On a Generalization of Hilbert-Type Integral Inequality

Abstract

By introducing some parameters, we establish generalizations of the Hilbert-type inequality. As applications, the reverse and its equivalent form are considered.

1. Introduction

Considerable attention has been given to Hilbert inequalities and Hilbert-type inequalities by several authors including Gao and Yang [1], Yang [2–4], Jichang and Debnath [5], Pachpatte [6], Zhao [7], Brnetić and Pečarić [8]. In 2007, Li et al. [9] gave a new inequality similar to Hilbert inequality for integrals:

If 𝑓(𝑥),𝑔(𝑥)≥0, 0<∫^∞₀𝑓²(𝑥)𝑑𝑥<∞,0<∫^∞₀𝑔²(𝑥)𝑑𝑥<∞, then one has

^∞₀|ln𝑥−ln𝑦|𝑥+𝑦+|𝑥−𝑦|𝑓(𝑥)𝑔(𝑦)𝑑𝑥𝑑𝑦<4[^∞₀𝑓²(𝑥)𝑑𝑥^∞₀𝑔²(𝑥)𝑑𝑥]^1/2, (1.1) where constant factor 4 is the best possible.

An equivalent inequality is

^∞₀[^∞₀|ln𝑥−ln𝑦|𝑥+𝑦+|𝑥−𝑦|𝑓(𝑥)𝑑𝑥]²𝑑𝑦<16^∞₀𝑓²(𝑥)𝑑𝑥. (1.2)

In this paper, by introducing some parameters we generalize (1.1), (1.2), and we obtain the reverse form for each of them. The equivalent forms are also considered.

2. Main Results

Lemma 2.1. Suppose that 𝜆>0, 𝑝>1(1/𝑝+1/𝑞=1), define weight functions 𝑤_𝜆(𝑥,𝑞), 𝑤_𝜆(𝑦,𝑝), respectively as
𝑤_𝜆(𝑥,𝑞)=^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|(𝑥𝑦)^𝜆/𝑞1𝑦^1−𝜆𝑤𝑑𝑦,_𝜆(𝑦,𝑝)=^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|(𝑦𝑥)^𝜆/𝑝1𝑥^1−𝜆𝑑𝑥. (2.1) One has 𝑤_𝜆(𝑥,𝑞)=𝑤_𝜆(𝑦,𝑝)=(1/2𝜆²)(𝑝²+𝑞²).

Proof. Letting 𝑡=𝑦^𝜆/𝑥^𝜆, we have
𝑤_𝜆=1(𝑥,𝑞)𝜆^2∞0|ln𝑡|1+𝑡+|1−𝑡|⋅𝑡^−1/𝑞1𝑑𝑡=𝜆²[¹₀−ln𝑡2⋅𝑡^−1/𝑞𝑑𝑡+^∞₁ln𝑡2𝑡⋅𝑡^−1/𝑞1𝑑𝑡]=2𝜆²(𝑝²+𝑞²). (2.2) By symmetry we have
𝑤_𝜆1(𝑦,𝑝)=2𝜆²(𝑝²+𝑞²). (2.3) The lemma is proved.

Lemma 2.2. Let 𝑝>1(𝑜𝑟0<𝑝<1), 1/𝑝+1/𝑞=1, 𝜆>0, and 0<𝜀<𝑞𝜆/2𝑝, setting
𝐽(𝜀)=^∞₁|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑥^{−[1+𝜀+(𝑝−1)(1−𝜆)]/𝑝}𝑦^{−[1+𝜀+(𝑞−1)(1−𝜆)]/𝑞}𝑑𝑥𝑑𝑦. (2.4) Then for 𝜀→0⁺, one gets
𝑝²+𝑞²2𝜆²𝜀𝑝[1+𝑜(1)]−𝑂(1)<𝐽(𝜀)<²+𝑞²2𝜆²𝜀[1+𝑜(1)]. (2.5)

Proof. Letting 𝑡=𝑦^𝜆/𝑥^𝜆, we have
1𝐽(𝜀)=𝜆^2∞11𝑥^1+𝜀[^∞_1/𝑥𝜆|ln𝑡|1+𝑡+|1−𝑡|⋅𝑡^{1/𝑝−1−𝜀/𝜆𝑞}=1𝑑𝑡]𝑑𝑥𝜆^2∞11𝑥^1+𝜀[^∞₀|ln𝑡|1+𝑡+|1−𝑡|⋅𝑡^{1/𝑝−1−𝜀/𝜆𝑞}−1𝑑𝑡]𝑑𝑥𝜆^2∞11𝑥^1+𝜀[^1/𝑥𝜆₀|ln𝑡|1+𝑡+|1−𝑡|⋅𝑡^{1/𝑝−1−𝜀/𝜆𝑞}=1𝑑𝑡]𝑑𝑥𝜆²𝜀^∞1|ln𝑡|1+𝑡+|1−𝑡|⋅𝑡^{1/𝑝−1−𝜀/𝜆𝑞}1𝑑𝑡−𝜆^2∞11𝑥^1+𝜀[^1/𝑥𝜆₀|ln𝑡|1+𝑡+|1−𝑡|⋅𝑡^{1/𝑝−1−𝜀/𝜆𝑞}𝑑𝑡]𝑑𝑥∶=𝐼₁−𝐼₂. (2.6) Now, observe that
^∞₀|ln𝑡|1+𝑡+|1−𝑡|⋅𝑡^{1/𝑝−1−𝜀/𝜆𝑞}𝑝𝑑𝑡=²+𝑞²2[1+𝑜(1)],(𝜀→0⁺),0<^1/𝑥𝜆₀|ln𝑡|1+𝑡+|1−𝑡|⋅𝑡^{1/𝑝−1−𝜀/𝜆𝑞}=𝑑𝑡_1/𝑥𝜆0−ln𝑡2⋅𝑡^{1/𝑝−1−𝜀/𝜆𝑞}<𝑑𝑡_1/𝑥𝜆0−ln𝑡2⋅𝑡^1/2𝑝−1𝑑𝑡=𝑝𝜆𝑥^{−𝜆/2𝑝}ln𝑥+2𝑝²𝑥^{−𝜆/2𝑝}, (2.7) then
𝐼₁=𝑝²+𝑞²2𝜆²𝜀[1+𝑜(1)],0<𝐼₂<1𝜆²[𝑝𝜆∫^∞₁𝑥^{−1−𝜆/2𝑝}ln𝑥𝑑𝑥+2𝑝²∫^∞₁𝑥^{−1−𝜆/2𝑝}𝑑𝑥]=8𝑝³𝜆³. (2.8) We get
𝑝𝐽(𝜀)>²+𝑞²2𝜆²𝜀[1+𝑜(1)]−𝑂(1). (2.9) On the other hand,
𝐽(𝜀)=^∞₁|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑥^{−[1+𝜀+(𝑝−1)(1−𝜆)]/𝑝}𝑦^{−[1+𝜀+(𝑞−1)(1−𝜆)]/𝑞}𝑑𝑥𝑑𝑦<^∞₁[^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑥^{−[1+𝜀+(𝑝−1)(1−𝜆)]/𝑝}𝑦^{−[1+𝜀+(𝑞−1)(1−𝜆)]/𝑞}=𝑝𝑑𝑦]𝑑𝑥₂+𝑞²2𝜆²𝜀[1+𝑜(1)]. (2.10) Hence, (2.5) is valid. The lemma is proved.

Theorem 2.3. Let 𝑝>1, 1/𝑝+1/𝑞=1, 𝜆>0, 𝑓(𝑥)≥0,𝑔(𝑦)≥0. If 0<∫^∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥<∞, 0<∫^∞₀𝑦^{(𝑞−1)(1−𝜆)}𝑔^𝑞(𝑦)𝑑𝑦<∞, then one has
∬^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|<𝑝𝑓(𝑥)𝑔(𝑦)𝑑𝑥𝑑𝑦₂+𝑞²2𝜆²(∫^∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥)^1/𝑝(∫^∞₀𝑦^{(𝑞−1)(1−𝜆)}𝑔^𝑞(𝑦)𝑑𝑦)^1/𝑞, (2.11)where constant factor (𝑝²+𝑞²)/2𝜆² is the best possible. In particular, for 𝜆=1, inequality (2.11) reduces to
^∞₀|ln𝑥−ln𝑦|𝑝𝑥+𝑦+|𝑥−𝑦|𝑓(𝑥)𝑔(𝑦)𝑑𝑥𝑑𝑦<²+𝑞²2(^∞₀𝑓^𝑝(𝑥)𝑑𝑥)^1/𝑝(^∞₀𝑔^𝑞(𝑦)𝑑𝑦)^1/𝑞. (2.12)

Proof. Applying Hölder’s inequality and Lemma 2.1, we have
∬^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|=∬𝑓(𝑥)𝑔(𝑦)𝑑𝑥𝑑𝑦_∞0[(|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|)^1/𝑝(𝑥𝑦)^{𝜆/𝑝𝑞}𝑥^{(1−𝜆)/𝑞}𝑦^{(1−𝜆)/𝑝}𝑓(𝑥)]×[(|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|)^1/𝑞(𝑦𝑥)^{𝜆/𝑝𝑞}𝑦^{(1−𝜆)/𝑝}𝑥^{(1−𝜆)/𝑞}𝑓(𝑦)]𝑑𝑥𝑑𝑦≤{∬^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|(𝑥𝑦)^𝜆/𝑞𝑥^{(𝑝/𝑞)(1−𝜆)}𝑦^1−𝜆𝑓^𝑝(𝑥)𝑑𝑥𝑑𝑦}^1/𝑝×{∬^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|(𝑦𝑥)^𝜆/𝑝𝑦^{(𝑞/𝑝)(1−𝜆)}𝑥^1−𝜆𝑔^𝑞(𝑦)𝑑𝑥𝑑𝑦}^1/𝑞≤𝑝²+𝑞²2𝜆²(∫^∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥)^1/𝑝(∫^∞₀𝑦^{(𝑞−1)(1−𝜆)}𝑔^𝑞(𝑦)𝑑𝑦)^1/𝑞. (2.13)
If (2.13) takes the form of equality, then there exist constants 𝐴 and 𝐵, such that they are not all zero, and
𝐴|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|(𝑥𝑦)^𝜆/𝑞𝑥^{(𝑝/𝑞)(1−𝜆)}𝑦^1−𝜆𝑓^𝑝(𝑥)=𝐵|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|(𝑦𝑥)^𝜆/𝑝𝑦^{(𝑞/𝑝)(1−𝜆)}𝑥^1−𝜆𝑔^𝑞(𝑦), (2.14) a.e. (𝑥,𝑦) in (0,∞)×(0,∞). It follows that there exists a constant 𝐶, such that
𝐴𝑥⋅𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)=𝐵𝑦⋅𝑦^{(𝑞−1)(1−𝜆)}𝑔^𝑞(𝑦)=𝐶,a.e.(𝑥,𝑦)in(0,∞)×(0,∞). (2.15) Without lose of generality, suppose 𝐴≠0, then we have
𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝1(𝑥)=𝐴𝑥, (2.16) which contradicts the fact that 0<∫^∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥<∞, hence (2.13) takes the form of strict inequality, so we obtain (2.11).
Assume that the constant factor (𝑝²+𝑞²)/2𝜆² in (2.11) is not the best possible, then there exists a positive number 𝑘 (with 𝑘<(𝑝²+𝑞²)/2𝜆²) such that (2.11) is still valid if one replaces (𝑝²+𝑞²)/2𝜆² by 𝑘. In particular, for 0<𝜀<𝑞𝜆/2𝑝, setting ^0𝑥0𝑓^02𝑑𝑐 and ^0𝑥0𝑔^02𝑑𝑐 as ^0𝑥0𝑓^02𝑑𝑐(𝑥)=𝑔^{0𝑥002𝑑𝑐}(𝑥)=0 for 𝑥∈(0,1), ^0𝑥0𝑓^02𝑑𝑐(𝑥)=𝑥^{−[1+𝜀+(𝑝−1)(1−𝜆)]/𝑝}, ^0𝑥0𝑔^02𝑑𝑐(𝑥)=𝑥^{−[1+𝜀+(𝑞−1)(1−𝜆)]/𝑞} for 𝑥∈[1,∞), then we have
𝑘(^∞₀𝑥𝑓^{(𝑝−1)(1−𝜆)0𝑥002𝑑𝑐𝑝}(𝑥)𝑑𝑥)^1/𝑝(^∞₀𝑦𝑔^{(𝑞−1)(1−𝜆)0𝑥002𝑑𝑐𝑞}(𝑦)𝑑𝑦)^1/𝑞=𝑘𝜀. (2.17) By using Lemma 2.2, we find
∬^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|^{0𝑥002𝑑𝑐}𝑓(𝑥)𝑔^{0𝑥002𝑑𝑐}(𝑦)𝑑𝑥𝑑𝑦=∬^∞₁|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑥^{−[1+𝜀+(𝑝−1)(1−𝜆)]/𝑝}𝑦^{−[1+𝜀+(𝑞−1)(1−𝜆)]/𝑞}>𝑝𝑑𝑥𝑑𝑦₂+𝑞²2𝜆²𝜀[1+𝑜(1)]−𝑂(1). (2.18) Therefore, we get
𝑝²+𝑞²2𝜆²𝜀𝑘[1+𝑜(1)]−𝑂(1)<𝜀 (2.19) or
𝑝²+𝑞²2𝜆²[1+𝑜(1)]−𝜀𝑂(1)<𝑘. (2.20) For 𝜀→0⁺, it follows that (𝑝²+𝑞²)/2𝜆²≤𝑘. This contradicts the fact that 𝑘<(𝑝²+𝑞²)/2𝜆². Hence, the constant factor in (2.11) is the best possible. Theorem 2.3 is proved.

Theorem 2.4. Let 0<𝑝<1, 1/𝑝+1/𝑞=1, 𝜆>0, 𝑓(𝑥)≥0,𝑔(𝑦)≥0. If 0<∫^∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥<∞, 0<∫^∞₀𝑦^{(𝑞−1)(1−𝜆)}𝑔^𝑞(𝑦)𝑑𝑦<∞, then one has
∬^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|>𝑝𝑓(𝑥)𝑔(𝑦)𝑑𝑥𝑑𝑦₂+𝑞²2𝜆²(∫^∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥)^1/𝑝(∫^∞₀𝑦^{(𝑞−1)(1−𝜆)}𝑔^𝑞(𝑦)𝑑𝑦)^1/𝑞, (2.21) where the constant factor (𝑝²+𝑞²)/2𝜆² is the best possible. In particular, for 𝜆=1, the inequality reduces to
^∞₀|ln𝑥−ln𝑦|𝑝𝑥+𝑦+|𝑥−𝑦|𝑓(𝑥)𝑔(𝑦)𝑑𝑥𝑑𝑦>²+𝑞²2(^∞₀𝑓^𝑝(𝑥)𝑑𝑥)^1/𝑝(^∞₀𝑔^𝑞(𝑦)𝑑𝑦)^1/𝑞. (2.22)

Proof. Applying reverse Hölder’s inequality and the same arguments as before, we have (2.21).
If the constant factor (𝑝²+𝑞²)/2𝜆² in (2.21) is not the best possible, then there exists a positive number ℎ (with ℎ>(𝑝²+𝑞²)/2𝜆²), such that (2.21) is still valid if one replaces (𝑝²+𝑞²)/2𝜆² by ℎ. In particular, for 0<𝜀<𝑞𝜆/2𝑝, setting ^0𝑥0𝑓^02𝑑𝑐 and ^0𝑥0𝑔^02𝑑𝑐 as in Theorem 2.3, we have
ℎ(^∞₀𝑥𝑓^{(𝑝−1)(1−𝜆)0𝑥002𝑑𝑐𝑝}(𝑥)𝑑𝑥)^1/𝑝(^∞₀𝑦𝑔^{(𝑞−1)(1−𝜆)0𝑥002𝑑𝑐𝑞}(𝑦)𝑑𝑦)^1/𝑞=ℎ𝜀. (2.23) By using Lemma 2.2, we find
∬^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|^{0𝑥002𝑑𝑐}𝑓(𝑥)𝑔^{0𝑥002𝑑𝑐}(𝑦)𝑑𝑥𝑑𝑦=∬^∞₁|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑥^{−[1+𝜀+(𝑝−1)(1−𝜆)]/𝑝}𝑦^{−[1+𝜀+(𝑞−1)(1−𝜆)]/𝑞}<𝑝𝑑𝑥𝑑𝑦₂+𝑞²2𝜆²𝜀[𝐼+𝑜(1)]. (2.24) Therefore, we get
𝑝²+𝑞²2𝜆²𝜀ℎ[1+𝑜(1)]>𝜀𝑝or²+𝑞²2𝜆²[1+𝑜(1)]>ℎ (2.25) for 𝜀→0⁺, and it follows that (𝑝²+𝑞²)/2𝜆²≥ℎ. This contradicts the fact that ℎ>(𝑝²+𝑞²)/2𝜆². Hence, the constant factor in (2.21) is the best possible. Theorem 2.4 is proved.

Theorem 2.5. If 𝑝>1, 1/𝑝+1/𝑞=1, 𝜆>0, 𝑓(𝑥)≥0, 0<∫^∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥<∞, then one has
^∞₀𝑦^𝜆−1[^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑓(𝑥)𝑑𝑥]^𝑝𝑝𝑑𝑦<(²+𝑞²2𝜆²)^𝑝∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥, (2.26)where the constant factor ((𝑝²+𝑞²)/2𝜆²)^𝑝 is the best possible. Inequality (2.26) is equivalent to (2.11).

Proof. Setting
𝑔(𝑦)=𝑦^𝜆−1[^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑓(𝑥)𝑑𝑥]^𝑝−1, (2.27) then by (2.11), we find
∫^∞₀𝑦^{(𝑞−1)(1−𝜆)}𝑔^𝑞(𝑦)𝑑𝑦=∫^∞₀𝑦^𝜆−1[∫^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑓(𝑥)𝑑𝑥]^𝑝𝑑𝑦=∬^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|≤𝑝𝑓(𝑥)𝑔(𝑦)𝑑𝑥𝑑𝑦₂+𝑞²2𝜆²(∫^∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥)^1/𝑝(∫^∞₀𝑦^𝜆−1[∫^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑓(𝑥)𝑑𝑥]^𝑝𝑑𝑦)^1/𝑞. (2.28) Hence, we obtain
^∞₀𝑦^{(𝑞−1)(1−𝜆)}𝑔^𝑞𝑝(𝑦)𝑑𝑦≤(²+𝑞²2𝜆²)^𝑝∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥, (2.29) Thus, by (2.11), both (2.28) and (2.29) keep the form of strict inequalities, then we have (2.26).
Applying Hölder’s inequality, we have
∬^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑓(𝑥)𝑔(𝑦)𝑑𝑥𝑑𝑦=∫^∞₀[𝑦^{(𝜆−1)/𝑝}∫^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑓(𝑥)𝑑𝑥][𝑦^{(1−𝜆)/𝑝}𝑔(𝑦)]𝑑𝑦≤{∫^∞₀𝑦^𝜆−1[∫^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑓(𝑥)𝑑𝑥]^𝑝𝑑𝑦}^1/𝑝×{∫^∞₀𝑦^{(𝑞−1)(1−𝜆)}𝑔^𝑞(𝑦)𝑑𝑦}^1/𝑞. (2.30) Therefore, by (2.26) we have (2.11). It follows that inequality (2.26) is equivalent to (2.11), and the constant factors in (2.26) are the best possible. The theorem is proved.

Theorem 2.6. If 0<𝑝<1, 1/𝑝+1/𝑞=1, 𝜆>0, 𝑓(𝑥)≥0, 0<∫^∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥<∞, then one has
^∞₀𝑦^𝜆−1[^∞₀|ln𝑥−ln𝑦|𝑥^𝜆+𝑦^𝜆+|𝑥^𝜆−𝑦^𝜆|𝑓(𝑥)𝑑𝑥]^𝑝𝑝𝑑𝑦>(²+𝑞²2𝜆²)^𝑝∞₀𝑥^{(𝑝−1)(1−𝜆)}𝑓^𝑝(𝑥)𝑑𝑥, (2.31) where the constant factor ((𝑝²+𝑞²)/2𝜆²)^𝑝 is the best possible. Inequality (2.31) is equivalent to (2.21).

The proof of Theorem 2.6 is similar to that of Theorem 2.5, so we omit it.

Acknowledgment

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