#### Abstract

We state and prove a new refined Boas-type inequality in a setting with a topological space and general -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 and prove that constant factors involved in their right-hand sides are the best possible.

#### 1. Introduction

Boas [1], proved that the inequality holds for all continuous convex functions , measurable nonnegative functions , and nondecreasing bounded functions , where , and the inner integral on the left-hand side of (1.1) is the Lebesgue-Stieltjes integral with respect to . After its author, inequality (1.1) was named the Boas inequality. In the case of a concave function , 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), ΔiΕΎmeΕ‘ija et al. [4] considered a general Borel measure on , such that , a convex function on a convex set , and a weight function on , and proved that the inequality holds for all measurable functions with values in , where and , . 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 -finite Borel measures and on a topological space and a Borel probability measure on . For a -balanced Borel set and the measures , defined for all Borel sets and by , he obtained the inequality where is a nonnegative convex function on a real interval , and is a Borel measurable function with values in . A weighted version of (1.3) was given by ΔiΕΎmeΕ‘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 PΓ³lya-Knopp's inequalities. In the sequel, we state their strengthened versions, obtained independently by Debnath et al. [7, 8] and ΔiΕΎmeΕ‘ija et al. [9, 10].

Let and such that . If , is a nonnegative function, , and then holds, while for the sign of inequality in (1.5) is reversed. Moreover, if and parameters and such that , then the inequality holds for all nonnegative functions such that , where For inequality (1.6) holds with the sign of inequality reversed. The constant 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 in (1.5), while for in (1.6) we get its dual inequality.

On the other hand, if , is a positive function, and then holds, while the inequality holds for , , and For in (1.9) and for in (1.10) we, respectively, get the classical PΓ³lya-Knopp's inequality and its dual inequality. Notice that the constant factors and , 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 -dimensional strengthened Hardy's inequality related to the setting with balls in centered at the origin (see [11] for details). Let such that , , and . Suppose that is a nonnegative measurable function and the function is defined on by where is a ball in centered at the origin and of radius , while denotes the Euclidean norm of . If , then the inequality holds, while for we have Terms and , respectively, denote the volumes of and . The constant is the best possible for both inequalities. Observe that the first natural generalization of the classical Hardy's inequality to balls in was given by Christ and Grafakos in [12].

Finally, here we state an -dimensional PΓ³lya-Knopp's inequality, related to (1.13) and (1.14), for a positive function on and as well as its dual inequality for a positive function on and

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 [13β15], 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 -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 , 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 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 , , and are taken to be equal to zero. As usual, by and we denote the Lebesgue measure on and , respectively, while by a weight function we mean a nonnegative measurable function on the actual set. An interval in is any convex subset of , while Int denotes the interior of an interval . In particular, . For , by we denote a ball in centered at the origin and of radius , that is, , where denotes the Euclidean norm of . By its dual set we mean the set . Finally, by we denote the surface of the unit ball and by its area. Using polar coordinates in , the volume of the ball is then

#### 2. A New Refined Weighted Boas-Type Inequality

To start, we recall some basic facts on convex functions. Let be an interval in and be a convex function. For , by we denote the subdifferential of at , that is, the set It is well known that for all . More precisely, at each point we have , , and the set on which is not differentiable is at most countable. Moreover, every function for which , whenever , is increasing on .

On the other hand, if is a concave function, that is, is convex, then denotes the superdifferential of at the point . For all , in this setting we have and . Notice that, although the symbol has two different notions, it will be clear from the context whether it applies to a convex or to a concave function . 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 be a finite Borel measure on . By , we denote its support, that is, the set of all such that holds for all open neighbourhoods of . Hence, Further, let be a topological space equipped with a continuous scalar multiplication , for and , such that Let a Borel set be -balanced, that is, let hold for all . For a Borel measurable function , we define its Hardy-Littlewood average, , as Finally, suppose that and are -finite Borel measures on . For and a Borel set , we define Obviously, is a -finite Borel measure on for all . Throughout this paper we assume that is absolutely continuous with respect to the measure , that is, , for each . As usual, by 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 be a finite Borel measure on and be defined by (2.2). Let and be -finite Borel measures on a topological space , be defined by (2.5), and let for all . Further, let be a -balanced Borel set and be a nonnegative function on , such that
**
Suppose that is a nonnegative convex function on an interval and is any function fulfilling , for all . If is a Borel measurable function with values in and is defined by (2.4), then for all and the inequality
**
holds. For a nonpositive concave function , relation (2.7) holds with
**
on its left-hand side.*

*Proof. *Since for all , it is not hard to see that , for all (see the proof of Theorem 2.1 in [6] for details). Suppose the function is convex and nonnegative. To prove inequality (2.7), observe that for arbitrary and we have , so
In particular, for , such that , and for , from (2.9) we get
On the other hand, if is not an open interval and is an endpoint of for some , then either for all or for all . Since
we conclude that for -a.e. , so in that case both sides of (2.10) are equal to 0. Hence, (2.10) holds for all and -a.e. . Multiplying it by and then integrating over and , we obtain the following sequence of inequalities:
By using Fubini's and the Radon-Nikodym theorems, the substitution , and the fact that the set is -balanced and the function is nonnegative, the first integral on the left-hand side of (2.12) becomes
Further, the second integral on the left-hand side in (2.12) reduces to
while the corresponding third integral is equal to 0 since by (2.11) we have
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 is a nonpositive concave function, then is a nonnegative convex function and (2.9) becomes
where is any function such that for all . 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 in Theorem 2.1 can be omitted only in a particular setting with cones in . More precisely, the following corollary holds.

Corollary 2.3. *If in Theorem 2.1 one has for -a.e. , then (2.7) holds for all convex functions on an interval . In this setting, inequality (2.7) holds also for all concave functions on , 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 , and are as in Theorem 2.1. If is a nonnegative convex function on an interval , then the inequality
**
holds for all measurable functions such that for all , where is defined by (2.4). For a nonpositive concave function , the sign of inequality in (2.7) is reversed. Moreover, if in Theorem 2.1 one has for -a.e. , then (2.7) holds for all convex functions on an interval . In that case, for all concave functions , 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 , , 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 centered at the origin and to their dual sets. The results obtained represent a new class of -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 .

Theorem 3.1. *Let be a finite Borel measure on , be defined by (2.2), be a -balanced Borel set in , and be a Borel subset of the unit sphere . Let be a nonnegative function on , be a nonnegative convex function on an interval , and be any function fulfilling , for all . Finally, let be a measurable function with values in . *(i)*If , , and , then
where .*(ii)*If , , and , then
where.*

*Proof. *Relation (3.1) is a direct consequence of Theorem 2.1, rewritten with **,** and , as well as with the function replaced with . Then we have ,
and , so relation (1.13) reduces to (3.1). The proof of (3.2) follows the same lines, by considering . In such setting we get
and .

By taking , that is, by setting a -balanced set to be a ball in centered at the origin or its corresponding dual set, and by choosing a suitable measure and a weight function , we obtain the following sequence of new refined strengthened inequalities of the Hardy type.

Theorem 3.2. *Let , , and , . *(i)*If , , and is a nonnegative measurable function on , then the inequality
holds, where
*(ii)*If , , and is a nonnegative measurable function on , then
where
**For 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 , , , , and , , that is, . In the case (i), let also , , and . Then we have ,, and
Replace the function in (2.7) with the function , . Then
where we applied the substitution . In this setting, by using polar coordinates, the first integral on the left-hand side of inequality (2.7) becomes
while the second integral on the left-hand side of (2.7) reduces to
Analogously, on the right-hand side of (2.7) we get
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 , , and . As in the case (i), then we have , , and
Inequality (3.7) now follows by rewriting (2.7) with the above parameters and with the function , instead of , 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 , , and , . *(i)*If , , is a nonnegative function on , and is defined by (3.6), then
*(ii)*If , , is a nonnegative function on , and is given by (3.8), then**
The constant is the best possible for both inequalities. For , the signs of inequality in (3.15) and (3.16) are reversed.*

*Proof. * We only need to prove that is the best possible constant for inequalities (3.15) and (3.16). Consider the case (i) first. For a sufficiently small , and the function defined by , the left-hand side of (3.15) is equal to
while on the right-hand side of (3.15) we get
Therefore, . Since , as , the constant is the best possible for (3.15). The proof that the constant is the best possible for (3.16) follows the same lines, considering the function , .

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

Corollary 3.4. *Let , , and , . Let be a nonnegative measurable function and and be defined by (3.6) and (3.8), respectively. If , then the inequality
**
holds, while for one has
**
The constant is the best possible for both inequalities. Moreover, for , 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 . *(i)*If , is a positive measurable function on , and
then the inequality
holds. *(ii)*If , is a positive measurable function on , and
then
*

*Proof. *Follows from Theorems 2.1 and 3.1 by considering , , , , and . To get (3.22), we also set , , and . In that case, we have , , and
Further, replace the function in (2.7) with the function . The first integral on the left-hand side of (2.7) then becomes
the corresponding second integral reduces to
while on the right-hand side of (2.7) we obtain
Finally, (3.22) holds by combining (3.26), (3.27), and (3.28) and then multiplying the whole inequality by .

Inequality (3.24) can be derived by a similar technique by taking , , and . Then , and
so (3.24) follows by replacing the function *f* in (2.7) with .

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

Theorem 3.6. *Let . *(i)*If , is a positive measurable function on , and is defined by (3.21), then the inequality
holds and the constant is the best possible.*(ii)* If , is a positive measurable function on , and is defined by (3.23), then the inequality
holds and the constant 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 , let the function be defined by . Calculating the left-hand side of (3.30) for , we obtain
On the other hand, the right-hand side of (3.30), rewritten for , can be estimated as
Since , as , is the best possible constant for inequality (3.30). The proof that is the best possible constant for (3.31) is similar, if the function , , is considered.

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

Corollary 3.7. *If is a positive measurable function on , and , are, respectively, defined by (3.21) and (3.23), then the inequalities
**
hold. The constants and 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 , and replaced with , and by letting . In particular, observe that .

Although being related to the setting with balls in 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 inequalities of both type coincide. In particular, inequality (3.5) for was obtained in [4]. It is given in the following corollary.

Corollary 3.9. *Let , be a nonnegative function on , and be such that , , and . If , then the inequality
**
holds, where is defined by (1.4). In the case when , 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 and it can be found in [4].

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

Corollary 3.10. *Let , be a positive function on , and be defined by (1.8). Then
**
holds. On the other hand, if , is a positive function on , and is defined by (1.11), then
**
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 is a -finite Borel measure on . By a kernel we mean a nonnegative measurable function , such that for -a.e. . For a -balanced set and a Borel measurable function , we define its Hardy-Littlewood average with the kernel , denoted by , as A related Boas-type inequality is given as follows.

Theorem 4.1. *Let be a -finite Borel measure on , let and be -finite Borel measures on a topological space , and let , defined by (2.5), be absolutely continuous with respect to the measure for all . Let be a -balanced set and be a nonnegative function on , such that
**
where is a nonnegative measurable function satisfying (4.1). Further, let be a nonnegative convex function on an interval . If is a measurable function, such that for all , and is defined by (4.2), then , for all , and the inequality
**
holds. For a nonpositive concave function , relation (4.4) holds with the sign of inequality reversed.*