Some New Refined General Boas-Type Inequalities

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 Pólya-Knopp-type inequalities related to balls in ℝ^𝑛 and prove that constant factors involved in their right-hand sides are the best possible.

1. Introduction

Boas [1], proved that the inequality^∞0Φ1𝑀^∞0𝑓(𝑡𝑥)𝑑𝑚(𝑡)𝑑𝑥𝑥≤^∞0Φ(𝑓(𝑥))𝑑𝑥𝑥(1.1) holds for all continuous convex functions Φ∶[0,∞⟩→ℝ, measurable nonnegative functions 𝑓∶ℝ₊→ℝ, and nondecreasing bounded functions 𝑚∶[0,∞⟩→ℝ, where 𝑀=𝑚(∞)−𝑚(0)>0, and the inner integral on the left-hand side of (1.1) is the Lebesgue-Stieltjes integral with respect to 𝑚. After its author, inequality (1.1) was named the Boas inequality. In the case of a concave function Φ, 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), Čižmešija et al. [4] considered a general Borel measure 𝜆 on ℝ₊, such that 𝐿=𝜆(ℝ₊)<∞, a convex function Φ on a convex set 𝐼⊆ℝ, and a weight function 𝑢 on ℝ₊, and proved that the inequality^∞0𝑢(𝑥)Φ(𝐴𝑓(𝑥))𝑑𝑥𝑥≤𝐿1^∞0𝑤(𝑥)Φ(𝑓(𝑥))𝑑𝑥𝑥(1.2) holds for all measurable functions 𝑓∶ℝ₊→ℝ with values in 𝐼, where 𝐴𝑓(𝑥)=(1/𝐿)∫^∞0𝑓(𝑡𝑥)𝑑𝜆(𝑡) and 𝑤(𝑥)=∫^∞0𝑢(𝑥/𝑡)𝑑𝜆(𝑡)<∞, 𝑥∈ℝ₊. 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 𝜇 and 𝜈 on a topological space 𝑋 and a Borel probability measure 𝜆 on ℝ₊. For a 𝜆-balanced Borel set Ω⊆𝑋 and the measures 𝜇_𝑡≪𝜈, defined for all Borel sets 𝑆⊆𝑋 and 𝑡∈supp𝜆 by 𝜇_𝑡(𝑆)=𝜇(𝑡⁻¹𝑆), he obtained the inequality_ΩΦ^∞0𝑓(𝑡𝐱)𝑑𝜆(𝑡)𝑑𝜇(𝐱)≤_ΩΦ(𝑓(𝐱))^∞0𝑑𝜇_𝑡𝑑𝜈(𝐱)𝑑𝜆(𝑡)𝑑𝜈(𝐱),(1.3) where Φ is a nonnegative convex function on a real interval 𝐼, and 𝑓∶Ω→ℝ is a Borel measurable function with values in 𝐼. A weighted version of (1.3) was given by Čižmešija 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 Čižmešija et al. [9, 10].

Let 0<𝑏≤∞ and 𝑝,𝑘∈ℝ such that 𝑝/(𝑘−1)>0. If 𝑝∈ℝ⧵[0,1], 𝑓 is a nonnegative function, 𝑥^{1−(𝑘/𝑝)}𝑓∈𝐿^𝑝(0,𝑏), and𝐹(𝑥)=^𝑥0𝑓(𝑡)𝑑𝑡,𝑥∈⟨0,𝑏⟩,(1.4) then^𝑏0𝑥^−𝑘𝐹^𝑝(𝑥)𝑑𝑥≤𝑝𝑘−1^𝑝^𝑏01−𝑥𝑏^{(𝑘−1)/𝑝}𝑥^𝑝−𝑘𝑓^𝑝(𝑥)𝑑𝑥(1.5) holds, while for 𝑝∈⟨0,1⟩ the sign of inequality in (1.5) is reversed. Moreover, if 0≤𝑏<∞ and parameters 𝑝∈ℝ⧵[0,1] and 𝑘∈ℝ such that 𝑝/(𝑘−1)<0, then the inequality^∞𝑏𝑥^−k𝐹^𝑝(𝑥)𝑑𝑥≤𝑝1−𝑘^𝑝^∞𝑏1−𝑥𝑏^{(1−𝑘)/𝑝}𝑥^𝑝−𝑘𝑓^𝑝(𝑥)𝑑𝑥(1.6) holds for all nonnegative functions 𝑓 such that 𝑥^{1−(𝑘/𝑝)}𝑓∈𝐿^𝑝(𝑏,∞), where 𝐹(𝑥)=^∞𝑥𝑓(𝑡)𝑑𝑡,𝑥∈⟨𝑏,∞⟩.(1.7) For 𝑝∈⟨0,1⟩ inequality (1.6) holds with the sign of inequality reversed. The constant |𝑝/(𝑘−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<𝑏≤∞, 𝑓∈𝐿¹(0,𝑏) is a positive function, and𝐺(𝑥)=exp1𝑥^𝑥0log𝑓(𝑡)𝑑𝑡,𝑥∈⟨0,𝑏⟩,(1.8) then^𝑏0𝐺(𝑥)𝑑𝑥≤𝑒^𝑏01−𝑥𝑏𝑓(𝑥)𝑑𝑥(1.9) holds, while the inequality^∞𝑏𝐺(𝑥)𝑑𝑥≤1𝑒^∞𝑏1−𝑥𝑏𝑓(𝑥)𝑑𝑥(1.10) holds for 0≤𝑏<∞, 0<𝑓∈𝐿¹(𝑏,∞), and𝐺(𝑥)=exp𝑥^∞𝑥log𝑓(𝑡)𝑑𝑡𝑡²,𝑥∈⟨𝑏,∞⟩.(1.11) For 𝑏=∞ in (1.9) and for 𝑏=0 in (1.10) we, respectively, get the classical Pólya-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 ℝ^𝑛 centered at the origin (see [11] for details). Let 𝑝,𝑘,𝑅∈ℝ such that 𝑝>1, 𝑘≠1, and 𝑅>0. Suppose that 𝑓 is a nonnegative measurable function and the function 𝐹 is defined on ℝ^𝑛 by 𝐹(𝐱)=⎧⎪⎨⎪⎩^𝐵(|𝐱|)𝑓(𝐲)𝑑𝐲,𝑘>1,_{ℝ𝑛⧵𝐵(|𝐱|)}𝑓(𝐲)𝑑𝐲,𝑘<1,(1.12) where 𝐵(|𝐱|) is a ball in ℝ^𝑛 centered at the origin and of radius |𝐱|, while |𝐱| denotes the Euclidean norm of 𝐱∈ℝ^𝑛. If 𝑝/(𝑘−1)>0, then the inequality_𝐵(𝑅)≤||𝐵(|𝐱𝑝|)||^−𝑘𝐹^𝑝(𝐱)𝑑𝐱𝑘−1^𝑝_𝐵(𝑅)1−||||𝐵𝐵((|𝑅𝐱|))||||^{(𝑘−1)/𝑝}||𝐵(|𝐱|)||^𝑝−𝑘𝑓^𝑝(𝐱)𝑑𝐱(1.13) holds, while for 𝑝/(𝑘−1)<0 we have_{ℝ𝑛⧵𝐵(𝑅)}≤||𝐵𝑝(|𝐱|)||^−𝑘𝐹^𝑝(𝐱)𝑑𝐱1−𝑘^𝑝_{ℝ𝑛⧵𝐵(𝑅)}1−||𝐵||𝐵((|𝐱𝑅|))||||^{(1−𝑘)/𝑝}||𝐵(|𝐱|)||^𝑝−𝑘𝑓^𝑝(𝐱)𝑑𝐱.(1.14) Terms |𝐵(|𝐱|)| and |𝐵(𝑅)|, respectively, denote the volumes of 𝐵(|𝐱|) and 𝐵(𝑅). The constant (𝑝/|𝑘−1|)^𝑝 is the best possible for both inequalities. Observe that the first natural generalization of the classical Hardy's inequality to balls in ℝ^𝑛 was given by Christ and Grafakos in [12].

Finally, here we state an 𝑛-dimensional Pólya-Knopp's inequality, related to (1.13) and (1.14),_𝐵(𝑅)𝐺(𝐱)𝑑𝐱<𝑒_𝐵(𝑅)1−||||𝐵𝐵((|𝑅𝐱|))||||𝑓(𝐱)𝑑𝐱(1.15) for a positive function 𝑓 on 𝐵(𝑅) and 𝐺(𝐱)=exp1||𝐵(|𝐱|)||_𝐵(|𝐱|)log𝑓(𝐲)𝑑𝐲,𝐱∈𝐵(𝑅),(1.16) as well as its dual inequality_{ℝ𝑛⧵𝐵(𝑅)}𝐺(𝐱)𝑑𝐱<1𝑒_ℝ^𝑛_{⧵𝐵(𝑅)}1−||||𝐵𝐵((|𝑅𝐱|))||||𝑓(𝐱)𝑑𝐱(1.17) for a positive function 𝑓 on ℝ^𝑛⧵𝐵(𝑅) and 𝐺(𝐱)=exp||𝐵(|𝐱|)||_{ℝ𝑛⧵𝐵(|𝐱|)}log𝑓(𝐲)𝑑𝐲||𝐵(|𝐱|)||²,𝐱∈ℝ^𝑛⧵𝐵(𝑅).(1.18)

Observe that all above-mentioned inequalities of the Hardy and Pólya-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 Pólya-Knopp-type inequalities related to balls in ℝ^𝑛, 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 Pólya-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 0⋅∞, 0/0,𝑎/∞(𝑎∈ℝ), and ∞/∞ are taken to be equal to zero. As usual, by 𝑑𝑥 and 𝑑𝐱 we denote the Lebesgue measure on ℝ and ℝ^𝑛(𝑛∈ℕ,𝑛≥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 𝐼⊆ℝ. In particular, ℝ₊=⟨0,∞⟩. For 𝑅>0, by 𝐵(𝑅) we denote a ball in ℝ^𝑛 centered at the origin and of radius 𝑅, that is, 𝐵(𝑅)={𝐱∈ℝ^𝑛∶|𝐱|≤𝑅}, where |𝐱| denotes the Euclidean norm of 𝐱∈ℝ^𝑛. By its dual set we mean the set ℝ^𝑛⧵𝐵(𝑅)={𝐱∈ℝ^𝑛∶|𝐱|>𝑅}. Finally, by 𝑆^𝑛−1 we denote the surface of the unit ball 𝐵(1) and by |𝑆^𝑛−1| its area. Using polar coordinates in ℝ^𝑛, the volume of the ball B(𝑅) is then ||𝐵(𝑅)||=_𝐵(𝑅)𝑑𝐱=_{|𝐱|≤𝑅}𝑑𝐱=^𝑅0𝑡^𝑛−1_𝑆𝑛−1=𝑑𝑆𝑑𝑡_𝑆𝑛−1^𝑅0𝑡^𝑛−1𝑑𝑡𝑑𝑆=𝑅^𝑛||𝑆^𝑛−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 Φ∶𝐼→ℝ be a convex function. For 𝑥∈𝐼, by 𝜕Φ(𝑥) we denote the subdifferential of Φ at 𝑥, that is, the set𝜕Φ(𝑥)={𝛼∈ℝ∶Φ(𝑦)−Φ(𝑥)−𝛼(𝑦−𝑥)≥0,𝑦∈𝐼}.(2.1) It is well known that 𝜕Φ(𝑥)≠∅ for all 𝑥∈Int𝐼. More precisely, at each point 𝑥∈Int𝐼 we have −∞<Φ^−(𝑥)≤Φ^+(𝑥)<∞, 𝜕Φ(𝑥)=[Φ^−(𝑥),Φ^+(𝑥)], and the set on which Φ is not differentiable is at most countable. Moreover, every function 𝜑∶𝐼→ℝ for which 𝜑(𝑥)∈𝜕Φ(𝑥), whenever 𝑥∈Int𝐼, is increasing on Int𝐼.

On the other hand, if Φ∶𝐼→ℝ is a concave function, that is, −Φ is convex, then 𝜕Φ(𝑥)={𝛼∈ℝ∶Φ(𝑥)−Φ(𝑦)−𝛼(𝑥−𝑦)≥0,𝑦∈𝐼} denotes the superdifferential of Φ at the point 𝑥∈𝐼. For all 𝑥∈Int𝐼, in this setting we have −∞<Φ^+(𝑥)≤Φ^−(𝑥)<∞ and 𝜕Φ(𝑥)=[Φ^+(𝑥),Φ^−(𝑥)]≠∅. Notice that, although the symbol 𝜕Φ(𝑥) has two different notions, it will be clear from the context whether it applies to a convex or to a concave function Φ. 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 𝜆 be a finite Borel measure on ℝ₊. By supp𝜆, we denote its support, that is, the set of all 𝑡∈ℝ₊ such that 𝜆(𝑁_𝑡)>0 holds for all open neighbourhoods 𝑁_𝑡 of 𝑡. Hence,𝐿=_supp𝜆𝑑𝜆(𝑡)=^∞0𝑑𝜆(𝑡)=𝜆ℝ₊<∞.(2.2) Further, let 𝑋 be a topological space equipped with a continuous scalar multiplication (𝑎,𝐱)↦𝑎𝐱∈𝑋, for 𝑎∈ℝ₊ and 𝐱∈𝑋, such that1𝐱=𝐱,𝑎(𝑏𝐱)=(𝑎𝑏)𝐱,𝐱∈𝑋,𝑎,𝑏∈ℝ₊.(2.3) Let a Borel set Ω⊆𝑋 be 𝜆-balanced, that is, let 𝑡Ω={𝑡𝐱∶𝐱∈Ω}⊆Ω hold for all 𝑡∈supp𝜆. For a Borel measurable function 𝑓∶Ω→ℝ, we define its Hardy-Littlewood average, 𝐴𝑓, as𝐴𝑓(𝐱)=1𝐿^∞0𝑓(𝑡𝐱)𝑑𝜆(𝑡),𝐱∈Ω.(2.4) Finally, suppose that 𝜇 and 𝜈 are 𝜎-finite Borel measures on 𝑋. For 𝑡>0 and a Borel set 𝑆⊆𝑋, we define𝜇_𝑡(𝑆)=𝜇1𝑡𝑆.(2.5) Obviously, 𝜇_𝑡 is a 𝜎-finite Borel measure on 𝑋 for all 𝑡∈ℝ₊. Throughout this paper we assume that 𝜇_𝑡 is absolutely continuous with respect to the measure 𝜈, that is, 𝜇_𝑡≪𝜈, for each 𝑡∈supp𝜆. As usual, by 𝑑𝜇_𝑡/𝑑𝜈 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 𝜆 be a finite Borel measure on ℝ₊ and 𝐿 be defined by (2.2). Let 𝜇 and 𝜈 be 𝜎-finite Borel measures on a topological space 𝑋, 𝜇_𝑡 be defined by (2.5), and let 𝜇_𝑡≪𝜈 for all 𝑡∈supp𝜆. Further, let Ω⊆𝑋 be a 𝜆-balanced Borel set and 𝑢 be a nonnegative function on 𝑋, such that 𝑣(𝐱)=^∞0𝑢1𝑡𝐱𝑑𝑑𝜇𝜈^𝑡(𝐱)𝑑𝜆(𝑡)<∞,𝐱∈Ω.(2.6) Suppose that Φ∶𝐼→ℝ is a nonnegative convex function on an interval 𝐼⊆ℝ and 𝜑∶𝐼→ℝ is any function fulfilling 𝜑(𝑥)∈𝜕Φ(𝑥), for all 𝑥∈Int𝐼. If 𝑓∶Ω→ℝ 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 1𝐿_Ω≥1𝑣(𝐱)Φ(𝑓(𝐱))𝑑𝜈(𝐱)−_Ω𝑢(𝐱)Φ(𝐴𝑓(𝐱))𝑑𝜇(𝐱)𝐿||||_Ω−𝑢(𝐱)^∞0||Φ(𝑓(𝑡𝐱))−Φ(𝐴𝑓(𝐱))||𝑑𝜆(𝑡)𝑑𝜇(𝐱)_Ω𝑢(𝐱)^∞0||𝜑(𝐴𝑓(𝐱))||⋅||𝑓(𝑡𝐱)−𝐴𝑓(𝐱)||𝑑𝜆(𝑡)𝑑𝜇(𝐱)||||(2.7) holds. For a nonpositive concave function Φ, relation (2.7) holds with _Ω𝑢(𝐱)Φ(𝐴𝑓(𝐱))𝑑𝜇(𝐱)−1𝐿_Ω𝑣(𝐱)Φ(𝑓(𝐱))𝑑𝜈(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 Φ is convex and nonnegative. To prove inequality (2.7), observe that for arbitrary 𝑟∈Int𝐼 and 𝑠∈𝐼 we have Φ(𝑠)−Φ(𝑟)−𝜑(𝑟)(𝑠−𝑟)≥0, so Φ(𝑠)−Φ(𝑟)−𝜑(𝑟)(𝑠−𝑟)=≥||||Φ||Φ(𝑠(𝑠))−−ΦΦ(𝑟(𝑟))−||−𝜑||𝜑(𝑟)(𝑟()𝑠||−⋅|𝑠𝑟)−||𝑟|||.(2.9) In particular, for 𝐱∈Ω, such that 𝐴𝑓(𝐱)∈Int𝐼, and for 𝑡∈suppΩ, from (2.9) we get Φ(𝑓≥(𝑡𝐱||||)Φ)−(𝑓Φ(𝑡(𝐱𝐴))𝑓−(𝐱Φ))(−𝐴𝑓𝜑((𝐱𝐴)𝑓)||−(𝐱)||𝜑)⋅(𝐴(𝑓𝑓(𝑡(𝐱𝐱)))−||⋅𝐴||𝑓𝑓((𝑡𝐱𝐱)))−𝐴𝑓(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(𝑡𝐱)−𝐴𝑓(𝐱)≥0 for all 𝑡∈supp𝜆 or 𝑓(𝑡𝐱)−𝐴𝑓(𝐱)≤0 for all 𝑡∈supp𝜆. Since ^∞0(𝑓(𝑡𝐱)−𝐴𝑓(𝐱))𝑑𝜆(𝑡)=0,(2.11) we conclude that 𝑓(𝑡𝐱)−𝐴𝑓(𝐱)=0 for 𝜆-a.e. 𝑡∈supp𝜆, so in that case both sides of (2.10) are equal to 0. Hence, (2.10) holds for all 𝐱∈Ω and 𝜆-a.e. 𝑡∈supp𝜆. Multiplying it by 𝑢(𝐱) and then integrating over ℝ₊ and Ω, we obtain the following sequence of inequalities: _Ω^∞0𝑢(𝐱)Φ(𝑓(𝑡𝐱))𝑑𝜆(𝑡)𝑑𝜇(𝐱)−_Ω^∞0−𝑢(𝐱)Φ(𝐴𝑓(𝐱))𝑑𝜆(𝑡)𝑑𝜇(𝐱)_Ω^∞0≥𝑢(𝐱)𝜑(𝐴𝑓(𝐱))(𝑓(𝑡𝐱)−𝐴𝑓(𝐱))𝑑𝜆(𝑡)𝑑𝜇(𝐱)_Ω^∞0≥𝑢(𝐱)||||Φ(𝑓(𝑡𝐱))−Φ(𝐴𝑓(𝐱))||−||𝜑(𝐴𝑓(𝐱))||⋅||𝑓(𝑡𝐱)−𝐴𝑓(𝐱)||||𝑑𝜆(𝑡)𝑑𝜇(𝐱)_Ω≥||||𝑢(𝐱)||||^∞0||Φ(𝑓(𝑡𝐱))−Φ(𝐴𝑓(𝐱))||𝑑𝜆(𝑡)−||𝜑(𝐴𝑓(𝐱))||^∞0||𝑓(𝑡𝐱)−𝐴𝑓(𝐱)||𝑑𝜆(𝑡)||||𝑑𝜇(𝐱)_Ω−𝑢(𝐱)^∞0||Φ(𝑓(𝑡𝐱))−Φ(𝐴𝑓(𝐱))||𝑑𝜆(𝑡)𝑑𝜇(𝐱)_Ω𝑢(𝐱)||𝜑(𝐴𝑓(𝐱))||^∞0||𝑓(𝑡𝐱)−𝐴𝑓(𝐱)||𝑑𝜆(𝑡)𝑑𝜇(𝐱)||||.(2.12) By using Fubini's and the Radon-Nikodym theorems, the substitution 𝐲=𝑡𝐱, and the fact that the set Ω is 𝜆-balanced and the function Φ is nonnegative, the first integral on the left-hand side of (2.12) becomes _Ω^∞0=𝑢(𝐱)Φ(𝑓(𝑡𝐱))𝑑𝜆(𝑡)𝑑𝜇(𝐱)_∞0_Ω=𝑢(𝐱)Φ(𝑓(𝑡𝐱))𝑑𝜇(𝐱)𝑑𝜆(𝑡)_∞0_𝑡Ω𝑢1𝑡𝐲Φ(𝑓(𝐲))𝑑𝜇^𝑡≤(𝐲)𝑑𝜆(𝑡)_∞0_Ω𝑢1𝑡𝐲Φ(𝑓(𝐲))𝑑𝜇^𝑡=(𝐲)𝑑𝜆(𝑡)_∞0_Ω𝑢1=𝑡𝐲Φ(𝑓(𝐲))𝑑𝑑𝜇𝜈^𝑡(𝐲)𝑑𝜈(𝐲)𝑑𝜆(𝑡)_Ω^∞0𝑢1=𝑡𝐲𝑑𝑑𝜇𝜈^𝑡(𝐲)𝑑𝜆(𝑡)Φ(𝑓(𝐲))𝑑𝜈(𝐲)_Ω𝑣(𝐲)Φ(𝑓(𝐲))𝑑𝜈(𝐲).(2.13) Further, the second integral on the left-hand side in (2.12) reduces to _Ω^∞0𝑢(𝐱)Φ(𝐴𝑓(𝐱))𝑑𝜆(𝑡)𝑑𝜇(𝐱)=𝐿_Ω𝑢(𝐱)Φ(𝐴𝑓(𝐱))𝑑𝜇(𝐱),(2.14) while the corresponding third integral is equal to 0 since by (2.11) we have _Ω^∞0=𝑢(𝐱)𝜑(𝐴𝑓(𝐱))(𝑓(𝑡𝐱)−𝐴𝑓(𝐱))𝑑𝜆(𝑡)𝑑𝜇(𝐱)_Ω𝑢(𝐱)𝜑(𝐴𝑓(𝐱))^∞0(𝑓(𝑡𝐱)−𝐴𝑓(𝐱))𝑑𝜆(𝑡)𝑑𝜇(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 Φ is a nonpositive concave function, then −Φ is a nonnegative convex function and (2.9) becomes Φ(𝑟)−Φ(𝑠)−𝜑(𝑟)(𝑟−𝑠)=≥||||Φ||Φ(𝑟(𝑠))−−ΦΦ(𝑠(𝑟))−||−𝜑||𝜑(𝑟)(𝑟()𝑟||−|𝑠−𝑠)||𝑟|||,(2.16) where 𝜑∶𝐼→ℝ is any function such that 𝜑(𝑥)∈𝜕Φ(𝑥)=[Φ^+(𝑥),Φ^−(𝑥)] for all 𝑥∈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 Φ 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 𝑡Ω=Ω for 𝜆-a.e. 𝑡∈supp𝜆, then (2.7) holds for all convex functions Φ on an interval 𝐼⊆ℝ. In this setting, inequality (2.7) holds also for all concave functions Φ on 𝐼⊆ℝ, 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 𝜆,𝐿,𝑋,𝜇,𝜈,𝜇_𝑡,Ω,𝑢, and 𝑣 are as in Theorem 2.1. If Φ is a nonnegative convex function on an interval 𝐼⊆ℝ, then the inequality _Ω𝑢(𝐱)Φ(𝐴𝑓(𝐱))𝑑𝜇(𝐱)≤1𝐿_Ω𝑣(𝐱)Φ(𝑓(𝐱))𝑑𝜈(𝐱)(2.17) holds for all measurable functions 𝑓∶Ω→ℝ such that 𝑓(𝐱)∈𝐼 for all 𝐱∈Ω, where 𝐴𝑓 is defined by (2.4). For a nonpositive concave function Φ, the sign of inequality in (2.7) is reversed. Moreover, if in Theorem 2.1 one has 𝑡Ω=Ω for 𝜆-a.e. 𝑡∈supp𝜆, then (2.7) holds for all convex functions Φ on an interval 𝐼⊆ℝ. In that case, for all concave functions Φ, 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 𝑢(𝐱)≡1, 𝐱∈Ω, inequality (2.17) reduces to (1.3).

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

In this section, we apply Theorem 2.1 to a particular multidimensional setting, namely, to balls in ℝ^𝑛 centered at the origin and to their dual sets. The results obtained represent a new class of 𝑛-dimensional Hardy and Pólya-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 ℝ^𝑛.

Theorem 3.1. Let 𝜆 be a finite Borel measure on ℝ₊, 𝐿 be defined by (2.2), Ω be a 𝜆-balanced Borel set in ℝ^𝑛, and 𝐵 be a Borel subset of the unit sphere 𝑆^𝑛−1. Let 𝑢 be a nonnegative function on ℝ^𝑛, Φ be a nonnegative convex function on an interval 𝐼⊆ℝ, and 𝜑∶𝐼→ℝ be any function fulfilling 𝜑(𝑥)∈𝜕Φ(𝑥), for all 𝑥∈Int𝐼. Finally, let 𝑓∶Ω→ℝ be a measurable function with values in 𝐼. (i)If supp𝜆⊆⟨0,1], 0<𝑅≤∞, and Ω=Ω₁={𝐱=𝜉𝑆∶𝑆∈𝐵,0≤𝜉<𝑅}, then 1𝐿_Ω1Φ(𝑓(𝐱))^1|𝐱|/𝑅𝑢1𝑡𝐱𝑡^1−𝑛𝑑𝜆(𝑡)𝑑𝐱|𝐱|−_Ω1𝑢(𝐱)Φ𝐴₁𝑓(𝐱)𝑑𝐱≥1|𝐱|𝐿||||_Ω1𝑢(𝐱)¹⁰||Φ(𝑓(𝑡𝐱))−Φ𝐴₁𝑓(𝐱)||𝑑𝜆(𝑡)𝑑𝐱−|𝐱|_Ω1𝑢(𝐱)¹⁰||𝜑𝐴₁𝑓(𝐱)||⋅||𝑓(𝑡𝐱)−𝐴₁𝑓(𝐱)||𝑑𝜆(𝑡)𝑑𝐱|𝐱|||||,(3.1) where 𝐴₁𝑓(𝐱)=∫¹⁰𝑓(𝑡𝐱)𝑑𝜆(𝑡),𝐱∈Ω₁.(ii)If supp𝜆⊆[1,∞⟩, 0≤𝑅<∞, and Ω=Ω₂={𝐱=𝜉𝑆∶𝑆∈𝐵,𝑅≤𝜉<∞}, then 1𝐿_Ω2Φ(𝑓(𝐱))^1|𝐱|/𝑅𝑢1𝑡𝐱𝑡^1−𝑛𝑑𝜆(𝑡)𝑑𝐱|𝐱|−_Ω2𝑢(𝐱)Φ𝐴₂𝑓(𝐱)𝑑𝐱≥1|𝐱|𝐿||||_Ω2𝑢(𝐱)^∞1||Φ(𝑓(𝑡𝐱))−Φ𝐴₂𝑓(𝐱)||𝑑𝜆(𝑡)𝑑𝐱−|𝐱|_Ω2𝑢(𝐱)^∞1||𝜑𝐴₂𝑓(𝐱)||⋅||𝑓(𝑡𝐱)−𝐴₂𝑓(𝐱)||𝑑𝜆(𝑡)𝑑𝐱|𝐱|||||,(3.2) where𝐴₂𝑓(𝐱)=∫^∞1𝑓(𝑡𝐱)𝑑𝜆(𝑡),𝐱∈Ω₂.

Proof. Relation (3.1) is a direct consequence of Theorem 2.1, rewritten with 𝑋=ℝ^𝑛,Ω=Ω₁,𝑑𝜇(𝐱)=𝜒_Ω1(𝐱)𝑑𝐱, and 𝑑𝜈(𝐱)=𝑑𝐱, as well as with the function 𝑢 replaced with 𝐱↦|𝐱|⁻¹𝑢(𝐱). Then we have (𝑑𝜇_𝑡/𝑑𝜈)(𝐱)=𝑡^−𝑛𝜒_𝑡Ω1(𝐱),𝑡∈⟨0,1], 𝑣(𝐱)=^1|𝐱|/𝑅𝑢1𝑡𝐱𝑡^1−𝑛𝑑𝜆(𝑡),𝐱∈Ω₁,(3.3) and 𝐴𝑓(𝐱)=𝐴₁𝑓(𝐱),𝑥∈Ω₁, so relation (1.13) reduces to (3.1). The proof of (3.2) follows the same lines, by considering Ω=Ω₂. In such setting we get 𝑣(𝐱)=^1|𝐱|/𝑅𝑢1𝑡𝐱𝑡^1−𝑛𝑑𝜆(𝑡),𝐱∈Ω₂,(3.4) and 𝐴𝑓=𝐴₂𝑓.

By taking 𝐵=𝑆^𝑛−1, that is, by setting a 𝜆-balanced set Ω to be a ball in ℝ^𝑛 centered at the origin or its corresponding dual set, and by choosing a suitable measure 𝜆 and a weight function 𝑢, we obtain the following sequence of new refined strengthened inequalities of the Hardy type.

Theorem 3.2. Let 𝑛∈ℕ, 𝑝∈ℝ⧵[0,1⟩, and 𝑘∈ℝ, 𝑘≠𝑛. (i)If 0<𝑅≤∞, 𝑝/(𝑘−𝑛)>0, and 𝑓 is a nonnegative measurable function on 𝐵(𝑅), then the inequality 𝑘−𝑝≥𝑛||||^𝑝𝑝_𝐵(𝑅)|𝐱|^𝑝−𝑘1−|𝑅𝐱|^{(𝑘−𝑛)/𝑝}𝑓^𝑝(𝐱)𝑑𝐱−_𝐵(𝑅)|𝐱|^−𝑘(𝐻𝑓(𝐱))^𝑝𝑑𝐱𝑘−𝑛^𝑝−1_𝐵(𝑅)|𝐱|^−𝑘¹⁰𝑡^{((𝑘−𝑛)/𝑝)−1}|||||𝐱|_𝑝𝑡^{𝑛−𝑘+𝑝}𝑓^𝑝(𝑡𝐱)−𝑘−𝑛𝑝^𝑝(𝐻𝑓(𝐱))^𝑝−||𝑝||||||𝑑𝑡𝑑𝐱_𝐵(𝑅)|𝐱|^−𝑘(𝐻𝑓(𝐱))^𝑝−1⋅¹⁰|||||𝐱|𝑓(𝑡𝐱)−𝑘−𝑛𝑝𝑡^{((𝑘−𝑛)/𝑝)−1}𝐻𝑓(𝐱)||||𝑑𝑡𝑑𝐱||||(3.5) holds, where 𝐻𝑓(𝐱)=|𝐱|¹⁰𝑓(𝑡𝐱)𝑑𝑡,𝐱∈𝐵(𝑅).(3.6)(ii)If 0≤𝑅<∞, 𝑝/(𝑘−𝑛)<0, and 𝑓 is a nonnegative measurable function on ℝ^𝑛⧵𝐵(𝑅), then ≥𝑛−||||𝑝𝑘𝑝^𝑝_{ℝ𝑛⧵𝐵(𝑅)}|𝐱|^𝑝−𝑘1−|𝑅𝐱|^{(𝑛−𝑘)/𝑝}𝑓^𝑝(𝐱)𝑑𝐱−_{ℝ𝑛⧵𝐵(𝑅)}|𝐱|^−𝑘𝐻𝑓(𝐱)^𝑝𝑑𝐱𝑛−𝑘^𝑝−1_{ℝ𝑛⧵𝐵(𝑅)}|𝐱|^−𝑘^∞1𝑡^{((𝑘−𝑛)/𝑝)−1}|||||𝐱|_𝑝𝑡^{𝑛−𝑘+𝑝}𝑓^𝑝(𝑡𝐱)−𝑛−𝑘𝑝^𝑝𝐻𝑓(𝐱)^𝑝−||𝑝||||||𝑑𝑡𝑑𝐱_{ℝ𝑛⧵𝐵(𝑅)}|𝐱|^−𝑘𝐻𝑓(𝐱)^𝑝−1⋅^∞1|||||𝐱|𝑓(𝑡𝐱)−𝑛−𝑘𝑝𝑡^{((𝑘−𝑛)/𝑝)−1}𝐻𝑓(𝐱)||||𝑑𝑡𝑑𝐱||||,(3.7) where 𝐻𝑓(𝐱)=|𝐱|^∞1𝑓(𝑡𝐱)𝑑𝑡,𝐱∈ℝ^𝑛⧵𝐵(𝑅).(3.8)For 𝑝∈⟨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=ℝ^𝑛, 𝐼=[0,∞⟩, 𝑢(𝐱)=|𝐱|^−𝑛, 𝑑𝜈(𝐱)=𝑑𝐱, and Φ(𝑥)=𝑥^𝑝, 𝑝≠0, that is, 𝜑(𝑥)=𝑝𝑥^𝑝−1. In the case (i), let also Ω=𝐵(𝑅^{(𝑘−𝑛)/𝑝}), 𝑑𝜆(𝑡)=𝜒_⟨0,1⟩(𝑡)𝑑𝑡, and 𝑑𝜇(𝐱)=𝜒_𝐵(R^{(𝑘−𝑛)/𝑝}₎(𝐱)𝑑𝐱. Then we have 𝐿=1,(𝑑𝜇_𝑡/𝑑𝜈)(𝐱)=𝑡^−𝑛𝜒_{𝐵(𝑡𝑅}^{(𝑘−𝑛)/𝑝}₎(𝐱), and 𝑣(𝐱)=¹⁰|||1𝑡𝐱|||^−𝑛𝑡^−𝑛𝜒_{𝐵(𝑡𝑅}^{(𝑘−𝑛)/𝑝}₎(𝐱)𝑑𝑡=|𝐱|^−𝑛1−𝑅|𝐱|^{(𝑘−𝑛)/𝑝},𝐱∈𝐵𝑅^{(𝑘−𝑛)/𝑝}.(3.9) Replace the function 𝑓 in (2.7) with the function 𝑔∶𝐵(𝑅^{(𝑘−𝑛)/𝑝})→ℝ, 𝑔(𝐱)=|𝐱|^{(𝑝/(𝑘−𝑛))−1}𝑓(|𝐱|^{(𝑝/(𝑘−𝑛))−1}𝐱). Then 𝐴𝑔(𝐱)=|𝐱|^{(𝑝/(𝑘−𝑛))−1}¹⁰𝑡^{(𝑝/(𝑘−𝑛))−1}𝑓𝑡^{𝑝/(𝑘−𝑛)}|𝐱|^{𝑝/(𝑘−𝑛)}𝐱=𝑘−𝑛|𝐱|𝑑𝑡𝑝|𝐱1|^{|𝐱|𝑝/(𝑘−𝑛)}₀𝑓𝑟|𝐱|𝐱𝑑𝑟,(3.10) where we applied the substitution 𝑟=(𝑡|𝐱|)^{𝑝/(𝑘−𝑛)}. In this setting, by using polar coordinates, the first integral on the left-hand side of inequality (2.7) becomes _{𝐵(𝑅(𝑘−𝑛)/𝑝)}|𝐱|^{𝑝((𝑝/(𝑘−𝑛))−1)−𝑛}1−𝑅|𝐱|^{(𝑘−𝑛)/𝑝}𝑓^𝑝|𝐱|^{(𝑝/(𝑘−𝑛))−1}=𝐱𝑑𝐱_𝑆𝑛−1𝑑𝑆^{𝑅(𝑘−𝑛)/𝑝}₀𝑟^{𝑝((𝑝/(𝑘−𝑛))−1)−1}1−𝑅𝑟^{(𝑘−𝑛)/𝑝}𝑓^𝑝𝑟^{𝑝/(𝑘−𝑛)}=𝑘−𝑛𝑆𝑑𝑟𝑝_𝑆^𝑛−1𝑑𝑆^𝑅0𝑡^{𝑛−1+𝑝−𝑘}1−𝑅𝑡^{(𝑘−𝑛)/𝑝}𝑓^𝑝=𝑘−𝑛(𝑡𝑆)𝑑𝑡𝑝_𝐵(𝑅)|𝐱|^𝑝−𝑘1−|𝑅𝐱|^{(𝑘−𝑛)/𝑝}𝑓^𝑝(𝐱)𝑑𝐱,(3.11) while the second integral on the left-hand side of (2.7) reduces to 𝑘−𝑝𝑛^𝑝_{𝐵(𝑅(𝑘−𝑛)/𝑝)}|𝐱|^{−𝑛−𝑝}^{|𝐱|𝑝/(𝑘−𝑛)}₀𝑓𝑟|𝐱|𝐱𝑑𝑟^𝑝=𝑘−𝑛𝑑𝐱𝑝^𝑝_𝑆𝑛−1𝑑𝑆^{𝑅(𝑘−𝑛)/𝑝}₀𝑡^−𝑝−1^{𝑡𝑝/(𝑘−𝑛)}₀=𝑘−𝑛𝑓(𝑟𝑆)𝑑𝑟^𝑝𝑑𝑡𝑝^𝑝+1_𝑆𝑛−1𝑑𝑆^𝑅0𝑠^{𝑛−𝑘−1}^𝑠0=𝑘−𝑛𝑓(𝑟𝑆)𝑑𝑟^𝑝𝑑𝑠𝑝^𝑝+1_𝐵(𝑅)|𝐱|^−𝑘(𝐻𝑓(𝐱))^𝑝𝑑𝐱.(3.12) Analogously, on the right-hand side of (2.7) we get ||||_{𝐵(𝑅(𝑘−𝑛)/𝑝)}|𝐱|^{−𝑛−𝑝}¹⁰||||𝑓_𝑝𝑡^{𝑝/(𝑘−𝑛)}|𝐱|^{𝑝/(𝑘−𝑛)}𝑡^{𝑝((𝑝/(𝑘−𝑛))−1)}|𝐱|^{𝑝(𝑝/(𝑘−𝑛))}𝐱−𝑘−𝑛|𝐱|𝑝^𝑝^{|𝐱|𝑝/(𝑘−𝑛)}₀𝑓𝑟|𝐱|𝐱𝑑𝑟^𝑝−𝑘−𝑛|||||𝑑𝑡𝑑𝐱𝑝^𝑝−1||𝑝||_{𝐵(𝑅(𝑘−𝑛)/𝑝)}|𝐱|^{−𝑛−𝑝}^{|𝐱|𝑝/(𝑘−𝑛)}₀𝑓𝑟|𝐱|𝐱𝑑𝑟^𝑝−1⋅¹⁰||||𝑡_{(𝑝/(𝑘−𝑛))−1}|𝐱|^{𝑝/(𝑘−𝑛)}𝑓𝑡^{𝑝/(𝑘−𝑛)}|𝐱|^{𝑝/(𝑘−𝑛)}𝐱|𝐱|−𝑘−𝑝𝑛^{|𝐱|𝑝/(𝑘−𝑛)}₀𝑓𝑟=||||𝑘−𝑛|𝐱|𝐱𝑑𝑟||||𝑑𝑡𝑑𝐱||||𝑝_𝑆^𝑛−1𝑑𝑆^𝑅0𝑧^{𝑛−𝑘−1}¹⁰||||𝑧_𝑝−𝑘−𝑡𝑛^{𝑝((𝑝/(𝑘−𝑛))−1)}𝑓^𝑝𝑡^{𝑝/(𝑘−𝑛)}𝑧𝑆𝑝^𝑝^𝑧0𝑓(𝑟𝑆)𝑑𝑟^𝑝−𝑘−𝑛||||𝑑𝑡𝑑𝑧𝑝^𝑝||𝑝||_𝑆𝑛−1𝑑𝑆^𝑅0𝑧^{𝑛−𝑘−1}^𝑧0𝑓(𝑟𝑆)𝑑𝑟^𝑝−1⋅¹⁰||||𝑡_{(𝑝/(𝑘−𝑛))−1}𝑧𝑓𝑡^{𝑝/(𝑘−𝑛)}𝑧𝑆−𝑘−𝑛𝑝^𝑧0=||||𝑘−𝑛𝑓(𝑟𝑆)𝑑𝑟||||𝑑𝑡𝑑𝑧||||𝑝²_𝐵(𝑅)|𝐱|^−𝑘¹⁰𝑡^{((𝑘−𝑛)/𝑝)−1}|||||𝐱|_𝑝𝑡^{𝑛−𝑘+𝑝}𝑓^𝑝(𝑡𝐱)−𝑘−𝑛𝑝^𝑝(𝐻𝑓(𝐱))^𝑝−𝑘−𝑛||||𝑑𝑡𝑑𝐱𝑝^𝑝+1||𝑝||_𝐵(𝑅)|𝐱|^−𝑘(𝐻𝑓(𝐱))^𝑝−1⋅¹⁰|||||𝐱|𝑓(𝑡𝐱)−𝑘−𝑛𝑝𝑡^{((𝑘−𝑛)/𝑝)−1}𝐻𝑓(𝐱)||||𝑑𝑡𝑑𝐱||||.(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 Ω=ℝ^𝑛⧵𝐵(𝑅^{(𝑛−𝑘)/𝑝}), 𝑑𝜆(𝑡)=𝜒_⟨1,∞⟩(𝑡)(𝑑𝑡/𝑡²), and 𝑑𝜇(𝐱)=𝜒_ℝ^𝑛_⧵𝐵(𝑅^{(𝑛−𝑘)/𝑝}₎(𝐱)𝑑𝐱. As in the case (i), then we have 𝐿=1, (𝑑𝜇_𝑡/𝑑𝜈)(𝐱)=𝑡^−𝑛𝜒_ℝ^𝑛_{⧵𝐵(𝑡𝑅}^{(𝑛−𝑘)/𝑝}₎(𝐱), and 𝑣(𝐱)=|𝐱|^−𝑛1−𝑅|𝐱|^{(𝑛−𝑘)/𝑝},𝐱∈ℝ^𝑛⧵𝐵𝑅^{(𝑛−𝑘)/𝑝}.(3.14) Inequality (3.7) now follows by rewriting (2.7) with the above parameters and with the function 𝑔∶ℝ^𝑛⧵𝐵(𝑅^{(𝑛−𝑘)/𝑝})→ℝ, 𝑔(𝐱)=|𝐱|^{(𝑝/(𝑛−𝑘))+1}𝑓(|𝐱|^{(𝑝/(𝑛−𝑘))−1}𝐱) 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 𝑛∈ℕ, 𝑝∈ℝ⧵[0,1⟩, and 𝑘∈ℝ, 𝑘≠𝑛. (i)If 0<𝑅≤∞, 𝑝/(𝑘−𝑛)>0, 𝑓 is a nonnegative function on 𝐵(𝑅), and 𝐻𝑓 is defined by (3.6), then _𝐵(𝑅)|𝐱|^−𝑘(𝐻𝑓(𝐱))^𝑝𝑑𝐱≤𝑝𝑘−𝑛^𝑝_𝐵(𝑅)|𝐱|^𝑝−𝑘1−|𝑅𝐱|^{(𝑘−𝑛)/𝑝}𝑓^𝑝(𝐱)𝑑𝐱.(3.15)(ii)If 0≤𝑅<∞, 𝑝/(𝑘−𝑛)<0, 𝑓 is a nonnegative function on ℝ^𝑛⧵𝐵(𝑅), and 𝐻𝑓 is given by (3.8), then_{ℝ𝑛⧵𝐵(𝑅)}≤|𝐱|𝑝^−𝑘𝐻𝑓(𝐱)^𝑝𝑑𝐱𝑛−𝑘^𝑝_{ℝ𝑛⧵𝐵(𝑅)}|𝐱|^𝑝−𝑘1−|𝑅𝐱|^{(𝑛−𝑘)/𝑝}𝑓^𝑝(𝐱)𝑑𝐱.(3.16) The constant |𝑝/(𝑘−𝑛)|^𝑝 is the best possible for both inequalities. For 𝑝∈⟨0,1], the signs of inequality in (3.15) and (3.16) are reversed.

Proof. We only need to prove that |𝑝/(𝑘−𝑛)|^𝑝 is the best possible constant for inequalities (3.15) and (3.16). Consider the case (i) first. For a sufficiently small 𝜀>0, and the function 𝑓_𝜀∶𝐵(𝑅)→ℝ defined by 𝑓_𝜀(𝐱)=|𝐱|^{((𝑘−𝑛+𝜀)/𝑝)−1}, the left-hand side of (3.15) is equal to 𝐿_𝜀=_𝐵(𝑅)|𝐱|^−𝑛+𝜀¹⁰𝑡^{((𝑘−𝑛+𝜀)/𝑝)−1}𝑑𝑡^𝑝𝑑𝐱=𝑝𝑘−𝑛+𝜀^𝑝_𝐵(𝑅)=𝑝|𝐱|^−𝑛+𝜀𝑑𝐱𝑘−𝑛+𝜀^𝑝_𝑆𝑛−1𝑑𝑆^𝑅0𝑟^𝜀−1𝑑𝑟=𝑝𝑘−𝑛+𝜀^𝑝||𝑆^𝑛−1||⋅𝑅^𝜀𝜀,(3.17) while on the right-hand side of (3.15) we get 𝑅_𝜀=𝑝𝑘−𝑛^𝑝_𝐵(𝑅)|𝐱|^−𝑛+𝜀≤𝑝1−|𝑅𝐱|^{(𝑘−𝑛)/𝑝}𝑑𝐱𝑘−𝑛^𝑝_𝐵(𝑅)|𝐱|^−𝑛+𝜀𝑑𝐱=𝑝𝑘−𝑛^𝑝||𝑆^𝑛−1||⋅𝑅^𝜀𝜀.(3.18) Therefore, 1≤𝑅_𝜀/𝐿_𝜀≤((𝑘−𝑛+𝜀)/(𝑘−𝑛))^𝑝. Since ((𝑘−𝑛+𝜀)/(𝑘−𝑛))^𝑝↘1, as 𝜀↘0, the constant (𝑝/(𝑘−𝑛))^𝑝 is the best possible for (3.15). The proof that the constant (𝑝/(𝑛−𝑘))^𝑝 is the best possible for (3.16) follows the same lines, considering the function 𝑓_𝜀∶ℝ^𝑛⧵𝐵(𝑅)→ℝ, 𝑓_𝜀(𝐱)=|𝐱|^{((𝑘−𝑛−𝜀)/𝑝)−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 𝑛∈ℕ, 𝑝∈ℝ⧵[0,1⟩, and 𝑘∈ℝ, 𝑘≠𝑛. Let 𝑓∶ℝ^𝑛→ℝ be a nonnegative measurable function and 𝐻𝑓 and 𝐻𝑓 be defined by (3.6) and (3.8), respectively. If 𝑝/(𝑘−𝑛)>0, then the inequality _ℝ|𝐱|^−𝑘(𝐻𝑓(𝐱))^𝑝𝑑𝐱≤𝑝𝑘−𝑛^𝑝_ℝ𝑛|𝐱|^𝑝−𝑘𝑓^𝑝(𝐱)𝑑𝐱(3.19) holds, while for 𝑝/(𝑘−𝑛)<0 one has _ℝ𝑛|𝐱|^−𝑘𝐻𝑓(𝐱)^𝑝𝑑𝐱≤𝑝𝑛−𝑘^𝑝_ℝ𝑛|𝐱|^𝑝−𝑘𝑓^𝑝(𝐱)𝑑𝐱.(3.20) The constant |𝑝/(𝑘−𝑛)|^𝑝 is the best possible for both inequalities. Moreover, for 𝑝∈⟨0,1], the signs of inequality in (3.19) and (3.20) are reversed.

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

Theorem 3.5. Let 𝑛∈ℕ. (i)If 0<𝑅≤∞, 𝑓 is a positive measurable function on 𝐵(𝑅), and 𝐺𝑓(𝐱)=exp¹⁰log𝑓(𝑡𝐱)𝑑𝑡,𝐱∈𝐵(𝑅),(3.21) then the inequality 𝑒^𝑛_𝐵(𝑅)1−|𝑅𝐱|𝑓(𝐱)𝑑𝐱−_𝐵(𝑅)≥||||𝐺𝑓(𝐱)𝑑𝐱_𝐵(𝑅)¹⁰||(𝑒𝑡)^𝑛𝑓(𝑡𝐱)−𝐺𝑓(𝐱)||𝑑𝑡𝑑𝐱−_𝐵(𝑅)𝐺𝑓(𝐱)¹⁰||||log(𝑒𝑡)^𝑛𝑓(𝑡𝐱)𝐺𝑓(𝐱)||||𝑑𝑡𝑑𝐱||||(3.22) holds. (ii)If 0≤𝑅<∞, 𝑓 is a positive measurable function on ℝ^𝑛⧵𝐵(𝑅), and G𝑓(𝐱)=exp^∞1log𝑓(𝑡𝐱)𝑑𝑡𝑡²,𝐱∈ℝ^𝑛⧵𝐵(𝑅),(3.23) then 𝑒^−𝑛_{ℝ𝑛⧵𝐵(R)}1−|𝑅𝐱|𝑓(𝐱)𝑑𝐱−_ℝ^𝑛_{⧵𝐵(𝑅)}≥||||𝐺𝑓(𝐱)𝑑𝐱_{ℝ𝑛⧵𝐵(𝑅)}^∞1|||𝑡𝑒^𝑛𝑓(𝑡𝐱)−𝐺𝑓(𝐱)|||𝑑𝑡𝑡²−𝑑𝐱_{ℝ𝑛⧵𝐵(𝑅)}𝐺𝑓(𝐱)^∞1||||log(𝑒𝑡)^𝑛𝑓(𝑡𝐱)𝐺𝑓(𝐱)||||𝑑𝑡²𝑡𝑑𝐱||||.(3.24)

Proof. Follows from Theorems 2.1 and 3.1 by considering 𝑋=ℝ^𝑛, 𝐼=ℝ, 𝑢(𝐱)=|𝐵(|𝐱|)|⁻¹, 𝑑𝜈(𝐱)=𝑑𝐱, and Φ(𝑥)=𝜑(𝑥)=𝑒^𝑥. To get (3.22), we also set Ω=𝐵(𝑅), 𝑑𝜆(𝑡)=𝜒_⟨0,1⟩(𝑡)𝑑𝑡, and 𝑑𝜇(𝐱)=𝜒_𝐵(𝑅)(𝐱)𝑑𝐱. In that case, we have 𝐿=1, (𝑑𝜇_𝑡/𝑑𝜈)(𝐱)=𝑡^−𝑛𝜒_{𝐵(𝑡𝑅)}(𝐱), and 𝑣(𝐱)=¹⁰1||𝐵||(1/𝑡)𝐱||||𝑡^−𝑛𝜒_{𝐵(𝑡𝑅)}(𝐱)𝑑𝑡=1||𝐵(|𝐱|)||¹⁰=1𝜒_{𝐵(𝑡𝑅)}(𝐱)𝑑𝑡||𝐵(|𝐱|)||1−|𝑅𝐱|,𝐱∈𝐵(𝑅).(3.25) Further, replace the function 𝑓 in (2.7) with the function 𝐱↦log(|𝐵(|𝐱|)|𝑓(𝐱)). The first integral on the left-hand side of (2.7) then becomes _𝐵(𝑅)1||𝐵(|𝐱|)||1−|𝑅𝐱|||𝐵(|𝐱|)||𝑓(𝐱)𝑑𝐱=_𝐵(𝑅)1−|𝑅𝐱|𝑓(𝐱)𝑑𝐱,(3.26) the corresponding second integral reduces to _𝐵(𝑅)1||𝐵(|𝐱|)||exp¹⁰=log𝑡^𝑛||𝐵(|𝐱|)||𝑓(𝑡𝐱)𝑑𝑡𝑑𝐱_𝐵(𝑅)1||𝐵(|𝐱|)||exp−𝑛+log||𝐵(|𝐱|)||+log¹⁰=𝑒_−𝑛log𝑓(𝑡𝐱)𝑑𝑡𝑑𝐱_𝐵(𝑅)exp¹⁰log𝑓(𝑡𝐱)𝑑𝑡𝑑𝐱=𝑒^−𝑛_𝐵(𝑅)𝐺𝑓(𝐱)𝑑𝐱,(3.27) while on the right-hand side of (2.7) we obtain ||||_𝐵(𝑅)1||𝐵(|𝐱|)||¹⁰−||𝑡^𝑛||𝐵(|𝐱|)||𝑓(𝑡𝐱)−𝑒^−𝑛||𝐵(|𝐱|)||𝐺𝑓(𝐱)||𝑑𝑡𝑑𝐱_𝐵(𝑅)1||𝐵(|𝐱|)||¹⁰⋅||||log𝑡_𝑛||𝑒^−𝑛||𝐵(|𝐱|)||𝐺𝑓(𝐱)||=||||||𝐵(|𝐱|)||𝑓(𝑡𝐱)+𝑛−log||𝐵(|𝐱|)||−¹⁰log𝑓(𝑡𝐱)𝑑𝑡||||𝑑𝑡𝑑𝐱_𝐵(𝑅)||||¹⁰−𝑒_−𝑛||𝑡^𝑛𝑓(𝑡𝐱)−𝑒^−𝑛𝐺𝑓(𝐱)||𝑑𝑡𝑑𝐱_𝐵(𝑅)=||||𝐺𝑓(𝐱)¹⁰||𝑛log𝑡+log||𝐵(|𝐱|)||+log(𝑓(𝑡𝐱))+𝑛−log||𝐵(|𝐱|)||−log𝐺𝑓(𝐱)||𝑑𝑡𝑑𝐱_𝐵(𝑅)||||¹⁰||𝑡^𝑛𝑓(𝑡𝐱)−𝑒^−𝑛𝐺𝑓(𝐱)||𝑑𝑡𝑑𝐱−𝑒^−𝑛_𝐵(𝑅)𝐺𝑓(𝐱)¹⁰||||log(𝑒𝑡)^𝑛𝑓(𝑡𝐱)𝐺𝑓(𝐱)||||𝑑𝑡𝑑𝐱||||.(3.28) Finally, (3.22) holds by combining (3.26), (3.27), and (3.28) and then multiplying the whole inequality by 𝑒^𝑛.
Inequality (3.24) can be derived by a similar technique by taking Ω=ℝ^𝑛⧵𝐵(𝑅), 𝑑𝜆(𝑡)=𝜒_[1,∞⟩(𝑡)(𝑑𝑡/𝑡²), and 𝑑𝜇(𝐱)=𝜒_ℝ^𝑛_{⧵𝐵(𝑅)}(𝐱)𝑑𝐱. Then 𝐿=1, (𝑑𝜇_𝑡/𝑑𝜈)(𝐱)=𝑡^−𝑛𝜒_ℝ^𝑛_{⧵𝐵(𝑡𝑅)}(𝐱) and 𝑣(𝐱)=^∞1|||𝐵1𝑡|𝐱||||⁻¹𝑡^−𝑛𝜒_ℝ^𝑛_{⧵𝐵(𝑡𝑅)}(𝐱)𝑑𝑡𝑡²=1||𝐵(|𝐱|)||^1|𝐱|/𝑅𝑑𝑡𝑡²=1||𝐵(|𝐱|)||1−|𝑅𝐱|,𝐱∈ℝ^𝑛⧵𝐵(𝑅),(3.29) so (3.24) follows by replacing the function f in (2.7) with 𝐱↦log(|𝐵(|𝐱|)|𝑓(𝐱)).

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

Theorem 3.6. Let 𝑛∈ℕ. (i)If 0<𝑅≤∞, 𝑓 is a positive measurable function on 𝐵(𝑅), and 𝐺𝑓 is defined by (3.21), then the inequality _𝐵(𝑅)𝐺𝑓(𝐱)𝑑𝐱≤𝑒^𝑛_𝐵(R)1−|𝑅𝐱|𝑓(𝐱)𝑑𝐱(3.30) holds and the constant 𝑒^𝑛 is the best possible.(ii) If 0≤𝑅<∞, 𝑓 is a positive measurable function on ℝ^𝑛⧵𝐵(𝑅), and 𝐺𝑓 is defined by (3.23), then the inequality _{ℝ𝑛⧵𝐵(𝑅)}𝐺𝑓(𝐱)𝑑𝐱≤𝑒^−𝑛_{ℝ𝑛⧵𝐵(𝑅)}1−|𝑅𝐱|𝑓(𝐱)𝑑𝐱(3.31) holds and the constant 𝑒^−𝑛 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 𝜀>0, let the function 𝑓_𝜀∶𝐵(𝑅)→ℝ be defined by 𝑓_𝜀(𝐱)=𝑒^−𝑛|𝐵(|𝐱|)|^𝜀−1. Calculating the left-hand side of (3.30) for 𝑓_𝜀, we obtain 𝐿_𝜀==𝑒_{𝐵(𝑅)−𝑛}exp¹⁰log𝑒^−𝑛||𝐵(|𝑡𝐱|)||^𝜀−1𝑑𝑡𝑑𝐱_𝐵(𝑅)=𝑒_−𝑛𝜀exp𝑛(𝜀−1)¹⁰log𝑡𝑑𝑡+(𝜀−1)log||𝐵(|𝐱|)||𝑑𝐱_𝐵(𝑅)||𝐵(|𝐱|)||^𝜀−1𝑑𝐱=𝑒^−𝑛𝜀||𝐵(𝑅)||^𝜀𝜀.(3.32) On the other hand, the right-hand side of (3.30), rewritten for 𝑓_𝜀, can be estimated as 𝑅_𝜀=𝑒^𝑛⋅𝑒^−𝑛_𝐵(𝑅)1−|𝑅𝐱|||𝐵(|𝐱|)||^𝜀−1𝑑𝐱≤_𝐵(𝑅)||𝐵(|𝐱|)||^𝜀−1𝑑𝐱=||𝐵(𝑅)||^𝜀𝜀.(3.33) Since 1≤(𝑅_𝜀/𝐿_𝜀)≤𝑒^𝑛𝜀↘1, as 𝜀↘0, 𝑒^𝑛 is the best possible constant for inequality (3.30). The proof that 𝑒^−𝑛 is the best possible constant for (3.31) is similar, if the function 𝑓_𝜀∶ℝ^𝑛⧵𝐵(𝑅)→ℝ, 𝑓_𝜀=𝑒^𝑛|𝐵(|𝐱|)|^−𝜀−1, is considered.

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

Corollary 3.7. If 𝑓 is a positive measurable function on ℝ^𝑛, and 𝐺𝑓, 𝐺𝑓 are, respectively, defined by (3.21) and (3.23), then the inequalities _ℝ𝑛𝐺𝑓(𝐱)𝑑𝐱≤𝑒^𝑛_ℝ𝑛𝑓(𝐱)𝑑𝐱,_ℝ𝑛𝐺𝑓(𝐱)𝑑𝐱≤𝑒^−𝑛_ℝ𝑛𝑓(𝐱)𝑑𝐱(3.34) hold. The constants 𝑒^𝑛 and 𝑒^−𝑛 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 𝑘=𝑝>𝑛, and 𝑓 replaced with 𝑓^1/𝑝, and by letting 𝑝→∞. In particular, observe that lim_𝑝→∞(𝑝/(𝑝−𝑛))^𝑝=𝑒^𝑛.

Although being related to the setting with balls in ℝ^𝑛 centered at the origin, inequalities (3.15), (3.16) and (3.30), (3.31) are not equivalent with the previously obtained Hardy- and Pólya-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 Pólya-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<𝑏≤∞, 𝑓 be a nonnegative function on ⟨0,𝑏⟩, and 𝑝,𝑘∈ℝ be such that 0≠𝑝≠1, 𝑘≠1, and 𝑝/(𝑘−1)>0. If 𝑝∈ℝ⧵[0,1], then the inequality 𝑘−𝑝≥1||||^𝑝^𝑏0𝑝1−𝑥𝑏^{(𝑘−1)/𝑝}𝑥^𝑝−𝑘𝑓^𝑝(𝑥)𝑑𝑥−^∞0𝑥^−𝑘𝐹^𝑝(𝑥)𝑑𝑥𝑘−1^𝑝−1^𝑏0𝑥^{((1−𝑘)/𝑝)−1}^𝑥0𝑡^{((𝑘−1)/𝑝)−1}⋅||||𝑡^{𝑝−𝑘+1}𝑓^𝑝(t)−𝑘−1𝑝^𝑝−||𝑝||𝑥^1−𝑘𝐹^𝑝(𝑥)||||𝑑𝑡𝑑𝑥_𝑏0𝑥^−𝑘𝐹^𝑝−1(𝑥)^𝑥0||||𝑓(𝑡)−𝑘−1𝑝⋅1𝑡𝑥𝑡^{(𝑘−1)/𝑝}𝐹(𝑥)||||𝑑𝑡𝑑𝑥||||(3.35) holds, where 𝐹(𝑥) is defined by (1.4). In the case when 𝑝∈⟨0,1⟩, the order of integrals on the left-hand side in (3.35) is reversed.

An inequality dual to (3.35) can be derived from (3.7) for 𝑛=1 and it can be found in [4].

Finally, for 𝑛=1, inequalities (3.22) and (3.24), respectively, reduce to the following result from [4].

Corollary 3.10. Let 0<𝑏≤∞, 𝑓 be a positive function on ⟨0,𝑏⟩, and 𝐺(𝑥) be defined by (1.8). Then 𝑒^𝑏01−𝑥𝑏𝑓(𝑥)𝑑𝑥−^𝑏0≥||||𝐺(𝑥)𝑑𝑥_𝑏0^𝑥0||𝑒𝑡𝑓(𝑡)−𝑥𝐺(𝑥)||𝑑𝑡𝑑𝑥𝑥²−^𝑏0𝐺(𝑥)^𝑥0||||log𝑒𝑡𝑓(𝑡)𝑥𝐺(𝑥)||||𝑑𝑡𝑑𝑥𝑥||||(3.36) holds. On the other hand, if 0≤𝑏<∞, 𝑓 is a positive function on ⟨𝑏,∞⟩, and 𝐺(𝑥) is defined by (1.11), then 1𝑒^∞𝑏1−𝑥𝑏𝑓(𝑥)𝑑𝑥−^∞𝑏≥||||𝐺(𝑥)𝑑𝑥_∞𝑏^∞𝑥|||1𝑒𝑡𝑓(𝑡)−𝑥𝐺(𝑥)|||𝑑𝑡²𝑡𝑑𝑥−^∞𝑏𝑥𝐺(𝑥)^𝑏𝑥||||log𝑡𝑓(𝑡)𝑒𝑥𝐺(𝑥)||||𝑑𝑡²𝑡𝑑𝑥||||(3.37) holds.

Having in mind Corollaries 3.9 and 3.10, our results in this paper can be considered as generalizations of the inequalities obtained in [4].

4. Concluding Remarks

To conclude this paper, we analyze Boas-type inequalities with kernels. Let the setting be as in Section 2, except that 𝜆 is a 𝜎-finite Borel measure on ℝ₊. By a kernel we mean a nonnegative measurable function 𝑘∶𝑋×𝑋→ℝ, such that𝐾(𝐱)=^∞0𝑘(𝐱,𝑡𝐱)𝑑𝜆(𝑡)<∞(4.1) for 𝜇-a.e. 𝐱∈𝑋. For a 𝜆-balanced set Ω⊆𝑋 and a Borel measurable function 𝑓∶Ω→ℝ, we define its Hardy-Littlewood average with the kernel 𝑘, denoted by 𝐴_𝑘𝑓, as𝐴_𝑘𝑓(𝐱)=1𝐾(𝐱)^∞0𝑘(𝐱,𝑡𝐱)𝑓(𝑡𝐱)𝑑𝜆(𝑡),𝐱∈Ω.(4.2) A related Boas-type inequality is given as follows.

Theorem 4.1. Let 𝜆 be a 𝜎-finite Borel measure on ℝ₊, let 𝜇 and 𝜈 be 𝜎-finite Borel measures on a topological space 𝑋, and let 𝜇_𝑡, defined by (2.5), be absolutely continuous with respect to the measure 𝜈 for all 𝑡∈supp𝜆. Let Ω⊆𝑋 be a 𝜆-balanced set and 𝑢 be a nonnegative function on 𝑋, such that 𝑣(𝐱)=^∞0𝑢(𝐱)𝑘((1/𝑡)𝐱,𝐱)𝐾((1/𝑡)𝐱)⋅𝑑𝑑𝜇𝜈^𝑡(𝐱)𝑑𝜆(𝑡)<∞,𝐱∈Ω,(4.3) where 𝑘∶𝑋×𝑋→ℝ is a nonnegative measurable function satisfying (4.1). Further, let Φ be a nonnegative convex function on an interval 𝐼⊆ℝ. If 𝑓∶Ω→ℝ is a measurable function, such that 𝑓(𝐱)∈𝐼 for all 𝐱∈Ω, and 𝐴_𝑘𝑓 is defined by (4.2), then 𝐴_𝑘𝑓(𝐱)∈𝐼, for all 𝐱∈Ω, and the inequality _Ω𝑢(𝐱)Φ𝐴_𝑘𝑓(𝐱)𝑑𝜇(𝐱)≤_Ω𝑣(𝐱)Φ(𝑓(𝐱))𝑑𝜈(𝐱)(4.4) holds. For a nonpositive concave function Φ, relation (4.4) holds with the sign of inequality reversed.

Proof. First, we need to prove that 𝐴_𝑘𝑓(𝐱)∈𝐼 for all 𝐱∈Ω. Otherwise, there exists 𝐱₀∈Ω such that 𝐴_𝑘𝑓(𝐱₀)∉𝐼. In that case, we have either 𝑓(𝑡𝐱₀)−𝐴_𝑘𝑓(𝐱₀)<0 for all 𝑡∈supp𝜆, or 𝑓(𝑡𝐱₀)−𝐴_𝑘𝑓(𝐱₀)>0 for all 𝑡∈supp𝜆. On the other hand, the identity 1𝐾𝐱₀^∞0𝑘𝐱₀,𝑡𝐱₀𝑓𝑡𝐱₀−𝐴_𝑘𝑓𝐱₀𝑑𝜆(𝑡)=0(4.5) and nonnegativity of 𝑘(𝐱₀,𝑡𝐱₀) for all 𝑡∈supp𝜆 yield that 𝑘𝑥₀,𝑡𝐱₀𝑓𝑡𝐱₀−𝐴_𝑘𝑓𝐱₀=0,𝑡∈supp𝜆.(4.6) Since 𝐾(𝐱₀)>0, there exists a set 𝐽⊆supp𝜆 such that 𝜆(𝐽)>0 and 𝑘(𝐱₀,𝑡𝐱₀)>0 for all 𝑡∈𝐽. Hence, 𝑓(𝑡𝐱₀)−𝐴_𝑘𝑓(𝐱₀)=0, 𝑡∈𝐽, so we came to a contradiction. Therefore, 𝐴_𝑘𝑓(𝐱)∈𝐼 for all 𝐱∈Ω.
By using Jensen's inequality, Fubini's theorem, the Radon-Nikodym theorem, the substitution 𝐲=𝑡𝐱, and the properties of the set Ω and the function Φ, we now obtain _Ω𝑢≤(𝐱)Φ𝐴_𝑘𝑓(𝐱)𝑑𝜇(𝐱)_Ω𝑢(𝐱)𝐾(𝐱)^∞0=𝑘(𝐱,𝑡𝐱)Φ(𝑓(𝑡𝐱))𝑑𝜆(𝑡)𝑑𝜇(𝐱)_∞0_Ω𝑢(𝐱)𝑘(𝐱,𝑡𝐱)=𝐾(𝐱)Φ(𝑓(𝑡𝐱))𝑑𝜇(𝐱)𝑑𝜆(𝑡)_∞0_𝑡Ω𝑢1𝑡𝐲𝑘𝐾(((1(/1𝑡)/𝑡𝐲),𝐲𝐲))Φ(𝑓(𝐲))𝑑𝜇^𝑡≤(𝐲)𝑑𝜆(𝑡)_∞0_Ω𝑢1𝑡𝐲𝑘𝐾(((1(/1𝑡)/𝑡𝐲),𝐲𝐲))Φ(𝑓(𝐲))𝑑𝜇^𝑡=(𝐲)𝑑𝜆(𝑡)_∞0_Ω𝑢1=𝑡𝐲𝑘𝐾(((1(/1𝑡)/𝑡𝐲),𝐲𝐲))Φ(𝑓(𝐲))𝑑𝑑𝜇𝜈^𝑡(𝐲)𝑑𝜈(𝐲)𝑑𝜆(𝑡)_Ω^∞0𝑢1=𝑡𝐲𝑘𝐾(((1(/1𝑡)/𝑡𝐲),𝐲𝐲))⋅𝑑𝑑𝜇𝜈^𝑡(𝐲)𝑑𝜆(𝑡)Φ(𝑓(𝐲))𝑑𝜈(𝐲)_Ω𝑣(𝐲)Φ(𝑓(𝐲))𝑑𝜈(𝐲),(4.7) so the proof is completed.

Moreover, applying similar reasoning as in Section 2, we get a refinement of the Boas-type inequality (4.4).

Theorem 4.2. Suppose that 𝜆 is a 𝜎-finite Borel measure on ℝ₊, 𝜇 and 𝜈 are 𝜎-finite Borel measures on a topological space 𝑋, and the measures 𝜇_𝑡, defined by (2.5), are absolutely continuous with respect to the measure 𝜈 for all 𝑡∈supp𝜆. Further, suppose that Ω⊆𝑋 is a 𝜆-balanced set, 𝑢 is a nonnegative function on 𝑋, and 𝑣 is defined on Ω by (4.3), where 𝑘∶𝑋×𝑋→𝑅 is a nonnegative measurable function satisfying (4.1). If Φ is a nonnegative convex function on an interval 𝐼⊆ℝ and 𝜑∶𝐼→ℝ is any function such that 𝜑(𝑥)∈𝜕Φ(𝑥) for all 𝑥∈Int𝐼, then the inequality _Ω𝑣≥(𝐱)||||Φ(𝑓(𝐱))𝑑𝜈(𝐱)−_Ω𝑢(𝐱)Φ𝐴_𝑘𝑓(𝐱)𝑑𝜇(𝐱)_Ω𝑢(𝐱)𝐾(𝐱)^∞0−𝑘(𝐱,𝑡𝐱)||Φ(𝑓(𝑡𝐱))−Φ𝐴_𝑘𝑓(𝐱)||𝑑𝜆(𝑡)𝑑𝜇(𝐱)_Ω𝑢(𝐱)𝐾(𝐱)^∞0𝑘(𝐱,𝑡𝐱)||𝜑𝐴_𝑘𝑓(𝐱)||⋅||𝑓(𝑡𝐱)−𝐴_𝑘𝑓(𝐱)||𝑑𝜆(𝑡)𝑑𝜇(𝐱)||||(4.8) holds for all measurable functions 𝑓∶Ω→ℝ, such that 𝑓(𝐱)∈𝐼 for all 𝐱∈Ω, and 𝐴_𝑘𝑓 defined by (4.2). For a nonpositive concave function Φ, relation (4.8) holds with _Ω𝑢(𝐱)Φ𝐴_𝑘𝑓(𝐱)𝑑𝜇(𝐱)−_Ω𝑣(𝐱)Φ(𝑓(𝐱))𝑑𝜈(𝐱)(4.9) on its left-hand side.

Proof. The proof follows the same lines as the proof of Theorem 2.1, considering the setting from Theorem 4.1, so we omit further details.

Acknowledgment

The research of the authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grants 058-1170889-1050 (A. Čižmešija), 117-1170889-0888 (J. Pečarić), and 082-0000000-0893 (D. Pokaz).