Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 659761 | https://doi.org/10.1155/2012/659761

A. ČiΕΎmeΕ‘ija, J. PečariΔ‡, D. Pokaz, "Some New Refined General Boas-Type Inequalities", Journal of Function Spaces, vol. 2012, Article ID 659761, 23 pages, 2012. https://doi.org/10.1155/2012/659761

Some New Refined General Boas-Type Inequalities

Academic Editor: Lars-Erik Persson
Received01 Feb 2011
Accepted23 Feb 2011
Published22 Jan 2012

Abstract

We state and prove a new refined Boas-type inequality in a setting with a topological space and general 𝜎-finite and finite Borel measures. As a consequence of the result obtained, we derive a new class of Hardy- and PΓ³lya-Knopp-type inequalities related to balls in ℝ𝑛 and prove that constant factors involved in their right-hand sides are the best possible.

1. Introduction

Boas [1], proved that the inequalityξ€œβˆž0Ξ¦ξ‚΅1π‘€ξ€œβˆž0𝑓(𝑑π‘₯)π‘‘π‘š(𝑑)𝑑π‘₯π‘₯β‰€ξ€œβˆž0Ξ¦(𝑓(π‘₯))𝑑π‘₯π‘₯(1.1) holds for all continuous convex functions Φ∢[0,βˆžβŸ©β†’β„, measurable nonnegative functions π‘“βˆΆβ„+→ℝ, and nondecreasing bounded functions π‘šβˆΆ[0,βˆžβŸ©β†’β„, where 𝑀=π‘š(∞)βˆ’π‘š(0)>0, and the inner integral on the left-hand side of (1.1) is the Lebesgue-Stieltjes integral with respect to π‘š. After its author, inequality (1.1) was named the Boas inequality. In the case of a concave function Ξ¦, it holds with the sign of inequality reversed.

Since its publication, the Boas inequality has been generalized in different ways. Here we mention only those that have guided us in our research. Namely, following some ideas of Kaijser et al. [2] (see also the paper [3] of Levinson), ČiΕΎmeΕ‘ija et al. [4] considered a general Borel measure πœ† on ℝ+, such that 𝐿=πœ†(ℝ+)<∞, a convex function Ξ¦ on a convex set πΌβŠ†β„, and a weight function 𝑒 on ℝ+, and proved that the inequalityξ€œβˆž0𝑒(π‘₯)Ξ¦(𝐴𝑓(π‘₯))𝑑π‘₯π‘₯≀1πΏξ€œβˆž0𝑀(π‘₯)Ξ¦(𝑓(π‘₯))𝑑π‘₯π‘₯(1.2) holds for all measurable functions π‘“βˆΆβ„+→ℝ with values in 𝐼, where βˆ«π΄π‘“(π‘₯)=(1/𝐿)∞0𝑓(𝑑π‘₯)π‘‘πœ†(𝑑) and βˆ«π‘€(π‘₯)=∞0𝑒(π‘₯/𝑑)π‘‘πœ†(𝑑)<∞, π‘₯βˆˆβ„+. In the same paper, a refinement of (1.2) is obtained.

Another generalization of (1.1) was recently given by Luor [5] in a setting with 𝜎-finite Borel measures πœ‡ and 𝜈 on a topological space 𝑋 and a Borel probability measure πœ† on ℝ+. For a πœ†-balanced Borel set Ξ©βŠ†π‘‹ and the measures πœ‡π‘‘β‰ͺ𝜈, defined for all Borel sets π‘†βŠ†π‘‹ and π‘‘βˆˆsuppπœ† by πœ‡π‘‘(𝑆)=πœ‡(π‘‘βˆ’1𝑆), he obtained the inequalityξ€œΞ©Ξ¦ξ‚΅ξ€œβˆž0ξ‚Άξ€œπ‘“(𝑑𝐱)π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)β‰€Ξ©ξ‚΅ξ€œΞ¦(𝑓(𝐱))∞0π‘‘πœ‡π‘‘ξ‚Άπ‘‘πœˆ(𝐱)π‘‘πœ†(𝑑)π‘‘πœˆ(𝐱),(1.3) where Ξ¦ is a nonnegative convex function on a real interval 𝐼, and π‘“βˆΆΞ©β†’β„ is a Borel measurable function with values in 𝐼. A weighted version of (1.3) was given by ČiΕΎmeΕ‘ija et al. [6, Theorem 2.1], along with a detailed analysis of properties of the related so-called Boas functional.

Notice that the Boas inequality (1.1) unifies some well-known classical inequalities, such as Hardy's and Pólya-Knopp's inequalities. In the sequel, we state their strengthened versions, obtained independently by Debnath et al. [7, 8] and Čižmeőija et al. [9, 10].

Let 0<π‘β‰€βˆž and 𝑝,π‘˜βˆˆβ„ such that 𝑝/(π‘˜βˆ’1)>0. If π‘βˆˆβ„β§΅[0,1], 𝑓 is a nonnegative function, π‘₯1βˆ’(π‘˜/𝑝)π‘“βˆˆπΏπ‘(0,𝑏), andξ€œπΉ(π‘₯)=π‘₯0𝑓(𝑑)𝑑𝑑,π‘₯∈⟨0,π‘βŸ©,(1.4) thenξ€œπ‘0π‘₯βˆ’π‘˜πΉπ‘ξ‚€π‘(π‘₯)𝑑π‘₯β‰€ξ‚π‘˜βˆ’1π‘ξ€œπ‘0ξ‚Έξ‚€π‘₯1βˆ’π‘ξ‚(π‘˜βˆ’1)/𝑝π‘₯π‘βˆ’π‘˜π‘“π‘(π‘₯)𝑑π‘₯(1.5) holds, while for π‘βˆˆβŸ¨0,1⟩ the sign of inequality in (1.5) is reversed. Moreover, if 0≀𝑏<∞ and parameters π‘βˆˆβ„β§΅[0,1] and π‘˜βˆˆβ„ such that 𝑝/(π‘˜βˆ’1)<0, then the inequalityξ€œβˆžπ‘π‘₯βˆ’k𝐹𝑝𝑝(π‘₯)𝑑π‘₯≀1βˆ’π‘˜π‘ξ€œβˆžπ‘ξ‚Έξ‚€π‘1βˆ’π‘₯(1βˆ’π‘˜)/𝑝π‘₯π‘βˆ’π‘˜π‘“π‘(π‘₯)𝑑π‘₯(1.6) holds for all nonnegative functions 𝑓 such that π‘₯1βˆ’(π‘˜/𝑝)π‘“βˆˆπΏπ‘(𝑏,∞), where ξ‚ξ€œπΉ(π‘₯)=∞π‘₯𝑓(𝑑)𝑑𝑑,π‘₯βˆˆβŸ¨π‘,∞⟩.(1.7) For π‘βˆˆβŸ¨0,1⟩ inequality (1.6) holds with the sign of inequality reversed. The constant |𝑝/(π‘˜βˆ’1)|𝑝 is the best possible for both inequalities, that is, it cannot be replaced with any smaller constant. The classical Hardy's inequality follows by taking 𝑏=∞ in (1.5), while for 𝑏=0 in (1.6) we get its dual inequality.

On the other hand, if 0<π‘β‰€βˆž, π‘“βˆˆπΏ1(0,𝑏) is a positive function, andξ‚΅1𝐺(π‘₯)=expπ‘₯ξ€œπ‘₯0ξ‚Άlog𝑓(𝑑)𝑑𝑑,π‘₯∈⟨0,π‘βŸ©,(1.8) thenξ€œπ‘0ξ€œπΊ(π‘₯)𝑑π‘₯≀𝑒𝑏0ξ‚€π‘₯1βˆ’π‘ξ‚π‘“(π‘₯)𝑑π‘₯(1.9) holds, while the inequalityξ€œβˆžπ‘ξ‚1𝐺(π‘₯)𝑑π‘₯β‰€π‘’ξ€œβˆžπ‘ξ‚€π‘1βˆ’π‘₯𝑓(π‘₯)𝑑π‘₯(1.10) holds for 0≀𝑏<∞, 0<π‘“βˆˆπΏ1(𝑏,∞), andπ‘₯ξ€œπΊ(π‘₯)=exp∞π‘₯log𝑓(𝑑)𝑑𝑑𝑑2ξ‚Ά,π‘₯βˆˆβŸ¨π‘,∞⟩.(1.11) For 𝑏=∞ in (1.9) and for 𝑏=0 in (1.10) we, respectively, get the classical PΓ³lya-Knopp's inequality and its dual inequality. Notice that the constant factors 𝑒 and 1/𝑒, respectively, involved in the right-hand sides of (1.9) and (1.10), are the best possible.

In this paper, we also make use of the following 𝑛-dimensional strengthened Hardy's inequality related to the setting with balls in ℝ𝑛 centered at the origin (see [11] for details). Let 𝑝,π‘˜,π‘…βˆˆβ„ such that 𝑝>1, π‘˜β‰ 1, and 𝑅>0. Suppose that 𝑓 is a nonnegative measurable function and the function 𝐹 is defined on ℝ𝑛 by ⎧βŽͺ⎨βŽͺβŽ©ξ€œπΉ(𝐱)=𝐡(|𝐱|)ξ€œπ‘“(𝐲)𝑑𝐲,π‘˜>1,ℝ𝑛⧡𝐡(|𝐱|)𝑓(𝐲)𝑑𝐲,π‘˜<1,(1.12) where 𝐡(|𝐱|) is a ball in ℝ𝑛 centered at the origin and of radius |𝐱|, while |𝐱| denotes the Euclidean norm of π±βˆˆβ„π‘›. If 𝑝/(π‘˜βˆ’1)>0, then the inequalityξ€œπ΅(𝑅)||||𝐡(|𝐱|)βˆ’π‘˜πΉπ‘(≀𝑝𝐱)π‘‘π±ξ‚π‘˜βˆ’1π‘ξ€œπ΅(𝑅)||||1βˆ’π΅(|𝐱|)||||𝐡(𝑅)(π‘˜βˆ’1)/𝑝||||𝐡(|𝐱|)π‘βˆ’π‘˜π‘“π‘(𝐱)𝑑𝐱(1.13) holds, while for 𝑝/(π‘˜βˆ’1)<0 we haveξ€œβ„π‘›β§΅π΅(𝑅)||||𝐡(|𝐱|)βˆ’π‘˜πΉπ‘(≀𝑝𝐱)𝑑𝐱1βˆ’π‘˜π‘ξ€œβ„π‘›β§΅π΅(𝑅)||||1βˆ’π΅(𝑅)||𝐡||ξ‚Ά(|𝐱|)(1βˆ’π‘˜)/𝑝||||𝐡(|𝐱|)π‘βˆ’π‘˜π‘“π‘(𝐱)𝑑𝐱.(1.14) Terms |𝐡(|𝐱|)| and |𝐡(𝑅)|, respectively, denote the volumes of 𝐡(|𝐱|) and 𝐡(𝑅). The constant (𝑝/|π‘˜βˆ’1|)𝑝 is the best possible for both inequalities. Observe that the first natural generalization of the classical Hardy's inequality to balls in ℝ𝑛 was given by Christ and Grafakos in [12].

Finally, here we state an 𝑛-dimensional PΓ³lya-Knopp's inequality, related to (1.13) and (1.14),ξ€œπ΅(𝑅)ξ€œπΊ(𝐱)𝑑𝐱<𝑒𝐡(𝑅)ξ‚΅||||1βˆ’π΅(|𝐱|)||||𝐡(𝑅)𝑓(𝐱)𝑑𝐱(1.15) for a positive function 𝑓 on 𝐡(𝑅) and ξ‚΅1𝐺(𝐱)=exp||||ξ€œπ΅(|𝐱|)𝐡(|𝐱|)ξ‚Άlog𝑓(𝐲)𝑑𝐲,𝐱∈𝐡(𝑅),(1.16) as well as its dual inequalityξ€œβ„π‘›β§΅π΅(𝑅)1𝐺(𝐱)𝑑𝐱<π‘’ξ€œβ„π‘›β§΅π΅(𝑅)ξ‚΅||||1βˆ’π΅(𝑅)||||𝐡(|𝐱|)𝑓(𝐱)𝑑𝐱(1.17) for a positive function 𝑓 on ℝ𝑛⧡𝐡(𝑅) and ||||ξ€œπΊ(𝐱)=exp𝐡(|𝐱|)ℝ𝑛⧡𝐡(|𝐱|)log𝑓(𝐲)𝑑𝐲||𝐡||(|𝐱|)2ξƒͺ,π±βˆˆβ„π‘›β§΅π΅(𝑅).(1.18)

Observe that all above-mentioned inequalities of the Hardy and PΓ³lya-Knopp type are just a small contribution to the rich theory of the Hardy-type inequalities. Many information about history and new developments regarding Hardy's and related Carleman's inequalities can be found, for example, in recent monographs [13–15], expository papers [2, 16, 17], and in exhaustive lists of references given therein.

Our aim in this paper is to state and prove a new weighted refined Boas-type inequality in a setting with topological spaces and finite and 𝜎-finite Borel measures and to show that our result refines and generalizes Luor's inequality (1.3). In addition, we extend the results obtained to a case with nonnegative kernels.

Moreover, as a consequence of our new refined Boas-type inequality, we derive a new class of Hardy- and PΓ³lya-Knopp-type inequalities related to balls in ℝ𝑛, along with their respective dual inequalities, and prove that constant factors involved in their right-hand sides are the best possible. Finally, we show that our Hardy's and PΓ³lya-Knopp's inequalities differ from (1.13), (1.14), (1.15) and (1.17), although for 𝑛=1 both classes coincide.

Conventions. Throughout this paper, all measures are assumed to be positive, all functions are assumed to be measurable on their respective domains and expressions of the form 0β‹…βˆž, 0/0,π‘Ž/∞(π‘Žβˆˆβ„), and ∞/∞ are taken to be equal to zero. As usual, by 𝑑π‘₯ and 𝑑𝐱 we denote the Lebesgue measure on ℝ and ℝ𝑛(π‘›βˆˆβ„•,𝑛β‰₯2), respectively, while by a weight function we mean a nonnegative measurable function on the actual set. An interval in ℝ is any convex subset of ℝ, while Int 𝐼 denotes the interior of an interval πΌβŠ†β„. In particular, ℝ+=⟨0,∞⟩. For 𝑅>0, by 𝐡(𝑅) we denote a ball in ℝ𝑛 centered at the origin and of radius 𝑅, that is, 𝐡(𝑅)={π±βˆˆβ„π‘›βˆΆ|𝐱|≀𝑅}, where |𝐱| denotes the Euclidean norm of π±βˆˆβ„π‘›. By its dual set we mean the set ℝ𝑛⧡𝐡(𝑅)={π±βˆˆβ„π‘›βˆΆ|𝐱|>𝑅}. Finally, by π‘†π‘›βˆ’1 we denote the surface of the unit ball 𝐡(1) and by |π‘†π‘›βˆ’1| its area. Using polar coordinates in ℝ𝑛, the volume of the ball B(𝑅) is then ||||=ξ€œπ΅(𝑅)𝐡(𝑅)ξ€œπ‘‘π±=|𝐱|β‰€π‘…ξ€œπ‘‘π±=𝑅0π‘‘π‘›βˆ’1ξ‚΅ξ€œπ‘†π‘›βˆ’1ξ‚Ά=ξ€œπ‘‘π‘†π‘‘π‘‘π‘†π‘›βˆ’1ξ‚΅ξ€œπ‘…0π‘‘π‘›βˆ’1𝑅𝑑𝑑𝑑𝑆=𝑛||π‘†π‘›βˆ’1||𝑛.(1.19)

2. A New Refined Weighted Boas-Type Inequality

To start, we recall some basic facts on convex functions. Let 𝐼 be an interval in ℝ and Ξ¦βˆΆπΌβ†’β„ be a convex function. For π‘₯∈𝐼, by πœ•Ξ¦(π‘₯) we denote the subdifferential of Ξ¦ at π‘₯, that is, the setπœ•Ξ¦(π‘₯)={π›Όβˆˆβ„βˆΆΞ¦(𝑦)βˆ’Ξ¦(π‘₯)βˆ’π›Ό(π‘¦βˆ’π‘₯)β‰₯0,π‘¦βˆˆπΌ}.(2.1) It is well known that πœ•Ξ¦(π‘₯)β‰ βˆ… for all π‘₯∈Int𝐼. More precisely, at each point π‘₯∈Int𝐼 we have βˆ’βˆž<Ξ¦ξ…žβˆ’(π‘₯)β‰€Ξ¦ξ…ž+(π‘₯)<∞, πœ•Ξ¦(π‘₯)=[Ξ¦ξ…žβˆ’(π‘₯),Ξ¦ξ…ž+(π‘₯)], and the set on which Ξ¦ is not differentiable is at most countable. Moreover, every function πœ‘βˆΆπΌβ†’β„ for which πœ‘(π‘₯)βˆˆπœ•Ξ¦(π‘₯), whenever π‘₯∈Int𝐼, is increasing on Int𝐼.

On the other hand, if Ξ¦βˆΆπΌβ†’β„ is a concave function, that is, βˆ’Ξ¦ is convex, then πœ•Ξ¦(π‘₯)={π›Όβˆˆβ„βˆΆΞ¦(π‘₯)βˆ’Ξ¦(𝑦)βˆ’π›Ό(π‘₯βˆ’π‘¦)β‰₯0,π‘¦βˆˆπΌ} denotes the superdifferential of Ξ¦ at the point π‘₯∈𝐼. For all π‘₯∈Int𝐼, in this setting we have βˆ’βˆž<Ξ¦ξ…ž+(π‘₯)β‰€Ξ¦ξ…žβˆ’(π‘₯)<∞ and πœ•Ξ¦(π‘₯)=[Ξ¦ξ…ž+(π‘₯),Ξ¦ξ…žβˆ’(π‘₯)]β‰ βˆ…. Notice that, although the symbol πœ•Ξ¦(π‘₯) has two different notions, it will be clear from the context whether it applies to a convex or to a concave function Ξ¦. For more details about convex and concave functions see, for example, the recent monograph [18].

We continue by introducing some necessary notation, related to a setting with topological spaces and Borel measures, in which we state and prove a new refined weighted inequality of the Boas type.

Let πœ† be a finite Borel measure on ℝ+. By suppπœ†, we denote its support, that is, the set of all π‘‘βˆˆβ„+ such that πœ†(𝑁𝑑)>0 holds for all open neighbourhoods 𝑁𝑑 of 𝑑. Hence,ξ€œπΏ=suppπœ†ξ€œπ‘‘πœ†(𝑑)=∞0ξ€·β„π‘‘πœ†(𝑑)=πœ†+ξ€Έ<∞.(2.2) Further, let 𝑋 be a topological space equipped with a continuous scalar multiplication (π‘Ž,𝐱)β†¦π‘Žπ±βˆˆπ‘‹, for π‘Žβˆˆβ„+ and π±βˆˆπ‘‹, such that1𝐱=𝐱,π‘Ž(𝑏𝐱)=(π‘Žπ‘)𝐱,π±βˆˆπ‘‹,π‘Ž,π‘βˆˆβ„+.(2.3) Let a Borel set Ξ©βŠ†π‘‹ be πœ†-balanced, that is, let 𝑑Ω={π‘‘π±βˆΆπ±βˆˆΞ©}βŠ†Ξ© hold for all π‘‘βˆˆsuppπœ†. For a Borel measurable function π‘“βˆΆΞ©β†’β„, we define its Hardy-Littlewood average, 𝐴𝑓, as1𝐴𝑓(𝐱)=πΏξ€œβˆž0𝑓(𝑑𝐱)π‘‘πœ†(𝑑),𝐱∈Ω.(2.4) Finally, suppose that πœ‡ and 𝜈 are 𝜎-finite Borel measures on 𝑋. For 𝑑>0 and a Borel set π‘†βŠ†π‘‹, we defineπœ‡π‘‘ξ‚€1(𝑆)=πœ‡π‘‘π‘†ξ‚.(2.5) Obviously, πœ‡π‘‘ is a 𝜎-finite Borel measure on 𝑋 for all π‘‘βˆˆβ„+. Throughout this paper we assume that πœ‡π‘‘ is absolutely continuous with respect to the measure 𝜈, that is, πœ‡π‘‘β‰ͺ𝜈, for each π‘‘βˆˆsuppπœ†. As usual, by π‘‘πœ‡π‘‘/π‘‘πœˆ we denote the related Radon-Nikodym derivative.

As announced, now we can state and prove the main result of this paper, a new refined weighted Boas-type inequality in the above setting.

Theorem 2.1. Let πœ† be a finite Borel measure on ℝ+ and 𝐿 be defined by (2.2). Let πœ‡ and 𝜈 be 𝜎-finite Borel measures on a topological space 𝑋, πœ‡π‘‘ be defined by (2.5), and let πœ‡π‘‘β‰ͺ𝜈 for all π‘‘βˆˆsuppπœ†. Further, let Ξ©βŠ†π‘‹ be a πœ†-balanced Borel set and 𝑒 be a nonnegative function on 𝑋, such that ξ€œπ‘£(𝐱)=∞0𝑒1π‘‘π±ξ‚π‘‘πœ‡π‘‘π‘‘πœˆ(𝐱)π‘‘πœ†(𝑑)<∞,𝐱∈Ω.(2.6) Suppose that Ξ¦βˆΆπΌβ†’β„ is a nonnegative convex function on an interval πΌβŠ†β„ and πœ‘βˆΆπΌβ†’β„ is any function fulfilling πœ‘(π‘₯)βˆˆπœ•Ξ¦(π‘₯), for all π‘₯∈Int𝐼. If π‘“βˆΆΞ©β†’β„ is a Borel measurable function with values in 𝐼 and 𝐴𝑓 is defined by (2.4), then 𝐴𝑓(𝐱)∈𝐼 for all 𝐱∈Ω and the inequality 1πΏξ€œΞ©ξ€œπ‘£(𝐱)Ξ¦(𝑓(𝐱))π‘‘πœˆ(𝐱)βˆ’Ξ©β‰₯1𝑒(𝐱)Ξ¦(𝐴𝑓(𝐱))π‘‘πœ‡(𝐱)𝐿||||ξ€œΞ©ξ€œπ‘’(𝐱)∞0||||βˆ’ξ€œΞ¦(𝑓(𝑑𝐱))βˆ’Ξ¦(𝐴𝑓(𝐱))π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)Ξ©ξ€œπ‘’(𝐱)∞0||||β‹…||||||||πœ‘(𝐴𝑓(𝐱))𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱)π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)(2.7) holds. For a nonpositive concave function Ξ¦, relation (2.7) holds with ξ€œΞ©1𝑒(𝐱)Ξ¦(𝐴𝑓(𝐱))π‘‘πœ‡(𝐱)βˆ’πΏξ€œΞ©π‘£(𝐱)Ξ¦(𝑓(𝐱))π‘‘πœˆ(𝐱)(2.8) on its left-hand side.

Proof. Since 𝑓(𝐱)∈𝐼 for all 𝐱∈Ω, it is not hard to see that 𝐴𝑓(𝐱)∈𝐼, for all 𝐱∈Ω (see the proof of Theorem 2.1 in [6] for details). Suppose the function Ξ¦ is convex and nonnegative. To prove inequality (2.7), observe that for arbitrary π‘ŸβˆˆInt𝐼 and π‘ βˆˆπΌ we have Ξ¦(𝑠)βˆ’Ξ¦(π‘Ÿ)βˆ’πœ‘(π‘Ÿ)(π‘ βˆ’π‘Ÿ)β‰₯0, so ||||β‰₯||||||βˆ’||||||.Ξ¦(𝑠)βˆ’Ξ¦(π‘Ÿ)βˆ’πœ‘(π‘Ÿ)(π‘ βˆ’π‘Ÿ)=Ξ¦(𝑠)βˆ’Ξ¦(π‘Ÿ)βˆ’πœ‘(π‘Ÿ)(π‘ βˆ’π‘Ÿ)Ξ¦(𝑠)βˆ’Ξ¦(π‘Ÿ)πœ‘(π‘Ÿ)β‹…|π‘ βˆ’π‘Ÿ|(2.9) In particular, for 𝐱∈Ω, such that 𝐴𝑓(𝐱)∈Int𝐼, and for π‘‘βˆˆsuppΞ©, from (2.9) we get β‰₯||||||βˆ’||||β‹…||||||.Ξ¦(𝑓(𝑑𝐱))βˆ’Ξ¦(𝐴𝑓(𝐱))βˆ’πœ‘(𝐴𝑓(𝐱))β‹…(𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱))Ξ¦(𝑓(𝑑𝐱))βˆ’Ξ¦(𝐴𝑓(𝐱))πœ‘(𝐴𝑓(𝐱))𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱)(2.10) On the other hand, if 𝐼 is not an open interval and 𝐴𝑓(𝐱) is an endpoint of 𝐼 for some 𝐱∈Ω, then either 𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱)β‰₯0 for all π‘‘βˆˆsuppπœ† or 𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱)≀0 for all π‘‘βˆˆsuppπœ†. Since ξ€œβˆž0(𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱))π‘‘πœ†(𝑑)=0,(2.11) we conclude that 𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱)=0 for πœ†-a.e. π‘‘βˆˆsuppπœ†, so in that case both sides of (2.10) are equal to 0. Hence, (2.10) holds for all 𝐱∈Ω and πœ†-a.e. π‘‘βˆˆsuppπœ†. Multiplying it by 𝑒(𝐱) and then integrating over ℝ+ and Ξ©, we obtain the following sequence of inequalities: ξ€œΞ©ξ€œβˆž0ξ€œπ‘’(𝐱)Ξ¦(𝑓(𝑑𝐱))π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)βˆ’Ξ©ξ€œβˆž0βˆ’ξ€œπ‘’(𝐱)Ξ¦(𝐴𝑓(𝐱))π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)Ξ©ξ€œβˆž0β‰₯ξ€œπ‘’(𝐱)πœ‘(𝐴𝑓(𝐱))(𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱))π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)Ξ©ξ€œβˆž0||||||βˆ’||||β‹…||||||β‰₯ξ€œπ‘’(𝐱)Ξ¦(𝑓(𝑑𝐱))βˆ’Ξ¦(𝐴𝑓(𝐱))πœ‘(𝐴𝑓(𝐱))𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱)π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)Ξ©||||ξ€œπ‘’(𝐱)∞0||||||||ξ€œΞ¦(𝑓(𝑑𝐱))βˆ’Ξ¦(𝐴𝑓(𝐱))π‘‘πœ†(𝑑)βˆ’πœ‘(𝐴𝑓(𝐱))∞0||||||||β‰₯||||ξ€œπ‘“(𝑑𝐱)βˆ’π΄π‘“(𝐱)π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)Ξ©ξ€œπ‘’(𝐱)∞0||||βˆ’ξ€œΞ¦(𝑓(𝑑𝐱))βˆ’Ξ¦(𝐴𝑓(𝐱))π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)Ξ©||||ξ€œπ‘’(𝐱)πœ‘(𝐴𝑓(𝐱))∞0||||||||.𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱)π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)(2.12) By using Fubini's and the Radon-Nikodym theorems, the substitution 𝐲=𝑑𝐱, and the fact that the set Ξ© is πœ†-balanced and the function Ξ¦ is nonnegative, the first integral on the left-hand side of (2.12) becomes ξ€œΞ©ξ€œβˆž0=ξ€œπ‘’(𝐱)Ξ¦(𝑓(𝑑𝐱))π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)∞0ξ€œΞ©=ξ€œπ‘’(𝐱)Ξ¦(𝑓(𝑑𝐱))π‘‘πœ‡(𝐱)π‘‘πœ†(𝑑)∞0ξ€œπ‘‘Ξ©π‘’ξ‚€1𝑑𝐲Φ(𝑓(𝐲))π‘‘πœ‡π‘‘β‰€ξ€œ(𝐲)π‘‘πœ†(𝑑)∞0ξ€œΞ©π‘’ξ‚€1𝑑𝐲Φ(𝑓(𝐲))π‘‘πœ‡π‘‘=ξ€œ(𝐲)π‘‘πœ†(𝑑)∞0ξ€œΞ©π‘’ξ‚€1𝑑𝐲Φ(𝑓(𝐲))π‘‘πœ‡π‘‘=ξ€œπ‘‘πœˆ(𝐲)π‘‘πœˆ(𝐲)π‘‘πœ†(𝑑)Ξ©ξ‚΅ξ€œβˆž0𝑒1π‘‘π²ξ‚π‘‘πœ‡π‘‘ξ‚Ά=ξ€œπ‘‘πœˆ(𝐲)π‘‘πœ†(𝑑)Ξ¦(𝑓(𝐲))π‘‘πœˆ(𝐲)Ω𝑣(𝐲)Ξ¦(𝑓(𝐲))π‘‘πœˆ(𝐲).(2.13) Further, the second integral on the left-hand side in (2.12) reduces to ξ€œΞ©ξ€œβˆž0ξ€œπ‘’(𝐱)Ξ¦(𝐴𝑓(𝐱))π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)=𝐿Ω𝑒(𝐱)Ξ¦(𝐴𝑓(𝐱))π‘‘πœ‡(𝐱),(2.14) while the corresponding third integral is equal to 0 since by (2.11) we have ξ€œΞ©ξ€œβˆž0=ξ€œπ‘’(𝐱)πœ‘(𝐴𝑓(𝐱))(𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱))π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)Ξ©ξ‚΅ξ€œπ‘’(𝐱)πœ‘(𝐴𝑓(𝐱))∞0ξ‚Ά(𝑓(𝑑𝐱)βˆ’π΄π‘“(𝐱))π‘‘πœ†(𝑑)π‘‘πœ‡(𝐱)=0.(2.15) Finally, (2.7) holds by combining (2.12), (2.13), (2.14), and (2.15).
It remains to prove the last part of the statement of Theorem 2.1. If Ξ¦ is a nonpositive concave function, then βˆ’Ξ¦ is a nonnegative convex function and (2.9) becomes ||||β‰₯||||||βˆ’||||||,Ξ¦(π‘Ÿ)βˆ’Ξ¦(𝑠)βˆ’πœ‘(π‘Ÿ)(π‘Ÿβˆ’π‘ )=Ξ¦(π‘Ÿ)βˆ’Ξ¦(𝑠)βˆ’πœ‘(π‘Ÿ)(π‘Ÿβˆ’π‘ )Ξ¦(𝑠)βˆ’Ξ¦(π‘Ÿ)πœ‘(π‘Ÿ)|π‘ βˆ’π‘Ÿ|(2.16) where πœ‘βˆΆπΌβ†’β„ is any function such that πœ‘(π‘₯)βˆˆπœ•Ξ¦(π‘₯)=[Ξ¦ξ…ž+(π‘₯),Ξ¦ξ…žβˆ’(π‘₯)] for all π‘₯∈Int𝐼. Following the same lines as in the proof for a convex function, we get (2.7) with swapped order of two integrals on its left-hand side.

Remark 2.2. Observe that a pair of inequalities interpolated between the left-hand side and the right-hand side of (2.12) provides other new refinements of (2.7).

It is important to notice that the condition on nonnegativity of the convex function Ξ¦ in Theorem 2.1 can be omitted only in a particular setting with cones in 𝑋. More precisely, the following corollary holds.

Corollary 2.3. If in Theorem 2.1 one has 𝑑Ω=Ξ© for πœ†-a.e. π‘‘βˆˆsuppπœ†, then (2.7) holds for all convex functions Ξ¦ on an interval πΌβŠ†β„. In this setting, inequality (2.7) holds also for all concave functions Ξ¦ on πΌβŠ†β„, but with swapped order of the integrals on its left-hand side.

As a consequence of Theorem 2.1, we get a weighted general Boas-type inequality obtained in [6]. It is given in the following corollary.

Corollary 2.4. Suppose that πœ†,𝐿,𝑋,πœ‡,𝜈,πœ‡π‘‘,Ξ©,𝑒, and 𝑣 are as in Theorem 2.1. If Ξ¦ is a nonnegative convex function on an interval πΌβŠ†β„, then the inequality ξ€œΞ©1𝑒(𝐱)Ξ¦(𝐴𝑓(𝐱))π‘‘πœ‡(𝐱)β‰€πΏξ€œΞ©π‘£(𝐱)Ξ¦(𝑓(𝐱))π‘‘πœˆ(𝐱)(2.17) holds for all measurable functions π‘“βˆΆΞ©β†’β„ such that 𝑓(𝐱)∈𝐼 for all 𝐱∈Ω, where 𝐴𝑓 is defined by (2.4). For a nonpositive concave function Ξ¦, the sign of inequality in (2.7) is reversed. Moreover, if in Theorem 2.1 one has 𝑑Ω=Ξ© for πœ†-a.e. π‘‘βˆˆsuppπœ†, then (2.7) holds for all convex functions Ξ¦ on an interval πΌβŠ†β„. In that case, for all concave functions Ξ¦, inequality (2.7) holds with the sign of inequality reversed.

Remark 2.5. Observe that Theorem 2.1 generalizes and refines the Boas-type inequality (1.3) obtained by Luor [5]. Namely, since the right-hand side of (2.7) is nonnegative, for 𝑒(𝐱)≑1, 𝐱∈Ω, inequality (2.17) reduces to (1.3).

3. New Multidimensional Hardy- and PΓ³lya-Knopp-Type Inequalities

In this section, we apply Theorem 2.1 to a particular multidimensional setting, namely, to balls in ℝ𝑛 centered at the origin and to their dual sets. The results obtained represent a new class of 𝑛-dimensional Hardy and PΓ³lya-Knopp-type inequalities, different from the existing inequalities (1.13), (1.14), (1.15), and (1.17). Moreover, the constant factors appearing on the right-hand sides of our relations are the best possible.

Our first result in this direction is a refinement of an inequality by Luor [5, relation (1.14)], related to cones in ℝ𝑛.

Theorem 3.1. Let πœ† be a finite Borel measure on ℝ+, 𝐿 be defined by (2.2), Ξ© be a πœ†-balanced Borel set in ℝ𝑛, and 𝐡 be a Borel subset of the unit sphere π‘†π‘›βˆ’1. Let 𝑒 be a nonnegative function on ℝ𝑛, Ξ¦ be a nonnegative convex function on an interval πΌβŠ†β„, and πœ‘βˆΆπΌβ†’β„ be any function fulfilling πœ‘(π‘₯)βˆˆπœ•Ξ¦(π‘₯), for all π‘₯∈Int𝐼. Finally, let π‘“βˆΆΞ©β†’β„ be a measurable function with values in 𝐼. (i)If suppπœ†βŠ†βŸ¨0,1], 0<π‘…β‰€βˆž, and Ξ©=Ξ©1={𝐱=πœ‰π‘†βˆΆπ‘†βˆˆπ΅,0β‰€πœ‰<𝑅}, then 1πΏξ€œΞ©1ξ‚΅ξ€œΞ¦(𝑓(𝐱))1|𝐱|/𝑅𝑒1𝑑𝐱𝑑1βˆ’π‘›ξ‚Άπ‘‘πœ†(𝑑)π‘‘π±βˆ’ξ€œ|𝐱|Ξ©1𝐴𝑒(𝐱)Ξ¦1𝑓(𝐱)𝑑𝐱β‰₯1|𝐱|𝐿||||ξ€œΞ©1ξ€œπ‘’(𝐱)10||𝐴Φ(𝑓(𝑑𝐱))βˆ’Ξ¦1ξ€Έ||𝑓(𝐱)π‘‘πœ†(𝑑)π‘‘π±βˆ’ξ€œ|𝐱|Ξ©1π‘’ξ€œ(𝐱)10||πœ‘ξ€·π΄1𝑓||β‹…||𝑓(𝐱)(𝑑𝐱)βˆ’π΄1𝑓||(𝐱)π‘‘πœ†(𝑑)𝑑𝐱||||,|𝐱|(3.1) where 𝐴1βˆ«π‘“(𝐱)=10𝑓(𝑑𝐱)π‘‘πœ†(𝑑),𝐱∈Ω1.(ii)If suppπœ†βŠ†[1,∞⟩, 0≀𝑅<∞, and Ξ©=Ξ©2={𝐱=πœ‰π‘†βˆΆπ‘†βˆˆπ΅,π‘…β‰€πœ‰<∞}, then 1πΏξ€œΞ©2ξ‚΅ξ€œΞ¦(𝑓(𝐱))1|𝐱|/𝑅𝑒1𝑑𝐱𝑑1βˆ’π‘›ξ‚Άπ‘‘πœ†(𝑑)π‘‘π±βˆ’ξ€œ|𝐱|Ξ©2𝐴𝑒(𝐱)Ξ¦2𝑓(𝐱)𝑑𝐱β‰₯1|𝐱|𝐿||||ξ€œΞ©2ξ€œπ‘’(𝐱)∞1||𝐴Φ(𝑓(𝑑𝐱))βˆ’Ξ¦2ξ€Έ||𝑓(𝐱)π‘‘πœ†(𝑑)π‘‘π±βˆ’ξ€œ|𝐱|Ξ©2π‘’ξ€œ(𝐱)∞1||πœ‘ξ€·π΄2𝑓||β‹…||𝑓(𝐱)(𝑑𝐱)βˆ’π΄2𝑓||(𝐱)π‘‘πœ†(𝑑)𝑑𝐱||||,|𝐱|(3.2) where𝐴2βˆ«π‘“(𝐱)=∞1𝑓(𝑑𝐱)π‘‘πœ†(𝑑),𝐱∈Ω2.

Proof. Relation (3.1) is a direct consequence of Theorem 2.1, rewritten with 𝑋=ℝ𝑛,Ξ©=Ξ©1,π‘‘πœ‡(𝐱)=πœ’Ξ©1(𝐱)𝑑𝐱, and π‘‘πœˆ(𝐱)=𝑑𝐱, as well as with the function 𝑒 replaced with 𝐱↦|𝐱|βˆ’1𝑒(𝐱). Then we have (π‘‘πœ‡π‘‘/π‘‘πœˆ)(𝐱)=π‘‘βˆ’π‘›πœ’π‘‘Ξ©1(𝐱),π‘‘βˆˆβŸ¨0,1], ξ€œπ‘£(𝐱)=1|𝐱|/𝑅𝑒1𝑑𝐱𝑑1βˆ’π‘›π‘‘πœ†(𝑑),𝐱∈Ω1,(3.3) and 𝐴𝑓(𝐱)=𝐴1𝑓(𝐱),π‘₯∈Ω1, so relation (1.13) reduces to (3.1). The proof of (3.2) follows the same lines, by considering Ξ©=Ξ©2. In such setting we get ξ€œπ‘£(𝐱)=1|𝐱|/𝑅𝑒1𝑑𝐱𝑑1βˆ’π‘›π‘‘πœ†(𝑑),𝐱∈Ω2,(3.4) and 𝐴𝑓=𝐴2𝑓.

By taking 𝐡=π‘†π‘›βˆ’1, that is, by setting a πœ†-balanced set Ξ© to be a ball in ℝ𝑛 centered at the origin or its corresponding dual set, and by choosing a suitable measure πœ† and a weight function 𝑒, we obtain the following sequence of new refined strengthened inequalities of the Hardy type.

Theorem 3.2. Let π‘›βˆˆβ„•, π‘βˆˆβ„β§΅[0,1⟩, and π‘˜βˆˆβ„, π‘˜β‰ π‘›. (i)If 0<π‘…β‰€βˆž, 𝑝/(π‘˜βˆ’π‘›)>0, and 𝑓 is a nonnegative measurable function on 𝐡(𝑅), then the inequality ξ‚€π‘ξ‚π‘˜βˆ’π‘›π‘ξ€œπ΅(𝑅)|𝐱|π‘βˆ’π‘˜ξƒ¬ξ‚΅1βˆ’|𝐱|𝑅(π‘˜βˆ’π‘›)/π‘ξƒ­π‘“π‘ξ€œ(𝐱)π‘‘π±βˆ’π΅(𝑅)|𝐱|βˆ’π‘˜(𝐻𝑓(𝐱))𝑝β‰₯||||ξ‚€π‘π‘‘π±ξ‚π‘˜βˆ’π‘›π‘βˆ’1ξ€œπ΅(𝑅)|𝐱|βˆ’π‘˜ξ€œ10𝑑((π‘˜βˆ’π‘›)/𝑝)βˆ’1|||||𝐱|π‘π‘‘π‘›βˆ’π‘˜+𝑝𝑓𝑝(𝑑𝐱)βˆ’π‘˜βˆ’π‘›π‘ξ‚Άπ‘(𝐻𝑓(𝐱))𝑝||||βˆ’||𝑝||ξ€œπ‘‘π‘‘π‘‘π±π΅(𝑅)|𝐱|βˆ’π‘˜(𝐻𝑓(𝐱))π‘βˆ’1β‹…ξ€œ10|||||𝐱|𝑓(𝑑𝐱)βˆ’π‘˜βˆ’π‘›π‘π‘‘((π‘˜βˆ’π‘›)/𝑝)βˆ’1||||||||𝐻𝑓(𝐱)𝑑𝑑𝑑𝐱(3.5) holds, where ξ€œπ»π‘“(𝐱)=|𝐱|10𝑓(𝑑𝐱)𝑑𝑑,𝐱∈𝐡(𝑅).(3.6)(ii)If 0≀𝑅<∞, 𝑝/(π‘˜βˆ’π‘›)<0, and 𝑓 is a nonnegative measurable function on ℝ𝑛⧡𝐡(𝑅), then ξ‚€π‘ξ‚π‘›βˆ’π‘˜π‘ξ€œβ„π‘›β§΅π΅(𝑅)|𝐱|π‘βˆ’π‘˜ξƒ¬ξ‚΅π‘…1βˆ’ξ‚Ά|𝐱|(π‘›βˆ’π‘˜)/π‘ξƒ­π‘“π‘ξ€œ(𝐱)π‘‘π±βˆ’β„π‘›β§΅π΅(𝑅)|𝐱|βˆ’π‘˜ξ‚€ξ‚ξ‚π»π‘“(𝐱)𝑝β‰₯||||ξ‚€π‘π‘‘π±ξ‚π‘›βˆ’π‘˜π‘βˆ’1ξ€œβ„π‘›β§΅π΅(𝑅)|𝐱|βˆ’π‘˜ξ€œβˆž1𝑑((π‘˜βˆ’π‘›)/𝑝)βˆ’1|||||𝐱|π‘π‘‘π‘›βˆ’π‘˜+𝑝𝑓𝑝(𝑑𝐱)βˆ’π‘›βˆ’π‘˜π‘ξ‚Άπ‘ξ‚€ξ‚ξ‚π»π‘“(𝐱)𝑝||||βˆ’||𝑝||ξ€œπ‘‘π‘‘π‘‘π±β„π‘›β§΅π΅(𝑅)|𝐱|βˆ’π‘˜ξ‚€ξ‚ξ‚π»π‘“(𝐱)π‘βˆ’1β‹…ξ€œβˆž1|||||𝐱|𝑓(𝑑𝐱)βˆ’π‘›βˆ’π‘˜π‘π‘‘((π‘˜βˆ’π‘›)/𝑝)βˆ’1||||||||,𝐻𝑓(𝐱)𝑑𝑑𝑑𝐱(3.7) where ξ‚ξ€œπ»π‘“(𝐱)=|𝐱|∞1𝑓(𝑑𝐱)𝑑𝑑,π±βˆˆβ„π‘›β§΅π΅(𝑅).(3.8)For π‘βˆˆβŸ¨0,1] relations (3.5) and (3.7) hold with swapped order of the integrals on their respective left-hand sides.

Proof. Follows from Theorems 2.1 and 3.1 by rewriting (2.7), that is, (3.1) and (3.2), with some particular parameters. Namely, let 𝑋=ℝ𝑛, 𝐼=[0,∞⟩, 𝑒(𝐱)=|𝐱|βˆ’π‘›, π‘‘πœˆ(𝐱)=𝑑𝐱, and Ξ¦(π‘₯)=π‘₯𝑝, 𝑝≠0, that is, πœ‘(π‘₯)=𝑝π‘₯π‘βˆ’1. In the case (i), let also Ξ©=𝐡(𝑅(π‘˜βˆ’π‘›)/𝑝), π‘‘πœ†(𝑑)=πœ’βŸ¨0,1⟩(𝑑)𝑑𝑑, and π‘‘πœ‡(𝐱)=πœ’π΅(R(π‘˜βˆ’π‘›)/𝑝)(𝐱)𝑑𝐱. Then we have 𝐿=1,(π‘‘πœ‡π‘‘/π‘‘πœˆ)(𝐱)=π‘‘βˆ’π‘›πœ’π΅(𝑑𝑅(π‘˜βˆ’π‘›)/𝑝)(𝐱), and ξ€œπ‘£(𝐱)=10|||1𝑑𝐱|||βˆ’π‘›π‘‘βˆ’π‘›πœ’π΅(𝑑𝑅(π‘˜βˆ’π‘›)/𝑝)(𝐱)𝑑𝑑=|𝐱|βˆ’π‘›ξ‚΅1βˆ’|𝐱|𝑅(π‘˜βˆ’π‘›)/𝑝𝑅,𝐱∈𝐡(π‘˜βˆ’π‘›)/𝑝.(3.9) Replace the function 𝑓 in (2.7) with the function π‘”βˆΆπ΅(𝑅(π‘˜βˆ’π‘›)/𝑝)→ℝ, 𝑔(𝐱)=|𝐱|(𝑝/(π‘˜βˆ’π‘›))βˆ’1𝑓(|𝐱|(𝑝/(π‘˜βˆ’π‘›))βˆ’1𝐱). Then 𝐴𝑔(𝐱)=|𝐱|(𝑝/(π‘˜βˆ’π‘›))βˆ’1ξ€œ10𝑑(𝑝/(π‘˜βˆ’π‘›))βˆ’1𝑓𝑑𝑝/(π‘˜βˆ’π‘›)|𝐱|𝑝/(π‘˜βˆ’π‘›)𝐱=|𝐱|π‘‘π‘‘π‘˜βˆ’π‘›π‘1ξ€œ|𝐱||𝐱|𝑝/(π‘˜βˆ’π‘›)0π‘“ξ‚΅π‘Ÿπ±ξ‚Ά|𝐱|π‘‘π‘Ÿ,(3.10) where we applied the substitution π‘Ÿ=(𝑑|𝐱|)𝑝/(π‘˜βˆ’π‘›). In this setting, by using polar coordinates, the first integral on the left-hand side of inequality (2.7) becomes ξ€œπ΅(𝑅(π‘˜βˆ’π‘›)/𝑝)|𝐱|𝑝((𝑝/(π‘˜βˆ’π‘›))βˆ’1)βˆ’π‘›ξ‚΅1βˆ’|𝐱|𝑅(π‘˜βˆ’π‘›)/𝑝𝑓𝑝|𝐱|(𝑝/(π‘˜βˆ’π‘›))βˆ’1𝐱=ξ€œπ‘‘π±π‘†π‘›βˆ’1ξ€œπ‘‘π‘†π‘…(π‘˜βˆ’π‘›)/𝑝0π‘Ÿπ‘((𝑝/(π‘˜βˆ’π‘›))βˆ’1)βˆ’1ξ‚΅π‘Ÿ1βˆ’π‘…(π‘˜βˆ’π‘›)/π‘ξ‚Άπ‘“π‘ξ€·π‘Ÿπ‘/(π‘˜βˆ’π‘›)𝑆=π‘‘π‘Ÿπ‘˜βˆ’π‘›π‘ξ€œπ‘†π‘›βˆ’1ξ€œπ‘‘π‘†π‘…0π‘‘π‘›βˆ’1+π‘βˆ’π‘˜ξ‚Έξ‚€π‘‘1βˆ’π‘…ξ‚(π‘˜βˆ’π‘›)/𝑝𝑓𝑝=(𝑑𝑆)π‘‘π‘‘π‘˜βˆ’π‘›π‘ξ€œπ΅(𝑅)|𝐱|π‘βˆ’π‘˜ξƒ¬ξ‚΅1βˆ’|𝐱|𝑅(π‘˜βˆ’π‘›)/𝑝𝑓𝑝(𝐱)𝑑𝐱,(3.11) while the second integral on the left-hand side of (2.7) reduces to ξ‚΅π‘˜βˆ’π‘›π‘ξ‚Άπ‘ξ€œπ΅(𝑅(π‘˜βˆ’π‘›)/𝑝)|𝐱|βˆ’π‘›βˆ’π‘ξƒ©ξ€œ|𝐱|𝑝/(π‘˜βˆ’π‘›)0π‘“ξ‚΅π‘Ÿπ±ξ‚Άξƒͺ|𝐱|π‘‘π‘Ÿπ‘=ξ‚΅π‘‘π±π‘˜βˆ’π‘›π‘ξ‚Άπ‘ξ€œπ‘†π‘›βˆ’1ξ€œπ‘‘π‘†π‘…(π‘˜βˆ’π‘›)/𝑝0π‘‘βˆ’π‘βˆ’1ξƒ©ξ€œπ‘‘π‘/(π‘˜βˆ’π‘›)0ξƒͺ𝑓(π‘Ÿπ‘†)π‘‘π‘Ÿπ‘=ξ‚΅π‘‘π‘‘π‘˜βˆ’π‘›π‘ξ‚Άπ‘+1ξ€œπ‘†π‘›βˆ’1ξ€œπ‘‘π‘†π‘…0π‘ π‘›βˆ’π‘˜βˆ’1ξ‚΅ξ€œπ‘ 0𝑓(π‘Ÿπ‘†)π‘‘π‘Ÿπ‘=ξ‚΅π‘‘π‘ π‘˜βˆ’π‘›π‘ξ‚Άπ‘+1ξ€œπ΅(𝑅)|𝐱|βˆ’π‘˜(𝐻𝑓(𝐱))𝑝𝑑𝐱.(3.12) Analogously, on the right-hand side of (2.7) we get ||||ξ€œπ΅(𝑅(π‘˜βˆ’π‘›)/𝑝)|𝐱|βˆ’π‘›βˆ’π‘ξ€œ10||||𝑓𝑝𝑑𝑝/(π‘˜βˆ’π‘›)|𝐱|𝑝/(π‘˜βˆ’π‘›)𝑑𝑝((𝑝/(π‘˜βˆ’π‘›))βˆ’1)|𝐱|𝑝(𝑝/(π‘˜βˆ’π‘›))π±ξ‚Άβˆ’ξ‚΅|𝐱|π‘˜βˆ’π‘›π‘ξ‚Άπ‘ξƒ©ξ€œ|𝐱|𝑝/(π‘˜βˆ’π‘›)0π‘“ξ‚΅π‘Ÿπ±ξ‚Άξƒͺ|𝐱|π‘‘π‘Ÿπ‘|||||βˆ’ξ‚΅π‘‘π‘‘π‘‘π±π‘˜βˆ’π‘›π‘ξ‚Άπ‘βˆ’1||𝑝||ξ€œπ΅(𝑅(π‘˜βˆ’π‘›)/𝑝)|𝐱|βˆ’π‘›βˆ’π‘ξƒ©ξ€œ|𝐱|𝑝/(π‘˜βˆ’π‘›)0π‘“ξ‚΅π‘Ÿ|𝐱ξƒͺ𝐱|π‘‘π‘Ÿπ‘βˆ’1β‹…ξ€œ10||||𝑑(𝑝/(π‘˜βˆ’π‘›))βˆ’1|𝐱|𝑝/(π‘˜βˆ’π‘›)𝑓𝑑𝑝/(π‘˜βˆ’π‘›)|𝐱|𝑝/(π‘˜βˆ’π‘›)π±ξ‚Άβˆ’|𝐱|π‘˜βˆ’π‘›π‘ξ€œ|𝐱|𝑝/(π‘˜βˆ’π‘›)0π‘“ξ‚΅π‘Ÿπ±ξ‚Ά||||||||=|||||𝐱|π‘‘π‘Ÿπ‘‘π‘‘π‘‘π±π‘˜βˆ’π‘›π‘ξ€œπ‘†π‘›βˆ’1ξ€œπ‘‘π‘†π‘…0π‘§π‘›βˆ’π‘˜βˆ’1ξ€œ10||||𝑧𝑝𝑑𝑝((𝑝/(π‘˜βˆ’π‘›))βˆ’1)𝑓𝑝𝑑𝑝/(π‘˜βˆ’π‘›)ξ€Έβˆ’ξ‚΅π‘§π‘†π‘˜βˆ’π‘›π‘ξ‚Άπ‘ξ‚΅ξ€œπ‘§0𝑓(π‘Ÿπ‘†)π‘‘π‘Ÿπ‘||||βˆ’ξ‚΅π‘‘π‘‘π‘‘π‘§π‘˜βˆ’π‘›π‘ξ‚Άπ‘||𝑝||ξ€œπ‘†π‘›βˆ’1ξ€œπ‘‘π‘†π‘…0π‘§π‘›βˆ’π‘˜βˆ’1ξ‚΅ξ€œπ‘§0𝑓(π‘Ÿπ‘†)π‘‘π‘Ÿπ‘βˆ’1β‹…ξ€œ10||||𝑑(𝑝/(π‘˜βˆ’π‘›))βˆ’1𝑑𝑧𝑓𝑝/(π‘˜βˆ’π‘›)ξ€Έβˆ’π‘§π‘†π‘˜βˆ’π‘›π‘ξ€œπ‘§0||||||||=||||𝑓(π‘Ÿπ‘†)π‘‘π‘Ÿπ‘‘π‘‘π‘‘π‘§π‘˜βˆ’π‘›π‘ξ‚Ά2ξ€œπ΅(𝑅)|𝐱|βˆ’π‘˜ξ€œ10𝑑((π‘˜βˆ’π‘›)/𝑝)βˆ’1|||||𝐱|π‘π‘‘π‘›βˆ’π‘˜+𝑝𝑓𝑝(𝑑𝐱)βˆ’π‘˜βˆ’π‘›π‘ξ‚Άπ‘(𝐻𝑓(𝐱))𝑝||||βˆ’ξ‚΅π‘‘π‘‘π‘‘π±π‘˜βˆ’π‘›π‘ξ‚Άπ‘+1||𝑝||ξ€œπ΅(𝑅)|𝐱|βˆ’π‘˜(𝐻𝑓(𝐱))π‘βˆ’1β‹…ξ€œ10|||||𝐱|𝑓(𝑑𝐱)βˆ’π‘˜βˆ’π‘›π‘π‘‘((π‘˜βˆ’π‘›)/𝑝)βˆ’1||||||||.𝐻𝑓(𝐱)𝑑𝑑𝑑𝐱(3.13) Finally, (3.5) holds by combining (3.11), (3.12), and (3.13).
To obtain relation (3.7), that is, the case (ii), we consider Ξ©=ℝ𝑛⧡𝐡(𝑅(π‘›βˆ’π‘˜)/𝑝), π‘‘πœ†(𝑑)=πœ’βŸ¨1,∞⟩(𝑑)(𝑑𝑑/𝑑2), and π‘‘πœ‡(𝐱)=πœ’β„π‘›β§΅π΅(𝑅(π‘›βˆ’π‘˜)/𝑝)(𝐱)𝑑𝐱. As in the case (i), then we have 𝐿=1, (π‘‘πœ‡π‘‘/π‘‘πœˆ)(𝐱)=π‘‘βˆ’π‘›πœ’β„π‘›β§΅π΅(𝑑𝑅(π‘›βˆ’π‘˜)/𝑝)(𝐱), and 𝑣(𝐱)=|𝐱|βˆ’π‘›ξ‚΅π‘…1βˆ’(π‘›βˆ’π‘˜)/𝑝|𝐱|,π±βˆˆβ„π‘›ξ€·π‘…β§΅π΅(π‘›βˆ’π‘˜)/𝑝.(3.14) Inequality (3.7) now follows by rewriting (2.7) with the above parameters and with the function π‘”βˆΆβ„π‘›β§΅π΅(𝑅(π‘›βˆ’π‘˜)/𝑝)→ℝ, 𝑔(𝐱)=|𝐱|(𝑝/(π‘›βˆ’π‘˜))+1𝑓(|𝐱|(𝑝/(π‘›βˆ’π‘˜))βˆ’1𝐱) instead of 𝑓, and by using analogous techniques as in the proof of inequality (3.5).

Observe that the right-hand sides of inequalities (3.5) and (3.7) are nonnegative. Moreover, the constants involved in their left-hand sides are shown to be the best possible. That result is given in the following theorem.

Theorem 3.3. Let π‘›βˆˆβ„•, π‘βˆˆβ„β§΅[0,1⟩, and π‘˜βˆˆβ„, π‘˜β‰ π‘›. (i)If 0<π‘…β‰€βˆž, 𝑝/(π‘˜βˆ’π‘›)>0, 𝑓 is a nonnegative function on 𝐡(𝑅), and 𝐻𝑓 is defined by (3.6), then ξ€œπ΅(𝑅)|𝐱|βˆ’π‘˜(𝐻𝑓(𝐱))π‘ξ‚€π‘π‘‘π±β‰€ξ‚π‘˜βˆ’π‘›π‘ξ€œπ΅(𝑅)|𝐱|π‘βˆ’π‘˜ξƒ¬ξ‚΅1βˆ’|𝐱|𝑅(π‘˜βˆ’π‘›)/𝑝𝑓𝑝(𝐱)𝑑𝐱.(3.15)(ii)If 0≀𝑅<∞, 𝑝/(π‘˜βˆ’π‘›)<0, 𝑓 is a nonnegative function on ℝ𝑛⧡𝐡(𝑅), and 𝐻𝑓 is given by (3.8), thenξ€œβ„π‘›β§΅π΅(𝑅)|𝐱|βˆ’π‘˜ξ‚€ξ‚ξ‚π»π‘“(𝐱)π‘β‰€ξ‚€π‘π‘‘π±ξ‚π‘›βˆ’π‘˜π‘ξ€œβ„π‘›β§΅π΅(𝑅)|𝐱|π‘βˆ’π‘˜ξƒ¬ξ‚΅π‘…1βˆ’ξ‚Ά|𝐱|(π‘›βˆ’π‘˜)/𝑝𝑓𝑝(𝐱)𝑑𝐱.(3.16) The constant |𝑝/(π‘˜βˆ’π‘›)|𝑝 is the best possible for both inequalities. For π‘βˆˆβŸ¨0,1], the signs of inequality in (3.15) and (3.16) are reversed.

Proof. We only need to prove that |𝑝/(π‘˜βˆ’π‘›)|𝑝 is the best possible constant for inequalities (3.15) and (3.16). Consider the case (i) first. For a sufficiently small πœ€>0, and the function π‘“πœ€βˆΆπ΅(𝑅)→ℝ defined by π‘“πœ€(𝐱)=|𝐱|((π‘˜βˆ’π‘›+πœ€)/𝑝)βˆ’1, the left-hand side of (3.15) is equal to πΏπœ€=ξ€œπ΅(𝑅)|𝐱|βˆ’π‘›+πœ€ξ‚΅ξ€œ10𝑑((π‘˜βˆ’π‘›+πœ€)/𝑝)βˆ’1𝑑𝑑𝑝𝑝𝑑𝐱=ξ‚π‘˜βˆ’π‘›+πœ€π‘ξ€œπ΅(𝑅)|𝐱|βˆ’π‘›+πœ€=ξ‚€π‘π‘‘π±ξ‚π‘˜βˆ’π‘›+πœ€π‘ξ€œπ‘†π‘›βˆ’1ξ€œπ‘‘π‘†π‘…0π‘Ÿπœ€βˆ’1ξ‚€π‘π‘‘π‘Ÿ=ξ‚π‘˜βˆ’π‘›+πœ€π‘||π‘†π‘›βˆ’1||β‹…π‘…πœ€πœ€,(3.17) while on the right-hand side of (3.15) we get π‘…πœ€=ξ‚€π‘ξ‚π‘˜βˆ’π‘›π‘ξ€œπ΅(𝑅)|𝐱|βˆ’π‘›+πœ€ξƒ¬ξ‚΅1βˆ’|𝐱|𝑅(π‘˜βˆ’π‘›)/π‘ξƒ­β‰€ξ‚€π‘π‘‘π±ξ‚π‘˜βˆ’π‘›π‘ξ€œπ΅(𝑅)|𝐱|βˆ’π‘›+πœ€ξ‚€π‘π‘‘π±=ξ‚π‘˜βˆ’π‘›π‘||π‘†π‘›βˆ’1||β‹…π‘…πœ€πœ€.(3.18) Therefore, 1β‰€π‘…πœ€/πΏπœ€β‰€((π‘˜βˆ’π‘›+πœ€)/(π‘˜βˆ’π‘›))𝑝. Since ((π‘˜βˆ’π‘›+πœ€)/(π‘˜βˆ’π‘›))π‘β†˜1, as πœ€β†˜0, the constant (𝑝/(π‘˜βˆ’π‘›))𝑝 is the best possible for (3.15). The proof that the constant (𝑝/(π‘›βˆ’π‘˜))𝑝 is the best possible for (3.16) follows the same lines, considering the function π‘“πœ€βˆΆβ„π‘›β§΅π΅(𝑅)→ℝ, π‘“πœ€(𝐱)=|𝐱|((π‘˜βˆ’π‘›βˆ’πœ€)/𝑝)βˆ’1.

If 𝑅=∞ in (3.15) and 𝑅=0 in (3.16), we immediately get the following new multidimensional Hardy-type inequality.

Corollary 3.4. Let π‘›βˆˆβ„•, π‘βˆˆβ„β§΅[0,1⟩, and π‘˜βˆˆβ„, π‘˜β‰ π‘›. Let π‘“βˆΆβ„π‘›β†’β„ be a nonnegative measurable function and 𝐻𝑓 and 𝐻𝑓 be defined by (3.6) and (3.8), respectively. If 𝑝/(π‘˜βˆ’π‘›)>0, then the inequality ξ€œβ„|𝐱|βˆ’π‘˜(𝐻𝑓(𝐱))π‘ξ‚€π‘π‘‘π±β‰€ξ‚π‘˜βˆ’π‘›π‘ξ€œβ„π‘›|𝐱|π‘βˆ’π‘˜π‘“π‘(𝐱)𝑑𝐱(3.19) holds, while for 𝑝/(π‘˜βˆ’π‘›)<0 one has ξ€œβ„π‘›|𝐱|βˆ’π‘˜ξ‚€ξ‚ξ‚π»π‘“(𝐱)π‘ξ‚€π‘π‘‘π±β‰€ξ‚π‘›βˆ’π‘˜π‘ξ€œβ„π‘›|𝐱|π‘βˆ’π‘˜π‘“π‘(𝐱)𝑑𝐱.(3.20) The constant |𝑝/(π‘˜βˆ’π‘›)|𝑝 is the best possible for both inequalities. Moreover, for π‘βˆˆβŸ¨0,1], the signs of inequality in (3.19) and (3.20) are reversed.

We continue our analysis by obtaining the corresponding refined PΓ³lya-Knopp-type inequality.

Theorem 3.5. Let π‘›βˆˆβ„•. (i)If 0<π‘…β‰€βˆž, 𝑓 is a positive measurable function on 𝐡(𝑅), and ξ‚΅ξ€œπΊπ‘“(𝐱)=exp10ξ‚Άlog𝑓(𝑑𝐱)𝑑𝑑,𝐱∈𝐡(𝑅),(3.21) then the inequality π‘’π‘›ξ€œπ΅(𝑅)ξ‚΅1βˆ’|𝐱|π‘…ξ‚Άξ€œπ‘“(𝐱)π‘‘π±βˆ’π΅(𝑅)β‰₯||||ξ€œπΊπ‘“(𝐱)𝑑𝐱𝐡(𝑅)ξ€œ10||(𝑒𝑑)𝑛||ξ€œπ‘“(𝑑𝐱)βˆ’πΊπ‘“(𝐱)π‘‘π‘‘π‘‘π±βˆ’π΅(𝑅)ξ€œπΊπ‘“(𝐱)10||||log(𝑒𝑑)𝑛𝑓(𝑑𝐱)||||||||𝐺𝑓(𝐱)𝑑𝑑𝑑𝐱(3.22) holds. (ii)If 0≀𝑅<∞, 𝑓 is a positive measurable function on ℝ𝑛⧡𝐡(𝑅), and ξ‚ξ‚΅ξ€œG𝑓(𝐱)=exp∞1log𝑓(𝑑𝐱)𝑑𝑑𝑑2ξ‚Ά,π±βˆˆβ„π‘›β§΅π΅(𝑅),(3.23) then π‘’βˆ’π‘›ξ€œβ„π‘›β§΅π΅(R)𝑅1βˆ’ξ‚Άξ€œ|𝐱|𝑓(𝐱)π‘‘π±βˆ’β„π‘›β§΅π΅(𝑅)β‰₯||||ξ€œπΊπ‘“(𝐱)𝑑𝐱ℝ𝑛⧡𝐡(𝑅)ξ€œβˆž1|||𝑑𝑒𝑛𝑓|||(𝑑𝐱)βˆ’πΊπ‘“(𝐱)𝑑𝑑𝑑2βˆ’ξ€œπ‘‘π±β„π‘›β§΅π΅(𝑅)ξ‚ξ€œπΊπ‘“(𝐱)∞1||||log(𝑒𝑑)𝑛𝑓(𝑑𝐱)||||𝐺𝑓(𝐱)𝑑𝑑𝑑2||||.𝑑𝐱(3.24)

Proof. Follows from Theorems 2.1 and 3.1 by considering 𝑋=ℝ𝑛, 𝐼=ℝ, 𝑒(𝐱)=|𝐡(|𝐱|)|βˆ’1, π‘‘πœˆ(𝐱)=𝑑𝐱, and Ξ¦(π‘₯)=πœ‘(π‘₯)=𝑒π‘₯. To get (3.22), we also set Ξ©=𝐡(𝑅), π‘‘πœ†(𝑑)=πœ’βŸ¨0,1⟩(𝑑)𝑑𝑑, and π‘‘πœ‡(𝐱)=πœ’π΅(𝑅)(𝐱)𝑑𝐱. In that case, we have 𝐿=1, (π‘‘πœ‡π‘‘/π‘‘πœˆ)(𝐱)=π‘‘βˆ’π‘›πœ’π΅(𝑑𝑅)(𝐱), and ξ€œπ‘£(𝐱)=101||𝐡||||ξ€Έ||𝑑(1/𝑑)π±βˆ’π‘›πœ’π΅(𝑑𝑅)1(𝐱)𝑑𝑑=||||ξ€œπ΅(|𝐱|)10πœ’π΅(𝑑𝑅)=1(𝐱)𝑑𝑑||||𝐡(|𝐱|)1βˆ’|𝐱|𝑅,𝐱∈𝐡(𝑅).(3.25) Further, replace the function 𝑓 in (2.7) with the function 𝐱↦log(|𝐡(|𝐱|)|𝑓(𝐱)). The first integral on the left-hand side of (2.7) then becomes ξ€œπ΅(𝑅)1||||𝐡(|𝐱|)1βˆ’|𝐱|𝑅||||ξ€œπ΅(|𝐱|)𝑓(𝐱)𝑑𝐱=𝐡(𝑅)ξ‚΅1βˆ’|𝐱|𝑅𝑓(𝐱)𝑑𝐱,(3.26) the corresponding second integral reduces to ξ€œπ΅(𝑅)1||||ξ‚΅ξ€œπ΅(|𝐱|)exp10𝑑log𝑛||||ξ€Έξ‚Ά=ξ€œπ΅(|𝐱|)𝑓(𝑑𝐱)𝑑𝑑𝑑𝐱𝐡(𝑅)1||||ξ‚΅||||ξ€œπ΅(|𝐱|)expβˆ’π‘›+log𝐡(|𝐱|)+log10ξ‚Άlog𝑓(𝑑𝐱)𝑑𝑑𝑑𝐱=π‘’βˆ’π‘›ξ€œπ΅(𝑅)ξ‚΅ξ€œexp10ξ‚Άlog𝑓(𝑑𝐱)𝑑𝑑𝑑𝐱=π‘’βˆ’π‘›ξ€œπ΅(𝑅)𝐺𝑓(𝐱)𝑑𝐱,(3.27) while on the right-hand side of (2.7) we obtain ||||ξ€œπ΅(𝑅)1||||ξ€œπ΅(|𝐱|)10||𝑑𝑛||𝐡||𝑓(|𝐱|)(𝑑𝐱)βˆ’π‘’βˆ’π‘›||𝐡||||βˆ’ξ€œ(|𝐱|)𝐺𝑓(𝐱)𝑑𝑑𝑑𝐱𝐡(𝑅)1||||ξ€œπ΅(|𝐱|)10||π‘’βˆ’π‘›||||||β‹…||||𝑑𝐡(|𝐱|)𝐺𝑓(𝐱)log𝑛||||ξ€Έ||||βˆ’ξ€œπ΅(|𝐱|)𝑓(𝑑𝐱)+π‘›βˆ’log𝐡(|𝐱|)10||||||||=||||ξ€œlog𝑓(𝑑𝐱)𝑑𝑑𝑑𝑑𝑑𝐱𝐡(𝑅)ξ€œ10||𝑑𝑛𝑓(𝑑𝐱)βˆ’π‘’βˆ’π‘›||𝐺𝑓(𝐱)π‘‘π‘‘π‘‘π±βˆ’π‘’βˆ’π‘›ξ€œπ΅(𝑅)ξ€œπΊπ‘“(𝐱)10||||||||||||||||=||||ξ€œπ‘›log𝑑+log𝐡(|𝐱|)+log(𝑓(𝑑𝐱))+π‘›βˆ’log𝐡(|𝐱|)βˆ’log𝐺𝑓(𝐱)𝑑𝑑𝑑𝐱𝐡(𝑅)ξ€œ10||𝑑𝑛𝑓(𝑑𝐱)βˆ’π‘’βˆ’π‘›||𝐺𝑓(𝐱)π‘‘π‘‘π‘‘π±βˆ’π‘’βˆ’π‘›ξ€œπ΅(𝑅)ξ€œπΊπ‘“(𝐱)10||||log(𝑒𝑑)𝑛𝑓(𝑑𝐱)||||||||.𝐺𝑓(𝐱)𝑑𝑑𝑑𝐱(3.28) Finally, (3.22) holds by combining (3.26), (3.27), and (3.28) and then multiplying the whole inequality by 𝑒𝑛.
Inequality (3.24) can be derived by a similar technique by taking Ξ©=ℝ𝑛⧡𝐡(𝑅), π‘‘πœ†(𝑑)=πœ’[1,∞⟩(𝑑)(𝑑𝑑/𝑑2), and π‘‘πœ‡(𝐱)=πœ’β„π‘›β§΅π΅(𝑅)(𝐱)𝑑𝐱. Then 𝐿=1, (π‘‘πœ‡π‘‘/π‘‘πœˆ)(𝐱)=π‘‘βˆ’π‘›πœ’β„π‘›β§΅π΅(𝑑𝑅)(𝐱) and ξ€œπ‘£(𝐱)=∞1|||𝐡1𝑑||||𝐱|βˆ’1π‘‘βˆ’π‘›πœ’β„π‘›β§΅π΅(𝑑𝑅)(𝐱)𝑑𝑑𝑑2=1||||ξ€œπ΅(|𝐱|)1|𝐱|/𝑅𝑑𝑑𝑑2=1||||𝑅𝐡(|𝐱|)1βˆ’ξ‚Ά|𝐱|,π±βˆˆβ„π‘›β§΅π΅(𝑅),(3.29) so (3.24) follows by replacing the function f in (2.7) with 𝐱↦log(|𝐡(|𝐱|)|𝑓(𝐱)).

As a direct consequence of Theorem 3.5, we obtain the following strengthened PΓ³lya-Knopp-type inequality.

Theorem 3.6. Let π‘›βˆˆβ„•. (i)If 0<π‘…β‰€βˆž, 𝑓 is a positive measurable function on 𝐡(𝑅), and 𝐺𝑓 is defined by (3.21), then the inequality ξ€œπ΅(𝑅)𝐺𝑓(𝐱)π‘‘π±β‰€π‘’π‘›ξ€œπ΅(R)ξ‚΅1βˆ’|𝐱|𝑅𝑓(𝐱)𝑑𝐱(3.30) holds and the constant 𝑒𝑛 is the best possible.(ii) If 0≀𝑅<∞, 𝑓 is a positive measurable function on ℝ𝑛⧡𝐡(𝑅), and 𝐺𝑓 is defined by (3.23), then the inequality ξ€œβ„π‘›β§΅π΅(𝑅)𝐺𝑓(𝐱)π‘‘π±β‰€π‘’βˆ’π‘›ξ€œβ„π‘›β§΅π΅(𝑅)𝑅1βˆ’ξ‚Ά|𝐱|𝑓(𝐱)𝑑𝐱(3.31) holds and the constant π‘’βˆ’π‘› is the best possible.

Proof. Since the right-hand sides of (3.22) and (3.24) are nonnegative, inequalities (3.30) and (3.31) are their respective direct consequences. Now, we discuss the best possible constant for (3.30). For arbitrary πœ€>0, let the function π‘“πœ€βˆΆπ΅(𝑅)→ℝ be defined by π‘“πœ€(𝐱)=π‘’βˆ’π‘›|𝐡(|𝐱|)|πœ€βˆ’1. Calculating the left-hand side of (3.30) for π‘“πœ€, we obtain πΏπœ€=ξ€œπ΅(𝑅)ξ‚΅ξ€œexp10𝑒logβˆ’π‘›||||𝐡(|𝑑𝐱|)πœ€βˆ’1𝑑𝑑𝑑𝐱=π‘’βˆ’π‘›ξ€œπ΅(𝑅)ξ‚΅ξ€œexp𝑛(πœ€βˆ’1)10||||ξ‚Άlog𝑑𝑑𝑑+(πœ€βˆ’1)log𝐡(|𝐱|)𝑑𝐱=π‘’βˆ’π‘›πœ€ξ€œπ΅(𝑅)||||𝐡(|𝐱|)πœ€βˆ’1𝑑𝐱=π‘’βˆ’π‘›πœ€||||𝐡(𝑅)πœ€πœ€.(3.32) On the other hand, the right-hand side of (3.30), rewritten for π‘“πœ€, can be estimated as π‘…πœ€=π‘’π‘›β‹…π‘’βˆ’π‘›ξ€œπ΅(𝑅)ξ‚΅1βˆ’|𝐱|𝑅||||𝐡(|𝐱|)πœ€βˆ’1ξ€œπ‘‘π±β‰€π΅(𝑅)||||𝐡(|𝐱|)πœ€βˆ’1||||𝑑𝐱=𝐡(𝑅)πœ€πœ€.(3.33) Since 1≀(π‘…πœ€/πΏπœ€)β‰€π‘’π‘›πœ€β†˜1, as πœ€β†˜0, 𝑒𝑛 is the best possible constant for inequality (3.30). The proof that π‘’βˆ’π‘› is the best possible constant for (3.31) is similar, if the function π‘“πœ€βˆΆβ„π‘›β§΅π΅(𝑅)→ℝ, π‘“πœ€=𝑒𝑛|𝐡(|𝐱|)|βˆ’πœ€βˆ’1, is considered.

For 𝑅=∞ in (3.30) and 𝑅=0 in (3.31), we get a new multidimensional PΓ³lya-Knopp-type inequality.

Corollary 3.7. If 𝑓 is a positive measurable function on ℝ𝑛, and 𝐺𝑓, 𝐺𝑓 are, respectively, defined by (3.21) and (3.23), then the inequalities ξ€œβ„π‘›πΊπ‘“(𝐱)π‘‘π±β‰€π‘’π‘›ξ€œβ„π‘›ξ€œπ‘“(𝐱)𝑑𝐱,ℝ𝑛𝐺𝑓(𝐱)π‘‘π±β‰€π‘’βˆ’π‘›ξ€œβ„π‘›π‘“(𝐱)𝑑𝐱(3.34) hold. The constants 𝑒𝑛 and π‘’βˆ’π‘› are the best possible.

Remark 3.8. Notice that (3.30) and (3.31), respectively, follow from (3.15) and (3.16) by rewriting those inequalities for π‘˜=𝑝>𝑛, and 𝑓 replaced with 𝑓1/𝑝, and by letting π‘β†’βˆž. In particular, observe that limπ‘β†’βˆž(𝑝/(π‘βˆ’π‘›))𝑝=𝑒𝑛.

Although being related to the setting with balls in ℝ𝑛 centered at the origin, inequalities (3.15), (3.16) and (3.30), (3.31) are not equivalent with the previously obtained Hardy- and PΓ³lya-Knopp-type inequalities (1.13), (1.14), (1.15), and (1.17). Therefore, our inequalities can be considered as a new class of generalizations of the classical Hardy's and PΓ³lya-Knopp's inequalities in a multidimensional setting. However, for 𝑛=1 inequalities of both type coincide. In particular, inequality (3.5) for 𝑛=1 was obtained in [4]. It is given in the following corollary.

Corollary 3.9. Let 0<π‘β‰€βˆž, 𝑓 be a nonnegative function on ⟨0,π‘βŸ©, and 𝑝,π‘˜βˆˆβ„ be such that 0≠𝑝≠1,