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Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013, Article ID 925464, 4 pageshttp://dx.doi.org/10.1155/2013/925464

Research ArticleA Generalization on Some New Types of Hardy-Hilbert’sIntegral Inequalities

Banyat Sroysang

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, PathumThani 12121, Thailand

Correspondence should be addressed to Banyat Sroysang; [email protected]

Received 15 May 2013; Accepted 17 September 2013

Academic Editor: Wilfredo Urbina

Copyright © 2013 Banyat Sroysang. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Sulaiman presented, in 2008, new kinds of Hardy-Hilbert’s integral inequality in which the weight function is homogeneous. Inthis paper, we present a generalization on the kinds of Hardy-Hilbert’s integral inequality.

1. Introduction and Preliminaries

For any two nonnegative measurable functions 𝑓 and 𝑔 suchthat

0 < ∫∞

0

𝑓2 (𝑥) 𝑑𝑥 < ∞, 0 < ∫∞

0

𝑔2 (𝑦) 𝑑𝑦 < ∞, (1)

we have the Hilbert’s integral inequality [1] that

∬∞

0

𝑓 (𝑥) 𝑔 (𝑦)

𝑥 + 𝑦𝑑𝑥 𝑑𝑦

< 𝜋(∫∞

0

𝑓2 (𝑥) 𝑑𝑥∫∞

0

𝑔2 (𝑦) 𝑑𝑦)1/2

.

(2)

The constant 𝜋 is the best possible. In 1925, Hardy [2]extended theHilbert’s integral inequality into the integral ine-quality as follows. If 𝑝 > 1, 1/𝑝 + 1/𝑞 = 1, and 𝑓, 𝑔 ≥ 0 suchthat

0 < ∫∞

0

𝑓𝑝 (𝑥) 𝑑𝑥 < ∞, 0 < ∫∞

0

𝑔𝑞 (𝑦) 𝑑𝑦 < ∞, (3)

then we have the Hardy-Hilbert’s integral inequality that

∬∞

0

𝑓 (𝑥) 𝑔 (𝑦)

𝑥 + 𝑦𝑑𝑥 𝑑𝑦

<𝜋

sin (𝜋/𝑝)(∫∞

0

𝑓𝑝 (𝑥) 𝑑𝑥)1/𝑝

(∫∞

0

𝑔𝑞 (𝑦) 𝑑𝑦)1/𝑞

.

(4)

The constant 𝜋/ sin(𝜋/𝑝) is the best possible. Both the twoinequalities are important in mathematical analysis and itsapplications [3].

In 1938, Widder [4] studied on the Stieltjes Transform𝑆𝑓(𝑦) = ∫

∞

0

𝑓(𝑥)/(𝑥 + 𝑦)𝑑𝑥.Now, we recall the beta function 𝐵 as follows:

𝐵 (𝑝, 𝑞) = ∫1

0

𝑡𝑝−1(1 − 𝑡)𝑞−1𝑑𝑡, where 𝑝, 𝑞 > 0. (5)

In 2001, Yang [5] extended the Hardy-Hilbert’s integralinequality into the following integral inequality. If 𝑝, 𝑞 > 0,𝜆 > 2 −min{𝑝, 𝑞}, 1/𝑝 + 1/𝑞 = 1, and 𝑓, 𝑔 ≥ 0 such that

0 < ∫∞

0

𝑥1−𝜆𝑓𝑝 (𝑥) 𝑑𝑥 < ∞,

0 < ∫∞

0

𝑦1−𝜆𝑔𝑞 (𝑦) 𝑑𝑦 < ∞,

(6)

then we have

∬∞

0

𝑓 (𝑥) 𝑔 (𝑦)

(𝑥 + 𝑦)𝜆

𝑑𝑥 𝑑𝑦

2 Journal of Function Spaces and Applications

We also recall that a nonnegative function 𝑓(𝑥, 𝑦) whichis said to be homogeneous function of degree 𝜆 if 𝑓(𝑡𝑥, 𝑡𝑦) =𝑡𝜆𝑓(𝑥, 𝑦) for all 𝑡 > 0. And we say that𝐾(𝑢, V) is increasing if𝐾(1, 𝑡) and𝐾(𝑡, 1) are increasing functions.

In 2008, Sulaiman [6] gave new integral inequality similarto the Hardy-Hilbert’s integral inequality. If 𝑎, 𝑏 > 0, 𝑝 > 1,1/𝑝+ 1/𝑞 = 1, 0 < 𝜆 ≤ min{(1 − 𝑏)𝑝/𝑞, (1 − 𝑎)𝑞/𝑝},𝐾(𝑢, V) isa positive increasing homogeneous function of degree 𝜆, and𝑓, 𝑔 ≥ 0 and

𝐹 (𝑥) = ∫𝑥

0

𝑓 (𝑡) 𝑑𝑡, 𝐺 (𝑥) = ∫𝑥

0

𝑔 (𝑡) 𝑑𝑡 ∀𝑥 > 0, (8)

then, for all 𝑇 > 0, we have

∬𝑇

0

𝐹 (𝑢)𝐺 (V)

𝐾 (𝑢, V)𝑑𝑢 𝑑V

≤ 𝑇𝛼 𝑝√𝑝𝐾1

𝑞√𝑞𝐾2(∫𝑇

0

(𝑇 − 𝑡) 𝐹𝑝−1

(𝑡) 𝑓 (𝑡) 𝑑𝑡)

1/𝑝

× (∫𝑇

0

(𝑇 − 𝑡) 𝐺𝑞−1

(𝑡) 𝑔 (𝑡) 𝑑𝑡)

1/𝑞

,

(9)

where

𝐾1= ∫1

0

𝑡𝑎−1

𝐾 (1, 𝑡)𝑑𝑡, 𝐾

2= ∫1

0

𝑡𝑏−1

𝐾 (𝑡, 1)𝑑𝑡. (10)

In this paper, we present a generalization of the integralinequality (9) and its applications. Next proposition will beused in the next section.

Proposition 1 (see [6]). Let𝑔 be a positive increasing function,and 𝑎, 𝑏 > 0. Then, for all 𝑥 ≥ 1, one has

𝑥−𝑎 ∫𝑥

0

𝑡𝑎−1

𝑔 (𝑡)𝑑𝑡 ≤ ∫

1

0

𝑡𝑎−1

𝑔 (𝑡)𝑑𝑡. (11)

2. Main Results

Theorem 2. Let 0 < 𝑎, 𝑏 < 1 < 𝑝, 1/𝑝 + 1/𝑞 = 1, 0 < 𝜆 ≤min{(1−𝑏)𝑝/𝑞, (1−𝑎)𝑞/𝑝}, and let𝐾(𝑢, V) be positive increas-ing homogeneous function of degree 𝜆, and 𝑓, 𝑔 ≥ 0 and

𝐹 (𝑥) = ∫𝑥

0

𝑓 (𝑡) 𝑑𝑡, 𝐺 (𝑥) = ∫𝑥

0

𝑔 (𝑡) 𝑑𝑡 ∀𝑥 > 0, (12)

and let 𝜓 be a function such that 𝜓(𝑥) ≥ 𝑥 for all 𝑥 > 0.Then, for all 𝑇 > 0, one has

∬𝑇

0

𝐹 (𝑢)𝐺 (V)

𝜓 (𝐾 (𝑢, V))𝑑𝑢 𝑑V

≤ 𝑇1−𝜆 𝑝√𝑝𝐾1


0

(𝑇 − 𝑡) 𝐹𝑝−1

(𝑡) 𝑓 (𝑡) 𝑑𝑡)

1/𝑝

× (∫𝑇

0

(𝑇 − 𝑡) 𝐺𝑞−1

(𝑡) 𝑔 (𝑡) 𝑑𝑡)

1/𝑞

,

(13)

where

𝐾1= ∫1

0

𝑡𝑎−1

𝐾 (1, 𝑡)𝑑𝑡, 𝐾

2= ∫1

0

𝑡𝑏−1

𝐾 (𝑡, 1)𝑑𝑡. (14)

Proof. Let 𝑇 > 0 and𝑀 = ∬𝑇0

𝐹(𝑢)𝐺(V)/𝜓(𝐾(𝑢, V))𝑑𝑢 𝑑V.By the Hölder inequality, the assumption of 𝜓, and the

Tonelli theorem, we have

𝑀 =∬𝑇

0

𝐹 (𝑢) V(𝑎−1)/𝑝

𝑢(𝑏−1)/𝑞𝜓1/𝑝 (𝐾 (𝑢, V))

×𝐺 (V) 𝑢(𝑏−1)/𝑞

V(𝑎−1)/𝑝𝜓1/𝑞 (𝐾 (𝑢, V))𝑑𝑢 𝑑V

≤ (∬𝑇

0

𝐹𝑝 (𝑢) V𝑎−1

𝑢(𝑏−1)𝑝/𝑞𝜓 (𝐾 (𝑢, V))𝑑𝑢 𝑑V)

1/𝑝

×(∬𝑇

0

𝐺𝑞 (V) 𝑢𝑏−1

V(𝑎−1)𝑞/𝑝𝜓 (𝐾 (𝑢, V))𝑑𝑢 𝑑V)

1/𝑞

= (∫𝑇

0

𝐹𝑝 (𝑢) ∫𝑇

0

𝑢(1−𝑏)𝑝/𝑞V𝑎−1

𝜓 (𝐾 (𝑢, V))𝑑V 𝑑𝑢)

1/𝑝

×(∫𝑇

0

𝐺𝑞 (V) ∫𝑇

0

𝑢𝑏−1V(1−𝑎)𝑞/𝑝

𝜓 (𝐾 (𝑢, V))𝑑𝑢 𝑑V)

1/𝑞

≤ (∫𝑇

0


0

𝑢(1−𝑏)𝑝/𝑞V𝑎−1

𝐾 (𝑢, V)𝑑V 𝑑𝑢)

1/𝑝

×(∫𝑇

0


0

𝑢𝑏−1V(1−𝑎)𝑞/𝑝

𝐾 (𝑢, V)𝑑𝑢 𝑑V)

1/𝑞

= (∫𝑇

0


0

𝑢(1−𝑏)(𝑝/𝑞)+𝑎−1(V/𝑢)𝑎−1

𝑢𝜆𝐾 (1, V/𝑢)𝑑V 𝑑𝑢)

1/𝑝

× (∫𝑇

0


0

V(1−𝑎)(𝑞/𝑝)+𝑏−1(𝑢/V)𝑏−1

V𝜆𝐾 (𝑢/V, 1)𝑑𝑢 𝑑V)

1/𝑞

.

(15)

Now, we put 𝑡 = V/𝑢 and 𝑑𝑡 = 𝑑V/𝑢 for the first integral,and then we put 𝑡 = 𝑢/V and 𝑑𝑡 = 𝑑𝑢/V for the secondintegral.

And, by Proposition 1, one has

𝑀 ≤ (∫𝑇

0

𝑢𝑎+(1−𝑏)(𝑝/𝑞)−𝜆𝐹𝑝 (𝑢) ∫𝑇/𝑢

0

𝑡𝑎−1

𝐾 (1, 𝑡)𝑑𝑡 𝑑𝑢)

1/𝑝

× (∫𝑇

0

V𝑏+(1−𝑎)(𝑞/𝑝)−𝜆𝐺𝑞 (V) ∫𝑇/𝑢

0

𝑡𝑏−1

𝐾 (𝑡, 1)𝑑𝑡 𝑑V)

1/𝑞

Journal of Function Spaces and Applications 3

= (𝑇𝑎+(1−𝑏)(𝑝/𝑞)−𝜆 ∫𝑇

0

𝐹𝑝 (𝑢) (𝑢

𝑇)𝑎+(1−𝑏)(𝑝/𝑞)−𝜆

× ∫𝑇/𝑢

0

𝑡𝑎−1


1/𝑝

× (𝑇𝑏+(1−𝑎)(𝑞/𝑝)−𝜆 ∫𝑇

0

𝐺𝑞 (V) (V

𝑇)𝑏+(1−𝑎)(𝑞/𝑝)−𝜆

× ∫𝑇/𝑢

0

𝑡𝑏−1


1/𝑞

≤ (𝑇𝑎+(1−𝑏)(𝑝/𝑞)−𝜆 ∫𝑇

0


𝑇)𝑎

×∫𝑇/𝑢

0

𝑡𝑎−1


1/𝑝

× (𝑇𝑏+(1−𝑎)(𝑞/𝑝)−𝜆 ∫𝑇

0

𝐺𝑞 (V) (V

𝑇)𝑏

×∫𝑇/𝑢

0

𝑡𝑏−1


1/𝑞

= 𝑇1−𝜆(∫𝑇

0


𝑇)𝑎

∫𝑇/𝑢

0

𝑡𝑎−1


1/𝑝

× (∫𝑇

0

𝐺𝑞 (V) (V

𝑇)𝑏

∫𝑇/𝑢

0

𝑡𝑏−1


1/𝑞

≤ 𝑇1−𝜆(∫𝑇

0

𝐹𝑝 (𝑢) ∫1

0

𝑡𝑎−1


1/𝑝

× (∫𝑇

0

𝐺𝑞 (V) ∫1

0

𝑡𝑏−1


1/𝑞

= 𝑇1−𝜆(𝐾1∫𝑇

0

𝐹𝑝 (𝑢) 𝑑𝑢)

1/𝑝

×(𝐾2∫𝑇

0

𝐺𝑞 (V) 𝑑V)1/𝑞

.

(16)

Then, by the assumption, one has

𝑀 ≤ 𝑇1−𝜆(𝐾1∫𝑇

0

∫𝑢

0

(𝐹𝑝 (𝑡))

𝑑𝑡 𝑑𝑢)

1/𝑝

× (𝐾2∫𝑇

0

∫V

0

(𝐺𝑞 (𝑡))

𝑑𝑡 𝑑V)1/𝑞

= 𝑇1−𝜆(𝑝𝐾1∫𝑇

0

∫𝑢

0

𝐹𝑝−1 (𝑡) 𝑓 (𝑡) 𝑑𝑡 𝑑𝑢)

1/𝑝

×(𝑞𝐾2∫𝑇

0

∫V

0

𝐺𝑞−1 (𝑡) 𝑓 (𝑡) 𝑑𝑡 𝑑V)1/𝑞


0

𝐹𝑝−1 (𝑡) 𝑓 (𝑡) ∫𝑇

𝑡

𝑑𝑢 𝑑𝑡)

1/𝑝

×(𝑞𝐾2∫𝑇

0

𝐺𝑞−1 (𝑡) 𝑓 (𝑡) ∫𝑇

𝑡

𝑑V 𝑑𝑡)1/𝑞


0

(𝑇 − 𝑡) 𝐹𝑝−1

(𝑡) 𝑓 (𝑡) 𝑑𝑡)

1/𝑝

×(𝑞𝐾2∫𝑇

0

(𝑇 − 𝑡) 𝐺𝑞−1

(𝑡) 𝑓 (𝑡) 𝑑𝑡)

1/𝑞

= 𝑇1−𝜆 𝑝√𝑝𝐾1

𝑞√𝑞𝐾2

×(∫𝑇

0

(𝑇 − 𝑡) 𝐹𝑝−1

(𝑡) 𝑓 (𝑡) 𝑑𝑡)

1/𝑝

× (∫𝑇

0

(𝑇 − 𝑡) 𝐺𝑞−1

(𝑡) 𝑔 (𝑡) 𝑑𝑡)

1/𝑞

.

(17)

This proof is completed.

3. Applications

Corollary 3. Let 0 < 𝑎, 𝑏 < 1 < 𝑝, 1/𝑝 + 1/𝑞 = 1 and 0 <𝜆 ≤ min{(1 − 𝑏)𝑝/𝑞, (1 − 𝑎)𝑞/𝑝}, and let 𝐾(𝑢, V) be a positiveincreasing homogeneous function of degree 𝜆, and𝑓, 𝑔 ≥ 0 and

𝐹 (𝑥) = ∫𝑥

0

𝑓 (𝑡) 𝑑𝑡, 𝐺 (𝑥) = ∫𝑥

0

𝑔 (𝑡) 𝑑𝑡 ∀𝑥 > 0. (18)

Then, for all 𝑇 > 0, one has

(a) ∬𝑇

0

𝐹 (𝑢)𝐺 (V)

𝐾 (𝑢, V)𝑑𝑢 𝑑V


𝑞√𝑞𝐾2

×(∫𝑇

0

(𝑇 − 𝑡) 𝐹𝑝−1

(𝑡) 𝑓 (𝑡) 𝑑𝑡)

1/𝑝

× (∫𝑇

0

(𝑇 − 𝑡) 𝐺𝑞−1

(𝑡) 𝑔 (𝑡) 𝑑𝑡)

1/𝑞

,

(19)

(b) ∬𝑇

0

𝐹 (𝑢)𝐺 (V)

1 + 𝐾 (𝑢, V)𝑑𝑢 𝑑V


𝑞√𝑞𝐾2

×(∫𝑇

0

(𝑇 − 𝑡) 𝐹𝑝−1

(𝑡) 𝑓 (𝑡) 𝑑𝑡)

1/𝑝

× (∫𝑇

0

(𝑇 − 𝑡) 𝐺𝑞−1

(𝑡) 𝑔 (𝑡) 𝑑𝑡)

1/𝑞

,

(20)

4 Journal of Function Spaces and Applications

(c) ∬𝑇

0

𝐹 (𝑢) 𝐺 (V)

(1 + 𝐾 (𝑢, V)) 𝐾 (𝑢, V)𝑑𝑢 𝑑V


𝑞√𝑞𝐾2

×(∫𝑇

0

(𝑇 − 𝑡) 𝐹𝑝−1

(𝑡) 𝑓 (𝑡) 𝑑𝑡)

1/𝑝

× (∫𝑇

0

(𝑇 − 𝑡) 𝐺𝑞−1

(𝑡) 𝑔 (𝑡) 𝑑𝑡)

1/𝑞

,

(21)

(d) ∬𝑇

0

𝐹 (𝑢) 𝐺 (V)

𝑒𝐾(𝑢,V)𝑑𝑢 𝑑V



0

(𝑇 − 𝑡) 𝐹𝑝−1

(𝑡) 𝑓 (𝑡) 𝑑𝑡)

1/𝑝

× (∫𝑇

0

(𝑇 − 𝑡)𝐺𝑞−1(𝑡)𝑔(𝑡)𝑑𝑡)

1/𝑞

,

(22)

where

𝐾1= ∫1

0

𝑡𝑎−1

𝐾 (1, 𝑡)𝑑𝑡, 𝐾

2= ∫1

0

𝑡𝑏−1

𝐾 (𝑡, 1)𝑑𝑡. (23)

Proof. (a) This follows from Theorem 2 where 𝜓(𝑥) = 𝑥 forall 𝑥.

(b) This follows fromTheorem 2 where 𝜓(𝑥) = 1 + 𝑥 forall 𝑥.

(c) This follows fromTheorem 2 where 𝜓(𝑥) = 𝑥 + 𝑥2 forall 𝑥.

(d) This follows fromTheorem 2 where 𝜓(𝑥) = 𝑒𝑥 for all𝑥.

4. Open Problem

In this section, we pose a question that is how to generalizethe integral inequality (13) if 𝜓 may not satisfy the property𝜓(𝑥) ≥ 𝑥 for all 𝑥 > 0.

Acknowledgments

The author would like to thank the referees for their usefulcomments and suggestions.

References

[1] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cam-bridge University Press, Cambridge, UK, 2nd edition, 1952.

[2] G. H. Hardy, “Notes on a theorem of Hilbert concerning seriesof positive terms,” Proceedings of the LondonMathematical Soci-ety, vol. 23, pp. 45–46, 1925.

[3] D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, InequalitiesInvolving Functions andTheir Integrals andDerivatives, vol. 53 ofMathematics and Its Applications (East European Series), KluwerAcademic Publishers, Dordrecht, The Netherlands, 1991.

[4] D. V. Widder, “The Stieltjes transform,” Transactions of theAmerican Mathematical Society, vol. 43, no. 1, pp. 7–60, 1938.

[5] B. Yang, “On Hardy-Hilbert’s integral inequality,” Journal ofMathematical Analysis and Applications, vol. 261, pp. 295–306,2001.

[6] W. T. Sulaiman, “A study on some new types of Hardy-Hilbert’sintegral inequalities,” Banach Journal of Mathematical Analysis,vol. 2, no. 1, pp. 16–20, 2008.

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