On Hardy-Pachpatte-Copson's Inequalities

Abstract

We establish new inequalities similar to Hardy-Pachpatte-Copson’s type inequalities. These results in special cases yield some of the recent results.

1. Introduction

The classical Hardy’s integral inequality is as follows.

Theorem A. If 𝑝>1, 𝑓(𝑥)≥0 for 0<𝑥<∞, and 𝐹(𝑥)=(1/𝑥)∫^𝑥₀𝑓(𝑡)𝑑𝑡, then ^∞₀𝐹(𝑥)^𝑝𝑝𝑑𝑥<𝑝−1^𝑝∞₀𝑓(𝑥)^𝑝𝑑𝑥, (1) unless 𝑓≡0. The constant is the best possible.

Theorem A was first proved by Hardy [1], in an attempt to give a simple proof of Hilbert's double series theorem (see [2]). One of the best known and interesting generalization of the inequality (1) given by Hardy [3] himself can be stated as follows.

Theorem B. If 𝑝>1,𝑚≠1,𝑓(𝑥)≥0 for 0<𝑥<∞, and 𝐹(𝑥) is defined by 𝐹(𝑥)=^𝑥₀𝑓(𝑡)𝐹(𝑥)=𝑑𝑡,𝑚>1;_∞𝑥𝑓(𝑡)𝑑𝑡,𝑚<1, (2) then ^∞₀𝑥^−𝑚𝐹(𝑥)^𝑝𝑝𝑑𝑥<|𝑚−1|^𝑝∞₀𝑥^𝑝−𝑚𝑓(𝑥)^𝑝𝑑𝑥, (3) unless 𝑓≡0. The constant is the best possible.

Inequalities (1) and (3) which later went by the name of Hardy’s inequalities led to a great many papers dealing with alternative proofs, various generalizations, and numerous variants and applications in analysis (see [4–15]). In particular, Pachpatte [4] established some generalizations of Hardy inequalities (1) and (3). Very recently, Leng and Feng [16] proved some new Hardy-type integral inequalities. In the present paper we establish new inequalities similar to Hardy's integral inequalities (1) and (3). These results provide some new estimates to these types of inequalities and in special cases yield some of the recent results.

2. Main Results

Our main results are given in the following theorems.

Theorem 1. Let 𝑎<𝑏<𝑅, 𝑐<𝑑<𝑅^′, 𝑝>1, 𝑞<1, and 𝛼>0 be constants. Let 𝑤(𝑥,𝑦) be positive and locally absolutely continuous in (𝑎,𝑏)×(𝑐,𝑑). Let ℎ(𝑥,𝑦) be a positive continuous function and let 𝐻(𝑥,𝑦)=∫^𝑥_𝑎∫^𝑦_𝑐ℎ(𝑠,𝑡)𝑑𝑠𝑑𝑡, for (𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑). Let 𝑓(𝑥,𝑦) be nonnegative and measurable on (𝑎,𝑏)×(𝑐,𝑑). If 𝐷1𝑥,𝑦=1−𝐻1−𝑞𝑥,𝑦ℎ1𝑥,𝑦𝑤𝑥,𝑦𝜕𝑤𝑥,𝑦𝐻𝜕𝑥log𝑅,𝑅^′𝐻+𝑝𝑥,𝑦𝐻1−𝑞𝑥,𝑦ℎ×1𝑥,𝑦𝑟𝑥,𝑦𝜕𝑟𝑥,𝑦𝐻𝜕𝑥log𝑅,𝑅^′𝐻≥1𝑥,𝑦𝛼, (4) for almost all (𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑), and if 𝐹(𝑥,𝑦) is defined by 𝐹=1𝑥,𝑦𝑟𝑥,𝑦^𝑥_𝑎^𝑦_𝑐𝑟(𝑠,𝑡)ℎ(𝑠,𝑡)𝑓(𝑠,𝑡)𝑑𝑠𝑑𝑡 (5) for (𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑), then ^𝑑_𝑐^𝑏_𝑎𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹ℎ×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞𝐹𝑥,𝑦^𝑝≤𝛼𝑝𝑑𝑥𝑑𝑦1−𝑞^𝑝×^𝑑_𝑐^𝑏_𝑎𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^𝑝−𝑞𝑤𝑥,𝑦𝐻𝑥,𝑦^𝑝−1×ℎ𝑥,𝑦^1−𝑝𝐺𝑥,𝑦^𝑝𝑑𝑥𝑑𝑦, (6) where ℎ𝑥,𝑦=^𝑦_𝑐ℎ()𝐺=1𝑥,𝑡𝑑𝑡,𝑥,𝑦𝑟𝑥,𝑦^𝑦_𝑐𝑟(𝑥,𝑡)ℎ(𝑥,𝑡)𝑓(𝑥,𝑡)𝑑𝑡. (7)

Remark 2. Let 𝑓(𝑥,𝑦), 𝑤(𝑥,𝑦), ℎ(𝑥,𝑦), and 𝑟(𝑥,𝑦) reduce to 𝑓(𝑥), 𝑤(𝑥), ℎ(𝑥), and 𝑟(𝑥), respectively, and with suitable modifications in Theorem 1, (6) changes to the following result: ^𝑏_𝑎𝑤(𝑥)𝐻(𝑥)⁻¹ℎ(𝑥)𝐻(𝑅)log𝐻(𝑥)^−𝑞𝐹(𝑥)^𝑝≤𝛼𝑝𝑑𝑥1−𝑞^𝑝×^𝑏_𝑎𝐻(𝑅)log𝐻(𝑥)^𝑝−𝑞×𝑤(𝑥)𝐻(𝑥)^𝑝−1ℎ(𝑥)𝑓(𝑥)^𝑝𝑑𝑥. (8)

This is just a new inequality established by Pachpatte [4].

Moreover, we note that the inequality established in Theorem 1 is the further generalizations of the inequality established by Copson [17].

Taking for 𝑤(𝑥)=𝑟(𝑥)=1, 𝐻(𝑅)=𝑅, and 𝛼=1 in (8), (8) changes to the following result: ^𝑏_𝑎𝐻(𝑥)⁻¹ℎ(𝑥)𝑅log𝐻(𝑥)^−𝑞𝐹(𝑥)^𝑝≤𝑝𝑑𝑥1−𝑞^𝑝×^𝑏_𝑎𝑅log𝐻(𝑥)^𝑝−𝑞𝐻(𝑥)^𝑝−1ℎ(𝑥)𝑓(𝑥)^𝑝𝑑𝑥. (9)

This is just a new inequality established by Love [7].

Let ℎ(𝑥)=1, 𝑎→0, 𝑏→∞, and log(𝑅/(𝑥−𝑎))=1 in (9); then (9) changes to the following result: ^∞₀𝑥⁻¹𝐹(𝑥)^𝑝𝑝𝑑𝑥≤1−𝑞^𝑝∞₀𝑥^𝑝−1𝑓(𝑥)^𝑝𝑑𝑥. (10)

This result is obtained in (3) stated in the Introduction.

Theorem 3. Let 𝑎<𝑏<𝑅, 𝑐<𝑑<𝑅^′, 𝑝>1, 𝑞>1, and 𝛽>0 be constants. Let 𝑤(𝑥,𝑦) be positive and locally absolutely continuous in (𝑎,𝑏)×(𝑐,𝑑). Let ℎ(𝑥,𝑦) be a positive continuous function and let 𝐻(𝑥,𝑦)=∫^𝑥_𝑎∫^𝑦_𝑐ℎ(𝑠,𝑡)𝑑𝑠𝑑𝑡, for (𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑). Let 𝑓(𝑥,𝑦) be nonnegative and measurable on (𝑎,𝑏)×(𝑐,𝑑). Let 𝐸1𝑥,𝑦=1−𝐻𝑞−1𝑥,𝑦ℎ1𝑥,𝑦𝑤𝑥,𝑦𝜕𝑤𝑥,𝑦𝐻𝜕𝑦log𝑅,𝑅^′𝐻+𝑝𝑥,𝑦𝐻𝑞−1𝑥,𝑦ℎ×1𝑥,𝑦𝑟𝑥,𝑦𝜕𝑟𝑥,𝑦𝐻𝜕𝑦log𝑅,𝑅^′𝐻≥1𝑥,𝑦𝛽, (11) for almost all (𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑). If 𝐹(𝑥,y) is defined by 𝐹=1𝑥,𝑦𝑟𝑥,𝑦^𝑏_𝑥^𝑑_𝑦𝑟(𝑠,𝑡)ℎ(𝑠,𝑡)𝑓(𝑠,𝑡)𝑑𝑠𝑑𝑡 (12) for (𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑), then ^𝑑_𝑐^𝑏_𝑎𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹ℎ×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞𝐹𝑥,𝑦^𝑝≤𝛽𝑝𝑑𝑥𝑑𝑦𝑞−1^𝑝×^𝑑_𝑐^𝑏_𝑎𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^𝑝−𝑞𝑤𝑥,𝑦𝐻𝑥,𝑦^𝑝−1×ℎ𝑥,𝑦^1−𝑝𝐿𝑥,𝑦^𝑝𝑑𝑥𝑑𝑦, (13) where ℎ𝑥,𝑦=^𝑥_𝑎ℎ𝑠,𝑦𝑑𝑠, (14)

Remark 4. Let 𝑓(𝑥,𝑦),𝑤(𝑥,𝑦),ℎ(𝑥,𝑦), and 𝑟(𝑥,𝑦) reduce to 𝑓(𝑥), 𝑤(𝑥), ℎ(𝑥), and 𝑟(𝑥), respectively, and with suitable modifications in Theorem 3, (13) changes to the following result: ^𝑏_𝑎𝑤(𝑥)𝐻(𝑥)⁻¹ℎ(𝑥)𝐻(𝑅)log𝐻(𝑥)^−𝑞𝐹(𝑥)^𝑝≤𝛽𝑝𝑑𝑥𝑞−1^𝑝𝑏_𝑎log𝐻(𝑅)𝐻(𝑥)^𝑝−𝑞×𝑤(𝑥)𝐻(𝑥)^𝑝−1ℎ(𝑥)𝑓(𝑥)^𝑝𝑑𝑥. (15)

This is just a new inequality established by Pachpatte [4].

On the other hand, we note that the inequality established in Theorem 3 is the further generalizations of the inequality established by Copson [17].

Taking for 𝑤(𝑥)=𝑟(𝑥)=1,𝐻(𝑅)=𝑅, and 𝛽=1 in (15), (15) changes to the following result: ^𝑏_𝑎𝐻(𝑥)⁻¹ℎ(𝑥)𝑅log𝐻(𝑥)^−𝑞𝐹(𝑥)^𝑝≤𝑝𝑑𝑥𝑞−1^𝑝𝑏_𝑎𝑅log𝐻(𝑥)^𝑝−𝑞×𝐻(𝑥)^𝑝−1ℎ(𝑥)𝑓(𝑥)^𝑝𝑑𝑥. (16)

This is just a new inequality established by Love [7].

3. Proof of Theorems

Proof of Theorem 1. If we let 𝑢(𝑥,𝑦)=𝑤(𝑥,𝑦)𝐹(𝑥,𝑦)^𝑝 and in view of 𝐹=1𝑥,𝑦𝑟𝑥,𝑦^𝑥_𝑎^𝑦_𝑐𝑟(𝑠,𝑡)ℎ(𝑠,𝑡)𝑓(𝑠,𝑡)𝑑𝑠𝑑𝑡 (17) for (𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑), then 𝜕𝑢𝑥,𝑦=𝜕𝑥𝜕𝑤𝑥,𝑦𝐹𝜕𝑥𝑥,𝑦^𝑝+𝑤𝑥,𝑦𝑝𝐹𝑥,𝑦^𝑝−1×1𝑟𝑥,𝑦^𝑦_𝑐𝑟()ℎ()𝑓()−𝑥,𝑡𝑥,𝑡𝑥,𝑡𝑑𝑡𝜕𝑟𝑥,𝑦/𝜕𝑥𝑟²×𝑥,𝑦_{𝑥𝑎𝑦𝑐}𝑟(𝑠,𝑡)ℎ(𝑠,𝑡)𝑓(𝑠,𝑡)𝑑𝑠𝑑𝑡. (18)
Let 𝜕𝑣𝑥,𝑦𝜕𝑥=𝐻𝑥,𝑦⁻¹ℎ𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞, (19) where ℎ(𝑥,𝑦)=∫^𝑦_𝑐ℎ(𝑥,𝑡)𝑑𝑡 and in view of 𝐻(𝑥,𝑦)=∫^𝑥_𝑎∫^𝑦_𝑐ℎ(𝑠,𝑡)𝑑𝑠𝑑𝑡, for (𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑), then 𝑣𝐻𝑥,𝑦=−log𝑅,𝑅^′/𝐻𝑥,𝑦^−𝑞+1.−𝑞+1 (20)
From (18), (20), and integrating by parts for 𝑥, we have ^𝑑_𝑐^𝑏_𝑎𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹ℎ×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞𝐹𝑥,𝑦^𝑝𝑑𝑥𝑑𝑦=−^𝑑_𝑐𝑤𝑥,𝑦𝐹𝑥,𝑦^𝑝×log𝐻𝑅,𝑅^′/𝐻𝑥,𝑦^−𝑞+1−𝑞+1^𝑥=𝑏_𝑥=𝑎−^𝑏_𝑎log𝐻𝑅,𝑅^′/𝐻𝑥,𝑦^−𝑞+1×−𝑞+1𝜕𝑤𝑥,𝑦𝐹𝜕𝑥𝑥,𝑦^𝑝+𝑤𝑥,𝑦𝑝𝐹𝑥,𝑦^𝑝−1×𝐺𝑥,𝑦−𝜕𝑟𝑥,𝑦/𝜕𝑥𝑟²×𝑥,𝑦_{𝑥𝑎𝑦𝑐}𝑟()ℎ()()𝑠,𝑡𝑠,𝑡×𝑓𝑠,𝑡𝑑𝑠𝑑𝑡𝑑𝑥𝑑𝑦, (21) where 𝐺=1𝑥,𝑦𝑟𝑥,𝑦^𝑦_𝑐𝑟(𝑥,𝑡)ℎ(𝑥,𝑡)𝑓(𝑥,𝑡)𝑑𝑡. (22)
If 𝑞<1, then we observe that ^𝑑_𝑐^𝑏_𝑎𝐷𝑥,𝑦𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹ℎ×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞𝐹𝑥,𝑦^𝑝≤𝑝𝑑𝑥𝑑𝑦1−𝑞^𝑑_𝑐^𝑏_𝑎𝑤𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞+1×𝐺𝑥,𝑦𝐹𝑥,𝑦^𝑝−1=𝑝𝑑𝑥𝑑𝑦×1−𝑞_{𝑑𝑐𝑏𝑎}𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^{−𝑞1/𝑝}𝐻×log𝑅,𝑅^′𝐻𝑤𝑥,𝑦𝑥,𝑦^1/𝑝×𝐻𝑥,𝑦^{(𝑝−1)/𝑝}×ℎ𝑥,𝑦^{−(𝑝−1)/𝑝}𝐺×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^{−𝑞(𝑝−1)/𝑝}×𝑤𝑥,𝑦^{(𝑝−1)/𝑝}×𝐻𝑥,𝑦⁻¹×ℎ𝑥,𝑦^{(𝑝−1)/𝑝}×𝐹𝑥,𝑦^𝑝−1𝑑𝑥𝑑𝑦. (23)
By applying Hölder’s inequality with indices 𝑝, 𝑝/(𝑝−1) on the right side of (23), we obtain ^𝑑_𝑐^𝑏_𝑎𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹ℎ×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞𝐹𝑥,𝑦^𝑝𝑝𝑑𝑥𝑑𝑦≤𝛼×1−𝑞_{𝑑𝑐𝑏𝑎}𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞×𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^𝑝×𝑤𝑥,𝑦𝐻𝑥,𝑦^𝑝−1×ℎ𝑥,𝑦^{−(𝑝−1)}𝐺𝑥,𝑦^𝑝𝑑𝑥𝑑𝑦^1/𝑝×^𝑑_𝑐^𝑏_𝑎𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹×ℎ𝑥,𝑦𝐹𝑥,𝑦^𝑝𝑑𝑥𝑑𝑦^{(𝑝−1)/𝑝}. (24)
Dividing both sides of (24) by the second integral factor on the right side of (24) and raising both sides to the 𝑝th power, we obtain ^𝑑_𝑐^𝑏_𝑎𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹ℎ×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞𝐹𝑥,𝑦^𝑝≤𝛼𝑝𝑑𝑥𝑑𝑦1−𝑞^𝑝×^𝑑_𝑐^𝑏_𝑎𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^𝑝−𝑞×𝑤𝑥,𝑦𝐻𝑥,𝑦^𝑝−1×ℎ𝑥,𝑦^{−(𝑝−1)}𝐺𝑥,𝑦^𝑝𝑑𝑥𝑑𝑦. (25)

Proof of Theorem 3. If we let 𝑢(𝑥,𝑦)=𝑤(𝑥,𝑦)𝐹(𝑥,𝑦)^𝑝 and in view of 𝐹=1𝑥,𝑦𝑟𝑥,𝑦^𝑏_𝑥^𝑑_𝑦𝑟(𝑠,𝑡)ℎ(𝑠,𝑡)𝑓(𝑠,𝑡)𝑑𝑠𝑑𝑡 (26) for (𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑), then 𝜕𝑢𝑥,𝑦=𝜕𝑦𝜕𝑤𝑥,𝑦𝐹𝜕𝑦𝑥,𝑦^𝑝+𝑤𝑥,𝑦𝑝𝐹𝑥,𝑦^𝑝−1×−1𝑟𝑥,𝑦^𝑑_𝑦𝑟()ℎ()𝑓()−𝑥,𝑡𝑥,𝑡𝑥,𝑡𝑑𝑡𝜕𝑟𝑥,𝑦/𝜕𝑦𝑟²𝑥,𝑦^𝑏_𝑥^𝑑_𝑦𝑟(𝑠,𝑡)ℎ(𝑠,𝑡)𝑓(𝑠,𝑡)𝑑𝑠𝑑𝑡. (27)
Let 𝜕𝑣𝑥,𝑦𝜕𝑦=𝐻𝑥,𝑦⁻¹ℎ𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞, (28) where ℎ(𝑥,𝑦)=∫^𝑥_𝑎ℎ(𝑠,𝑦)𝑑𝑠 and in view of 𝐻(𝑥,𝑦)=∫^𝑥_𝑎∫^𝑦_𝑐ℎ(𝑠,𝑡)𝑑𝑠𝑑𝑡, for (𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑), then 𝑣𝐻𝑥,𝑦=−log𝑅,𝑅^′/𝐻𝑥,𝑦^−𝑞+1.−𝑞+1 (29)
From (27), (29), and integrating by parts for 𝑦, we have ^𝑑_𝑐^𝑏_𝑎𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹ℎ×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞𝐹𝑥,𝑦^𝑝𝑑𝑥𝑑𝑦=−^𝑏_𝑎𝑤𝑥,𝑦𝐹𝑥,𝑦^𝑝×log𝐻𝑅,𝑅^′/𝐻𝑥,𝑦^−𝑞+1−𝑞+1^𝑦=𝑑_𝑦=𝑐−^𝑑_𝑐log𝐻𝑅,𝑅^′/𝐻𝑥,𝑦^−𝑞+1×−𝑞+1𝜕𝑤𝑥,𝑦𝐹𝜕𝑦𝑥,𝑦^𝑝+𝑤𝑥,𝑦𝑝𝐹𝑥,𝑦^𝑝−1×𝐿𝑥,𝑦−𝜕𝑟𝑥,𝑦𝜕𝑦𝑟²×𝑥,𝑦_{𝑏𝑥𝑑𝑦}𝑟()ℎ()()𝑠,𝑡𝑠,𝑡×𝑓𝑠,𝑡𝑑𝑠𝑑𝑡𝑑𝑦𝑑𝑥, (30) where 𝐿1𝑥,𝑦=−𝑟𝑥,𝑦^𝑏_𝑥𝑟𝑠,𝑦ℎ𝑠,𝑦𝑓𝑠,𝑦𝑑𝑠. (31)
If 𝑞>1, then we observe that ^𝑑_𝑐^𝑏_𝑎𝐸𝑥,𝑦𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹ℎ×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞𝐹𝑥,𝑦^𝑝≤𝑝𝑑𝑥𝑑𝑦𝑞−1^𝑑_𝑐^𝑏_𝑎𝑤𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞+1×𝐿𝑥,𝑦𝐹𝑥,𝑦^𝑝−1=𝑝𝑑𝑥𝑑𝑦×𝑞−1_{𝑑𝑐𝑏𝑎}𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^{−𝑞1/𝑝}𝐻×log𝑅,𝑅^′𝐻𝑤𝑥,𝑦𝑥,𝑦^1/𝑝𝐻𝑥,𝑦^{(𝑝−1)/𝑝}×ℎ𝑥,𝑦^{−(𝑝−1)/𝑝}𝐿×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^{−𝑞(𝑝−1)/𝑝}×𝑤𝑥,𝑦^{(𝑝−1)/𝑝}×𝐻𝑥,𝑦⁻¹×ℎ𝑥,𝑦^{(𝑝−1)/𝑝}×𝐹𝑥,𝑦^𝑝−1𝑑𝑥𝑑𝑦. (32)
By applying Hölder’s inequality with indices 𝑝, 𝑝/(𝑝−1) on the right side of (32), we obtain ^𝑑_𝑐^𝑏_𝑎𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹ℎ×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞𝐹𝑥,𝑦^𝑝𝑝𝑑𝑥𝑑𝑦≤𝛽×𝑞−1_{𝑑𝑐𝑏𝑎}𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞×𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^𝑝𝑤𝑥,𝑦𝐻𝑥,𝑦^𝑝−1×ℎ𝑥,𝑦^{−(𝑝−1)}𝐿𝑥,𝑦^𝑝𝑑𝑥𝑑𝑦^1/𝑝×^𝑑_𝑐^𝑏_𝑎𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞×𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹×ℎ𝑥,𝑦𝐹𝑥,𝑦^𝑝𝑑𝑥𝑑𝑦^{(𝑝−1)/𝑝}. (33)
Dividing both sides of (33) by the second integral factor on the right side of (33) and raising both sides to the 𝑝th power, we obtain ^𝑑_𝑐^𝑏_𝑎𝑤𝑥,𝑦𝐻𝑥,𝑦⁻¹ℎ×𝐻𝑥,𝑦log𝑅,𝑅^′𝐻𝑥,𝑦^−𝑞×𝐹𝑥,𝑦^𝑝≤𝛽𝑝𝑑𝑥𝑑𝑦𝑞−1^𝑝×^𝑑_𝑐^𝑏_𝑎𝐻log𝑅,𝑅^′𝐻𝑥,𝑦^𝑝−𝑞𝑤𝑥,𝑦𝐻𝑥,𝑦^𝑝−1×ℎ𝑥,𝑦^{−(𝑝−1)}𝐿𝑥,𝑦^𝑝𝑑𝑥𝑑𝑦. (34)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Chang-Jian Zhao’s research is supported by National Natural Science Foundation of China (11371334). Wing-Sum Cheung’s research is partially supported by a HKU URG grant.