,

(1.Preparatory Department of Primary Education, Changsha Normal University, Changsha 410100;2.Department of Science and Information, Shaoyang University,Shaoyang 422000 China)

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A Hilbert-Type Integral Inequality with the Inhomogeneous Kernel and Multi-Parameters

HUANGLin1*,LIUQiong2

(1.Preparatory Department of Primary Education, Changsha Normal University, Changsha 410100;2.Department of Science and Information, Shaoyang University,Shaoyang 422000 China)

In this paper, by means of weight function and the technique of real analysis, and introducing multi-parameters and some special functions to jointly characterize the constant factor, a Hilbert-type integral inequality with the inhomogeneous kernel and multi-parameters and it’s equivalent form are given. Their constant factors are proved be the best possible, and its application is discussed.

Hilbert-type integral inequality; weight function; the best constant factor; inhomogeneous kernel; multi-parameters

1 Introduction

For convenience, Ifθ(x)(>0)ismeasurablefunction,ρ≥1,thefunctionspacesaresetas:

and

Iff,g∈L2(0,),‖f‖2,‖g‖2>0,thenwehavethefollowingHilbert’sintegralinequality[1]:

(1.1)

(1.2)

(1.3)

(1.4)

Inthispaper,bymeansofweightfunctionandthetechniqueofrealanalysis,aHilbert-typeintegralinequalitywiththeinhomogeneouskernelandMulti-parametersisgivenasfollows:

(1.5)

2 Some Lemmas

We need the following definitions[11]:

(2.1)

(2.2)

Lemma 2.1 Letmbe a positive integer, then we have the summation formulas[11]:

(2.3)

Lemma 2.2 Leta>-1,Re(s)>0,thentheLaplaceintegraltransformofthepowerfunctionxaasfollows[12]:

(2.4)

Lemma 2.3 Ifx>1,wehave

(2.5)

Proof Because

therefore

(2.6)

by(2.6),wefind

thenwehave

(2.7)

where

(2.8)

Particularly,whenη=2m(m=1,2,…),Γ(η)=Γ(2m)=(2m-1)!,by(2.3),wefind

(2.9)

wheretheBm′saretheBernoullinumbers.

Proof Settingαxλ1yλ2=u,thenby(2.4)and(2.6),wehave:

thenwehave:

(2.10)

(2.11)

Proof We easily get:

SinceF(u)=uη+1(1-tanhu)iscontinuousin(0,),(u)=0,(u)=0,thereexistsM>0,satisfyingF(u)≤M,byFubini’stheorem[13],wehave:

3 Main results and applications

(3.1)

(3.2)

Ifinequality(3.2)keepstheformofanequality,thenaccordingto[14]thereexisttwoconstantsAandB, such that they are not all zero and:

(3.3)

Proof Setting a bounded measurable function as:

since0<‖f‖p,φ<,thereexistsn0∈N, such that 0<φ(x)<(n≥n0),setting:

whenn≥n0,by(3.1)wefind:

(3.4)

(3.5)

Itfollows0<‖f‖p,φ<.Forn→,by(3.1),both(3.4)and(3.5)stillkeeptheformofstrictinequalities,hence,wehaveinequality(3.3).

Theinequalityis(3.1),whichisequivalentto(3.3).

Bytakingthespecialparametervaluesin(3.1)and(3.3),somemeaningfulinequalitiesareobtained:

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

Comments:Veryunfortunately,wecannotgetaHilbert-typeintegralinequalitywiththekernelofthehyperbolictangentfunctionby(3.1).

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責任編輯：龍順潮

2015-02-18

國家自然科學基金項目(11171280);湖南省教育廳科學研究項目(10C1186)

黃琳(1964— )，女，江西 上饒人，副教授.E-mail：13787312290@163.com

一個多參數(shù)非齊次核Hilbert型積分不等式

黃 琳1*， 劉 瓊2

(1.長沙師范學院 初等教育預科部，湖南 長沙 410100；2.邵陽學院 理學與信息科學系，湖南 邵陽 422000)

利用權正數(shù)方法和實分析技巧，引入多參數(shù)和一些特殊函數(shù)聯(lián)合刻畫常數(shù)因子，得到一個多參數(shù)非齊次核Hilbert型積分不等式和它的等價式，證明了它們的常數(shù)因子是最佳的，并討論了其應用.

Hilbert型積分不等式;權函數(shù);最佳常數(shù)因子；非齊次核；多參數(shù)

O178

A

1000-5900(2015)03-0001-08