Some new refinements of strengthened Hardy and Pólya – Knopp ’ s inequalities

We prove a new general one-dimensional inequality for convex functions and Hardy–Littlewood averages. Furthermore, we apply this result to unify and refine the so-called Boas’s inequality and the strengthened inequalities of the Hardy–Knopp–type, deriving their new refinements as special cases of the obtained general relation. In particular, we get new refinements of strengthened versions of the well-known Hardy and Pólya–Knopp’s inequalities.


Introduction
To begin with, we recall some well-known classical integral inequalities.If p > 1, k = 1 , and the function F is defined on R + = 0, ∞ by then the highly important Hardy's integral inequality holds for all non-negative functions f , such that x 1− k p f ∈ L p (R + ).This relation was obtained by G. H. Hardy [12] in 1928, although he announced its version with k = p > 1 already in 1920, [10], and then proved it in 1925, [11].In [12], Hardy also pointed out that if k and F fulfill the conditions of the above result, but 0 < p < 1 , then the sign of inequality in (1.1) is reversed, that is, holds.On the other hand, the first unweighted Hardy-type inequality for p < 0 was considered by K. Knopp [20] in 1928, but in a discrete setting, for sequences of positive real numbers, while general weighted integral Hardytype inequalities for exponents p, q < 0 and 0 < p, q < 1 were first studied much later, by P. R. Beesack and H. P. Heinig [1] and H. P. Heinig [14].
Another important classical integral inequality is the so-called Pólya-Knopp's inequality, which holds for all positive functions f ∈ L 1 (R + ).This result was first published by K. Knopp [20] in 1928, but it was certainly known before since Hardy himself (see [11, p. 156]) claimed that it was G. Pólya who pointed it out to him earlier.Note that the discrete version of (1.3) is surely due to T. Carleman, [3].
It is important to observe that relations (1.1) and (1.3) are closely related since (1.3) can be obtained from (1.1) by rewriting it with the function f replaced with f 1/p and letting p → ∞.Therefore, Pólya-Knopp's inequality may be considered as a limiting case of Hardy's inequality.Moreover, the constants p |k−1| p and e , respectively appearing on the right-hand sides of (1.1) and (1.3), are the best possible, that is, neither of them can be replaced with any smaller constant.
On the other hand, obviously unaware of the mentioned more general Boas's result for Hardy-Littlewood averages, in 2002, S. Kaijser et al. [18] established the so-called general Hardy-Knopp-type inequality for positive functions f : R + −→ R, where Φ is a convex function on R + .By taking Φ(x) = x p and Φ(x) = e x , they obtained an elegant new proof of inequalities (1.1) and (1.3) and showed that both Hardy and Pólya-Knopp's inequality can be derived by using only a convexity argument.Later on, A. Čižmešija et al. [9] generalized the relation (1.5) to the so-called strengthened Hardy-Knopp-type inequality by adding a weight function and truncating the range of integration to 0, b .They also obtained a related dual inequality, that is, an inequality with the outer integrals taken over b, ∞ and with the inner integral on the lefthand side taken over x, ∞ .These general inequalities provided an unified treatment of the strengthened Hardy and Pólya-Knopp's inequalities from [7,8] and [32,33].
Finally, we mention a recent paper [29] by L.-E.Persson and J. A. Oguntuase.They obtained a class of refinements of Hardy's inequality (1.1) related to an arbitrary b ∈ R + and the outer integrals on both hand sides of (1.1) taken over 0, b or b, ∞ .These results extend those of D. T. Shum [31] and C. O. Imoru [15,16] and cover all admissible parameters p, k ∈ R, p = 0, k = 1.Namely, let f be a non-negative integrable function on 0, b , while for p ∈ 0, 1] inequality (1.6) holds in the reversed direction.On the contrary, if f is a non-negative integrable function on b, ∞ , F (x) = ∞ x f (t) dt, and p k−1 < 0, then and prove a new weighted Boas-type inequality for this setting.Further, we point out that our result unifies, generalizes and refines relations (1.4) and (1.5), as well as the strengthened Hardy-Knopp-type inequalities from [9].More precisely, applying the obtained general relation with some particular weights and a measure λ, we derive new refinements of the above inequalities.Finally, as their special cases we get new refinements of the strengthened versions of Hardy and Pólya-Knopp's inequalities, completely different from (1.6) and (1.7) and even hardly comparable with these inequalities.
The paper is organized in the following way.After this Introduction, in Section 2 we introduce some necessary notation and state, prove and discuss a general refined weighted Boas-type inequality.As its particular cases, in the same section we obtain a new refinement of inequality (1.4), as well as refinements of (1.5) and of the strengthened weighted Hardy-Knopp-type inequalities.Refinements of the strengthened Hardy and Pólya-Knopp's inequalities are presented in the concluding Section 3 of the paper, along with some final remarks.
Conventions.Throughout this paper, all measures are assumed to be positive, all functions are assumed to be measurable, and expressions of the and ∞ ∞ are taken to be equal to zero.As usual, by dx we denote the Lebesgue measure on R, by a weight function (shortly: a weight) we mean a non-negative measurable function on the actual interval, while an interval in R is any convex subset of R. Finally, by Int I we denote the interior of an interval I ⊆ R.

The main results
First, we introduce some necessary notation and, for reader's convenience, recall some basic facts about 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 while 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 .For more details about convex functions see e.g. a recent monograph [26].
On the other hand, for a finite Borel measure λ on R + , that is, having property (1.8), and a Borel measurable function f : R + −→ R, by Af we denote its Hardy-Littlewood average, defined in terms of the Lebesgue integral as where L is defined by (1.8).Now, we can state and prove the main result of this paper.It is given in the following theorem.
Theorem 2.1.Let λ be a finite Borel measure on R + , L be defined by (1.8), and let u and v be non-negative measurable functions on R + , where Let Φ be a continuous convex function on an interval I ⊆ R and ϕ : Proof.
For a fixed x ∈ R + , let h x : R + −→ R be defined by Observing that f (R + ) ⊆ I and that I is an interval in R, we have h x (t) > 0 for all t ∈ R + , or h x (t) < 0 for all t ∈ R + , that is, the function h x is either strictly positive or strictly negative.Since this contradicts (2.4), we have proved that Af (x) ∈ I , for all x ∈ R + .Note that if Af (x) is an endpoint of I for some x ∈ R + (in cases when I is not an open interval), then h x (or −h x ) will be a nonnegative function whose integral over R + , with respect to the measure λ, is equal to 0. Therefore, h x ≡ 0, that is, f (tx) = Af (x) holds for λa.e.t ∈ R + .To prove inequality (2.3), observe that for all r ∈ Int I and s ∈ I we have where ϕ : I −→ R is any function such that ϕ(x) ∈ ∂Φ(x) for x ∈ Int I , and hence Especially, in the case when Af (x) ∈ Int I , by substituting r = Af (x) and s = f (tx) in (2.5), for all t ∈ R + we get On the other hand, the above analysis provides (2.6) to hold even if Af (x) is an endpoint of I , since in that case both sides of inequality (2.6) are equal to 0 for λ-a.e.t ∈ R + .Multiplying (2.6) by u(x)  x , then integrating it over R 2 + with respect to the measures dλ(t) and dx x , and applying Fubini's theorem, we obtain the following sequence of inequalities: Again, by using Fubini's theorem and the substitution y = tx, the first integral on the left-hand side of (2.7) becomes while for the second integral we have Finally, considering (2.4), we similarly get 3) holds by combining (2.7), (2.8), (2.9), and (2.10).
Remark 2.1.Observe that (2.7) provides a pair of inequalities interpolated between the left-hand side and the right-hand side of (2.3), that is, further new refinements of (2.3).
where ϕ is a real function on for all x ∈ Int I .Therefore, in this setting (2.3) holds by its left-hand side replaced with Moreover, if Φ is an affine function, then (2.3) becomes equality.
Since the right-hand side of (2.3) is non-negative, as an immediate consequence of Theorem 2.1 and Remark 2.2 we get the following result, a weighted Boas's inequality.
Corollary 2.1.Suppose λ is a finite Borel measure on R + , L is defined by (1.8), u is a non-negative measurable function on R + , and the function v is defined on R + by (2.2).If Φ is a continuous convex function on an interval I ⊆ R, then the inequality holds for all measurable functions f : R + −→ R, such that f (x) ∈ I for all x ∈ R + , where Af (x) is defined by (2.1).For a concave function Φ, the sign of inequality in relation (2.11) is reversed.
In the sequel, we analyze some important particular cases of Theorem 2.1 and Corollary 2.1 and compare them with some results previously known from the literature.Namely, by setting u(x) ≡ 1 , we obtain a refined Boas-type inequality with Af (x) defined by the Lebesgue integral.
Corollary 2.2.Let λ be a finite Borel measure on R + and L be defined by (1.8).Then the inequality holds for all continuous convex functions Φ on an interval I ⊆ R, real functions ϕ on I , such that ϕ(x) ∈ ∂Φ(x) for x ∈ Int I , and all measurable real functions f on R + , such that f (x) ∈ I for all x ∈ R + , and Af (x) defined by (2.1).If the function Φ is concave, then (2.1) holds with on its left-hand side.

Corollary 2.3.
If λ is a finite Borel measure on R + , L is defined by (1.8), Φ is a continuous convex function on an interval I ⊆ R, f is a measurable real function on R + with values in I , and Af (x) is defined by (2.1), then If Φ is concave, then the sign of inequality in (2.13) is reversed.
Remark 2.3.Let m : [0, ∞ −→ R be a non-decreasing bounded function and M = m(∞) − m(0) > 0 .It is well-known that m induces a finite Borel measure λ on R + (and vice versa), such that the related Lebesgue and Lebesgue-Stieltjes integrals are equivalent.Thus, all the above results from this section can be interpreted as for Af (x) defined by the Lebesgue-Stieltjes integral with respect to m, that is, as Therefore, our results refine and generalize Boas's inequality (1.4).Namely, we obtained a refinement of its weighted version.
To conclude this section, we consider measures λ which yield refinements of the Hardy-Knopp-type inequalities mentioned in the Introduction.Especially, for dλ(t) = χ [0,1] (t) dt we obtain a refinement of a weighted version of (1.5).Theorem 2.2.Let u be a non-negative function on R + , such that the function t → u(t) t 2 is locally integrable in R + , and let If a real-valued function Φ is convex on an interval I ⊆ R and ϕ : holds for all functions f on R + with values in I and for Hf (x) defined by on its left-hand side.
Proof.Follows directly from Theorem 2.1 and Remark 2.2, rewritten with the measure dλ(t) = χ [0,1] (t) dt.In this setting, we have L = 1, Moreover, for a concave function Φ relation (2.16) holds with the reversed sign of inequality.This result, the so-called weighted Hardy-Knopp-type inequality, was already obtained in [9, Theorem 1], while its particular case (1.5), originally proved in [18], follows by setting u(x) ≡ 1 .Therefore, (2.14) may be regarded as a refined weighted inequality of the Hardy-Knopp type and relation (2.3) as its generalization.
On the other hand, a dual result to Theorem 2.2 can be derived by considering (2.3) If Φ is a convex function on an interval I ⊆ R and ϕ : holds for all functions f on R + with values in I and for Hf (x) defined by for x ∈ R + .In the case when Φ is concave, (2.17) holds if its left-hand side is replaced with and L = 1 , so (2.17) holds.
Remark 2.5.As in Remark 2.4, note that for a convex function Φ and functions u , w , f , and Hf from the statement of Theorem 2.3, we have while for a concave Φ relation (2.19) holds with the inequality sign ≥ .Since as a consequence of Theorem 2.1 and Theorem 2.3 we derived a dual inequality to (2.16), relation (2.17) can be considered as a refined dual weighted Hardy-Knopp-type inequality and (2.3) as its generalization.
Finally, as special cases of Theorem 2.2 and Theorem 2.3, we formulate refinements of the strengthened Hardy-Knopp-type inequalities.
t 2 is locally integrable in 0, b , and the function w is defined by holds for all functions f : 0, b −→ R with values in I and Hf defined on 0, b by (2.15).If Φ is a concave function, the order of integrals on the left-hand side of (2.20) is reversed.
Proof.Let û , ŵ , and f be defined on R + by û , where c ∈ I is arbitrary.Since these functions naturally extend u , w , and f to act on R + , they evidently fulfill the conditions of Theorem 2.2, considered with û , ŵ , and f instead of u , w , and f respectively.Therefore, (2.14) holds and in this setting it becomes (2.20).
Remark 2.6.Since the right-hand side of (2.20) is non-negative, Corollary 2.4 improves a result from [9, Theorem 1].Hence, it can be considered as a refined strengthened Hardy-Knopp-type inequality.
Remark 2.7.For u(x) ≡ 1, we have w This relation provides a basis for results in the following section.
A dual result to inequality (2.20) is given in the next corollary.
Let Φ be a convex function on an interval I ⊆ R and ϕ : I −→ R be such that ϕ(x) ∈ ∂Φ(x) for all x ∈ Int I .Then the inequality holds for all functions f on b, ∞ with values in I and Hf defined by (2.18).For a concave function Φ, the order of integrals on the left-hand side of (2.22) is reversed.

Refinements of strengthened Hardy and Pólya-Knopp's inequalities
In the previous section, obtained inequalities were discussed with respect to a measure λ and a weight function u , while a convex function Φ remained unspecified.On the contrary, here we consider two particular convex (or concave) functions, namely Φ(x) = x p and Φ(x) = e x , and derive some new refinements of the well-known Hardy and Pólya-Knopp's inequalities, as well as their strengthened versions.Moreover, we show that they are just special cases of the results mentioned.
We start with new refinements of Hardy's inequality, so let p ∈ R, p = 0, and Φ(x) = x p .Obviously, ϕ(x) = Φ (x) = px p−1 , x ∈ R + , and the function Φ is convex for p ∈ R \ [0, 1 , concave for p ∈ 0, 1], and affine for p = 1 .On the other hand, for a locally integrable function f : R + −→ R, as in the Introduction, we denote A new refined strengthened Hardy's inequality is given in the following corollary.
Corollary 3.1.Let 0 < b ≤ ∞ and p, k ∈ R be such that p = 0, k = 1, and p k−1 > 0 .Let f be a non-negative function on 0 holds.In the case when p ∈ 0, 1 , the order of integrals of the right-hand side of inequality (3.2) is reversed.
Proof.First, let either p ≥ 1, k > 1, or p < 0, k < 1, and let Φ(x) = x p and ϕ(x) = px p−1 .According to Corollary 2.4 and Remark 2.7, then (2.21) holds.Rewriting it for a 2) follows by a sequence of substitutions such as s = x p k−1 .The remaining case, that is, when p ∈ 0, 1 and k > 1 , is a direct consequence of Corollary 2.4 and Remark 2.7.Now, we state and prove a refined strengthened dual Hardy's inequality.
holds for all non-negative functions f on b, ∞ .In the case when p ∈ 0, 1 , the order of integrals on the left-hand side of inequality (3.3) is reversed.

Proof.
As in the proof of Corollary 3.1, we use Φ( Therefore, we obtained a refinement of the classical Hardy's inequality (1.1).Also note that for p = 1 relations (3.2) and (3.2) are trivial since their both sides are equal to 0.
Remark 3.2.Observe that Corollary 3.1 and Corollary 3.2 provide new and original refinements of Hardy's inequality although the idea to strengthen and refine (1.1) is not new and results of such type already exist in the literature.As in the Introduction, here we just mention the papers [15,16], [31], and a recent paper [29].It is important to emphasize that our results are completely different from those given in these papers and even hardly comparable with (1.6) and (1.7).A third type of refinements of a similar form can be found in another recent paper [30], where a fairly new concept of superquadratic function was used in a crucial way.
Finally, we consider Φ(x) = e x to obtain refinements of the strengthened Pólya-Knopp's inequality and of its dual.For a positive function f on R + and x ∈ R + , we denote

Corollary 2 . 5 .
For b ∈ R, b ≥ 0 , let u : b, ∞ −→ R be a non-negative locally integrable function in b, ∞ and the function w be given by

Remark 3 . 1 .− 1 p ∞ 0 x 0 x
, and rewrite inequality (2.23) for b and f respectively replaced with a = b 1−k p and x → f (x p 1−k )x p 1−k +1 .Relation (3.3) then follows by a sequence of substitutions of the form s = x p 1−k .Note that we again distinguish two cases.The first one, which yields (3.3), holds when either p ≥ 1, k < 1, or p < 0, k > 1 .In the other one, with p ∈ 0, 1 and k < 1 , the order of integrals on the left-hand side of inequality (3.3) is reversed.Observe that for b = ∞ the left-hand side of (3.2) reads p k p−k f p (x) dx − ∞ −k F p (x) dx , while for b = 0 on the left-hand side of (3.3) we have p 1 − k p ∞ 0 x p−k f p (x) dx − ∞ 0 x −k F p (x) dx .

2 .Corollary 3 . 3 .Corollary 3 . 4 .Remark 3 . 3 .
Related results are given in the following two corollaries.Let 0 < b ≤ ∞ and f be a positive function on 0, b .Then e If 0 ≤ b < ∞ and f is a positive function on b, Note that for b = ∞ in (3.4) we have a refined Pólya-Knopp's inequality, while for b = 0 relation (3.5) becomes its refined dual inequality.
2.14)holds.Let a convex function Φ and functions u , w , f , and Hf be as in Theorem 2.2.Observing that the right-hand side of relation(2.14)