Some New Refined General Boas-Type Inequalities

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 Polya-Knopp-type inequalities related to balls in ℝ𝑛 and prove that constant factors 
involved in their right-hand sides are the best possible.


Introduction
, proved that the inequality 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 holds for all measurable functions f : R → R with values in I, where Af x 1/ L ∞ 0 f tx dλ t and w x ∞ 0 u 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 ∈ supp λ by μ t S μ t −1 S , he obtained the inequality 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 P ólya-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 , and x p−k f p x dx 1.5 holds, while for p ∈ 0, 1 the sign of inequality in 1.5 is reversed.
holds for all nonnegative functions f such that x 1− k/p f ∈ L p b, ∞ , where 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 ∈ L 1 0, b is a positive function, and holds, while the inequality For b ∞ in 1.9 and for b 0 in 1.10 we, respectively, get the classical P ólya-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 where B |x| is a ball in R n centered at the origin and of radius |x|, while |x| denotes the Euclidean norm of holds, while for p/ k − 1 < 0 we have Finally, here we state an n-dimensional P ólya-Knopp's inequality, related to 1.13 and 1.14 , for a positive function f on B R and as well as its dual inequality for a positive function f on R n \ B R and Observe that all above-mentioned inequalities of the Hardy and P ólya-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 P ólya-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 P ólya-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 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 1.19

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 It is well known that ∂Φ x / ∅ for all x ∈ Int I.More precisely, at each point 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 ∈ Int I, is increasing on Int I.
On the other hand, if Φ : 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 supp λ, 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, Further, let X be a topological space equipped with a continuous scalar multiplication a, x → ax ∈ X, for a ∈ R and x ∈ X, such that 1x x, Let a Borel set Ω ⊆ X be λ-balanced, that is, let tΩ {tx : x ∈ Ω} ⊆ Ω hold for all t ∈ supp λ.
For a Borel measurable function f : Ω → R, we define its Hardy-Littlewood average, Af, as Finally, suppose that μ and ν are σ-finite Borel measures on X.For t > 0 and a Borel set S ⊆ X, we define 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 ∈ supp λ.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 ∈ supp λ.Further, let Ω ⊆ X be a λ-balanced Borel set and u be a nonnegative function on X, such that

2.6
Suppose that Φ : I → R is a nonnegative convex function on an interval I ⊆ R and ϕ : with values in I and Af is defined by 2.4 , then Af x ∈ I for all x ∈ Ω and the inequality holds.For a nonpositive concave function Φ, relation 2.7 holds with 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 ∈ Int I and s In particular, for x ∈ Ω, such that Af x ∈ Int I, and for t ∈ supp Ω, from 2.9 we get 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 we conclude that f tx − Af x 0 for λ-a.e.t ∈ supp λ, 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 ∈ supp λ.Multiplying it by u x and then integrating over R and Ω, we obtain the following sequence of inequalities:

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

2.13
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
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 where ϕ : I → R is any function such that ϕ x ∈ ∂Φ x Φ x , Φ − x for all x ∈ Int I. 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 ∈ supp λ, 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

New Multidimensional Hardy-and P ólya-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 P ólya-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 ∈ Int I. Finally, let f : Ω → R be a measurable function with values in I.

3.1
where Proof.Relation 3.1 is a direct consequence of Theorem 2.1, rewritten with X R n , Ω Ω 1 , dμ x χ Ω 1 x dx, and dν x dx, as well as with the function u replaced with x → |x| −1 u x .Then we have dμ t /dν x t −n χ tΩ 1 x , t ∈ 0, 1 , 13 reduces to 3.1 .The proof of 3.2 follows the same lines, by considering Ω Ω 2 .In such setting we get and Af A 2 f.
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.
and f is a nonnegative measurable function on B R , then the inequality holds, where where

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 −n χ B tR k−n /p x , and 3.9 Replace the function f in 2.7 with the function g 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

3.11
while the second integral on the left-hand side of 2.7 reduces to

3.12
Analogously, on the right-hand side of 2.7 we get
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/t 2 , 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 −n χ R n \B tR n−k /p x , and

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.
, and Hf is given by 3.8 , then

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 3.17 while on the right-hand side of 3.15 we get 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

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 P ólya-Knopp-type inequality.
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 P ólya-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 P ólya-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.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 .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.

Corollary 3 . 4 .
15 and R 0 in 3.16 , we immediately get the following new multidimensional Hardy-type inequality.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 Hf be defined by 3.6 and 3.8 , respectively.If p/ k − n > 0, then the inequality

Theorem 3 . 5 .x exp 1 0
Let n ∈ N. i If 0 < R ≤ ∞, f is a positive measurable function on B R , and Gf log f tx dt , x ∈ B R , 3.21 then the inequality e

Corollary 3 . 10 .
Let 0 < b ≤ ∞, f be a positive function on 0, b , and G x be defined by 1the other hand, if 0 ≤ b < ∞, f is a positive function on b, ∞ , and G x is defined by 1Journal of Function Spaces and Applications holds for all measurable functions f : Ω → R, such that f x ∈ I for all x ∈ Ω, and A k f defined by 4.2 .For a nonpositive concave function Φ, relation 4.8 holds with Ω u x Φ A k f x dμ x − Ω v x Φ f x dν x 4.9 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 .