A Generalization on Some New Types of Hardy-Hilbert’s Integral Inequalities

Abstract

Sulaiman presented, in 2008, new kinds of Hardy-Hilbert’s integral inequality in which the weight function is homogeneous. In this 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 such that 0<^∞₀𝑓²(𝑥)𝑑𝑥<∞,0<^∞₀𝑔²𝑦𝑑𝑦<∞, (1) we have the Hilbert’s integral inequality [1] that ^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦𝑑𝑥𝑑𝑦<𝜋^∞₀𝑓²(𝑥)𝑑𝑥^∞₀𝑔²𝑦𝑑𝑦^1/2. (2) The constant 𝜋 is the best possible. In 1925, Hardy [2] extended the Hilbert’s integral inequality into the integral inequality as follows. If 𝑝>1, 1/𝑝+1/𝑞=1, and 𝑓,𝑔≥0 such that 0<^∞₀𝑓^𝑝(𝑥)𝑑𝑥<∞,0<^∞₀𝑔^𝑞𝑦𝑑𝑦<∞, (3) then we have the Hardy-Hilbert’s integral inequality that ^∞₀𝑓(𝑥)𝑔𝑦<𝜋𝑥+𝑦𝑑𝑥𝑑𝑦sin𝜋/𝑝^∞₀𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑔^𝑞𝑦𝑑𝑦^1/𝑞. (4) The constant 𝜋/sin(𝜋/𝑝) is the best possible. Both the two inequalities are important in mathematical analysis and its applications [3].

In 1938, Widder [4] studied on the Stieltjes Transform 𝑆_𝑓(𝑦)=∫^∞₀𝑓(𝑥)/(𝑥+𝑦)𝑑𝑥.

Now, we recall the beta function 𝐵 as follows: 𝐵𝑝,𝑞=¹₀𝑡^𝑝−1(1−𝑡)^𝑞−1𝑑𝑡,where𝑝,𝑞>0. (5)

In 2001, Yang [5] extended the Hardy-Hilbert’s integral inequality 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<^∞₀𝑥^1−𝜆𝑓^𝑝(𝑥)𝑑𝑥<∞,0<^∞₀𝑦^1−𝜆𝑔^𝑞𝑦𝑑𝑦<∞, (6)

then we have ^∞₀𝑓(𝑥)𝑔𝑦𝑥+𝑦^𝜆𝑑𝑥𝑑𝑦<𝑘_𝜆𝑝^∞₀𝑥^1−𝜆𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝∞₀𝑦^1−𝜆𝑔^𝑞𝑦𝑑𝑦^1/𝑞, (7) where 𝑘_𝜆(𝑝)=𝐵(1+(𝜆−2)/𝑝,1+(𝜆−2)/𝑝). The constant 𝑘_𝜆(𝑝) is the best possible.

We also recall that a nonnegative function 𝑓(𝑥,𝑦) which is said to be homogeneous function of degree 𝜆 if 𝑓(𝑡𝑥,𝑡𝑦)=𝑡^𝜆𝑓(𝑥,𝑦) for all 𝑡>0. And we say that 𝐾(𝑢,𝑣) is increasing if 𝐾(1,𝑡) and 𝐾(𝑡,1) are increasing functions.

In 2008, Sulaiman [6] gave new integral inequality similar to 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) is a positive increasing homogeneous function of degree 𝜆, and 𝑓,𝑔≥0 and 𝐹(𝑥)=^𝑥₀𝑓(𝑡)𝑑𝑡,𝐺(𝑥)=^𝑥₀𝑔(𝑡)𝑑𝑡∀𝑥>0, (8) then, for all 𝑇>0, we have ^𝑇₀𝐹(𝑢)𝐺(𝑣)𝐾(𝑢,𝑣)𝑑𝑢𝑑𝑣≤𝑇^𝛼𝑝𝑝𝐾₁^𝑞𝑞𝐾₂^𝑇₀(𝑇−𝑡)𝐹^𝑝−1(𝑡)𝑓(𝑡)𝑑𝑡^1/𝑝×^𝑇₀(𝑇−𝑡)𝐺^𝑞−1(𝑡)𝑔(𝑡)𝑑𝑡^1/𝑞, (9) where 𝐾₁=¹₀𝑡^𝑎−1𝐾(1,𝑡)𝑑𝑡,𝐾₂=¹₀𝑡^𝑏−1𝐾(𝑡,1)𝑑𝑡. (10)

In this paper, we present a generalization of the integral inequality (9) and its applications. Next proposition will be used in the next section.

Proposition 1 (see [6]). Let 𝑔 be a positive increasing function, and 𝑎,𝑏>0. Then, for all 𝑥≥1, one has 𝑥^−𝑎𝑥₀𝑡^𝑎−1𝑔(𝑡)𝑑𝑡≤¹₀𝑡^𝑎−1𝑔(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 let 𝐾(𝑢,𝑣) be positive increasing homogeneous function of degree 𝜆, and 𝑓,𝑔≥0 and 𝐹(𝑥)=^𝑥₀𝑓(𝑡)𝑑𝑡,𝐺(𝑥)=^𝑥₀𝑔(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 ^𝑇₀𝐹(𝑢)𝐺(𝑣)𝜓(𝐾(𝑢,𝑣))𝑑𝑢𝑑𝑣≤𝑇^1−𝜆𝑝𝑝𝐾₁^𝑞𝑞𝐾₂^𝑇₀(𝑇−𝑡)𝐹^𝑝−1(𝑡)𝑓(𝑡)𝑑𝑡^1/𝑝×^𝑇₀(𝑇−𝑡)𝐺^𝑞−1(𝑡)𝑔(𝑡)𝑑𝑡^1/𝑞, (13) where 𝐾₁=¹₀𝑡^𝑎−1𝐾(1,𝑡)𝑑𝑡,𝐾₂=¹₀𝑡^𝑏−1𝐾(𝑡,1)𝑑𝑡. (14)

Proof. Let 𝑇>0 and 𝑀=∬^𝑇₀𝐹(𝑢)𝐺(𝑣)/𝜓(𝐾(𝑢,𝑣))𝑑𝑢𝑑𝑣.
By the Hölder inequality, the assumption of 𝜓, and the Tonelli theorem, we have 𝑀=^𝑇₀𝐹(𝑢)𝑣^{(𝑎−1)/𝑝}𝑢^{(𝑏−1)/𝑞}𝜓^1/𝑝(𝐾(×𝐺(𝑣)𝑢𝑢,𝑣))_{(𝑏−1)/𝑞}𝑣^{(𝑎−1)/𝑝}𝜓^1/𝑞(𝐾(≤𝑢,𝑣))𝑑𝑢𝑑𝑣_𝑇0𝐹^𝑝(𝑢)𝑣^𝑎−1𝑢^{(𝑏−1)𝑝/𝑞}𝜓(𝐾(𝑢,𝑣))𝑑𝑢𝑑𝑣^1/𝑝×^𝑇₀𝐺^𝑞(𝑣)𝑢^𝑏−1𝑣^{(𝑎−1)𝑞/𝑝}𝜓(𝐾(𝑢,𝑣))𝑑𝑢𝑑𝑣^1/𝑞=^𝑇₀𝐹^𝑝(𝑢)^𝑇₀𝑢^{(1−𝑏)𝑝/𝑞}𝑣^𝑎−1𝜓(𝐾(𝑢,𝑣))𝑑𝑣𝑑𝑢^1/𝑝×^𝑇₀𝐺^𝑞(𝑣)^𝑇₀𝑢^𝑏−1𝑣^{(1−𝑎)𝑞/𝑝}𝜓(𝐾(𝑢,𝑣))𝑑𝑢𝑑𝑣^1/𝑞≤^𝑇₀𝐹^𝑝(𝑢)^𝑇₀𝑢^{(1−𝑏)𝑝/𝑞}𝑣^𝑎−1𝐾(𝑢,𝑣)𝑑𝑣𝑑𝑢^1/𝑝×^𝑇₀𝐺^𝑞(𝑣)^𝑇₀𝑢^𝑏−1𝑣^{(1−𝑎)𝑞/𝑝}𝐾(𝑢,𝑣)𝑑𝑢𝑑𝑣^1/𝑞=^𝑇₀𝐹^𝑝(𝑢)^𝑇₀𝑢^{(1−𝑏)(𝑝/𝑞)+𝑎−1}(𝑣/𝑢)^𝑎−1𝑢^𝜆𝐾(1,𝑣/𝑢)𝑑𝑣𝑑𝑢^1/𝑝×^𝑇₀𝐺^𝑞(𝑣)^𝑇₀𝑣^{(1−𝑎)(𝑞/𝑝)+𝑏−1}(𝑢/𝑣)^𝑏−1𝑣^𝜆𝐾(𝑢/𝑣,1)𝑑𝑢𝑑𝑣^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 second integral.
And, by Proposition 1, one has 𝑀≤^𝑇₀𝑢^{𝑎+(1−𝑏)(𝑝/𝑞)−𝜆}𝐹^𝑝(𝑢)₀^𝑇/𝑢𝑡^𝑎−1𝐾(1,𝑡)𝑑𝑡𝑑𝑢^1/𝑝×^𝑇₀𝑣^{𝑏+(1−𝑎)(𝑞/𝑝)−𝜆}𝐺^𝑞(𝑣)₀^𝑇/𝑢𝑡^𝑏−1𝐾(𝑡,1)𝑑𝑡𝑑𝑣^1/𝑞=𝑇^{𝑎+(1−𝑏)(𝑝/𝑞)−𝜆𝑇}₀𝐹^𝑝(𝑢𝑢)𝑇^{𝑎+(1−𝑏)(𝑝/𝑞)−𝜆}×₀^𝑇/𝑢𝑡^𝑎−1𝐾(1,𝑡)𝑑𝑡𝑑𝑢^1/𝑝×𝑇^{𝑏+(1−𝑎)(𝑞/𝑝)−𝜆𝑇}₀𝐺^𝑞(𝑣)𝑣𝑇^{𝑏+(1−𝑎)(𝑞/𝑝)−𝜆}×₀^𝑇/𝑢𝑡^𝑏−1𝐾(𝑡,1)𝑑𝑡𝑑𝑣^1/𝑞≤𝑇^{𝑎+(1−𝑏)(𝑝/𝑞)−𝜆𝑇}₀𝐹^𝑝(𝑢)𝑢𝑇^𝑎×₀^𝑇/𝑢𝑡^𝑎−1𝐾(1,𝑡)𝑑𝑡𝑑𝑢^1/𝑝×𝑇^{𝑏+(1−𝑎)(𝑞/𝑝)−𝜆𝑇}₀𝐺^𝑞(𝑣)𝑣𝑇^𝑏×₀^𝑇/𝑢𝑡^𝑏−1𝐾(𝑡,1)𝑑𝑡𝑑𝑣^1/𝑞=𝑇^1−𝜆𝑇₀𝐹^𝑝(𝑢𝑢)𝑇^𝑎₀^𝑇/𝑢𝑡^𝑎−1𝐾(1,𝑡)𝑑𝑡𝑑𝑢^1/𝑝×^𝑇₀𝐺^𝑞(𝑣)𝑣𝑇^𝑏₀^𝑇/𝑢𝑡^𝑏−1𝐾(𝑡,1)𝑑𝑡𝑑𝑣^1/𝑞≤𝑇^1−𝜆𝑇₀𝐹^𝑝(𝑢)¹₀𝑡^𝑎−1𝐾(1,𝑡)𝑑𝑡𝑑𝑢^1/𝑝×^𝑇₀𝐺^𝑞(𝑣)¹₀𝑡^𝑏−1𝐾(𝑡,1)𝑑𝑡𝑑𝑣^1/𝑞=𝑇^1−𝜆𝐾₁^𝑇₀𝐹^𝑝(𝑢)𝑑𝑢^1/𝑝×𝐾₂^𝑇₀𝐺^𝑞(𝑣)𝑑𝑣^1/𝑞. (16)
Then, by the assumption, one has 𝑀≤𝑇^1−𝜆𝐾₁^𝑇₀^𝑢₀𝐹^𝑝(𝑡)^′𝑑𝑡𝑑𝑢^1/𝑝×𝐾₂^𝑇₀^𝑣₀𝐺^𝑞(𝑡)^′𝑑𝑡𝑑𝑣^1/𝑞=𝑇^1−𝜆𝑝𝐾₁^𝑇₀^𝑢₀𝐹^𝑝−1(𝑡)𝑓(𝑡)𝑑𝑡𝑑𝑢^1/𝑝×𝑞𝐾₂^𝑇₀^𝑣₀𝐺^𝑞−1(𝑡)𝑓(𝑡)𝑑𝑡𝑑𝑣^1/𝑞=𝑇^1−𝜆𝑝𝐾₁^𝑇₀𝐹^𝑝−1(𝑡)𝑓(𝑡)^𝑇_𝑡𝑑𝑢𝑑𝑡^1/𝑝×𝑞𝐾₂^𝑇₀𝐺^𝑞−1(𝑡)𝑓(𝑡)^𝑇_𝑡𝑑𝑣𝑑𝑡^1/𝑞=𝑇^1−𝜆𝑝𝐾₁^𝑇₀(𝑇−𝑡)𝐹^𝑝−1(𝑡)𝑓(𝑡)𝑑𝑡^1/𝑝×𝑞𝐾₂^𝑇₀(𝑇−𝑡)𝐺^𝑞−1(𝑡)𝑓(𝑡)𝑑𝑡^1/𝑞=𝑇^1−𝜆𝑝𝑝𝐾₁^𝑞𝑞𝐾₂×^𝑇₀(𝑇−𝑡)𝐹^𝑝−1(𝑡)𝑓(𝑡)𝑑𝑡^1/𝑝×^𝑇₀(𝑇−𝑡)𝐺^𝑞−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 positive increasing homogeneous function of degree 𝜆, and 𝑓,𝑔≥0 and 𝐹(𝑥)=^𝑥₀𝑓(𝑡)𝑑𝑡,𝐺(𝑥)=^𝑥₀𝑔(𝑡)𝑑𝑡∀𝑥>0. (18)
Then, for all 𝑇>0, one has(a)^𝑇₀𝐹(𝑢)𝐺(𝑣)𝐾(𝑢,𝑣)𝑑𝑢𝑑𝑣≤𝑇^1−𝜆𝑝𝑝𝐾₁^𝑞𝑞𝐾₂×^𝑇₀(𝑇−𝑡)𝐹^𝑝−1(𝑡)𝑓(𝑡)𝑑𝑡^1/𝑝×^𝑇₀(𝑇−𝑡)𝐺^𝑞−1(𝑡)𝑔(𝑡)𝑑𝑡^1/𝑞, (19)(b)^𝑇₀𝐹(𝑢)𝐺(𝑣)1+𝐾(𝑢,𝑣)𝑑𝑢𝑑𝑣≤𝑇^1−𝜆𝑝𝑝𝐾₁^𝑞𝑞𝐾₂×^𝑇₀(𝑇−𝑡)𝐹^𝑝−1(𝑡)𝑓(𝑡)𝑑𝑡^1/𝑝×^𝑇₀(𝑇−𝑡)𝐺^𝑞−1(𝑡)𝑔(𝑡)𝑑𝑡^1/𝑞, (20)(c)^𝑇₀𝐹(𝑢)𝐺(𝑣)(1+𝐾(𝑢,𝑣))𝐾(𝑢,𝑣)𝑑𝑢𝑑𝑣≤𝑇^1−𝜆𝑝𝑝𝐾₁^𝑞𝑞𝐾₂×^𝑇₀(𝑇−𝑡)𝐹^𝑝−1(𝑡)𝑓(𝑡)𝑑𝑡^1/𝑝×^𝑇₀(𝑇−𝑡)𝐺^𝑞−1(𝑡)𝑔(𝑡)𝑑𝑡^1/𝑞, (21)(d)^𝑇₀𝐹(𝑢)𝐺(𝑣)𝑒^{𝐾(𝑢,𝑣)}𝑑𝑢𝑑𝑣≤𝑇^1−𝜆𝑝𝑝𝐾₁^𝑞𝑞𝐾₂^𝑇₀(𝑇−𝑡)𝐹^𝑝−1(𝑡)𝑓(𝑡)𝑑𝑡^1/𝑝×^𝑇₀(𝑇−𝑡)𝐺^𝑞−1(𝑡)𝑔(𝑡)𝑑𝑡^1/𝑞, (22) where 𝐾₁=¹₀𝑡^𝑎−1𝐾(1,𝑡)𝑑𝑡,𝐾₂=¹₀𝑡^𝑏−1𝐾(𝑡,1)𝑑𝑡. (23)

Proof. (a) This follows from Theorem 2 where 𝜓(𝑥)=𝑥 for all 𝑥.
(b) This follows from Theorem 2 where 𝜓(𝑥)=1+𝑥 for all 𝑥.
(c) This follows from Theorem 2 where 𝜓(𝑥)=𝑥+𝑥² for all 𝑥.
(d) This follows from Theorem 2 where 𝜓(𝑥)=𝑒^𝑥 for all 𝑥.

4. Open Problem

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

Acknowledgments

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