, Italy, for financial support to carry out this work within the framework of the associate scheme of the Centre. The research of the second and fifth authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grants 058-1170889-1050 (second author) and 082-0000000-0893 (fifth author). All authors express their gratitude to the careful referee whose advices improved the final version of this paper.
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