A Hilbert's Inequality with a Best Constant Factor

Abstract

We give a new Hilbert's inequality with a best constant factor and some parameters.

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Zheng Zeng 1 and Zi-tian Xie 2

Recommended by Yong Zhou

1, Department of Mathematics, Shaoguan University, Shaoguan, Guangdong 512005, China
2, Department of Mathematics, Zhaoqing University, Zhaoqing, Guangdong 526061, China

Received 6 February 2009; Revised 3 May 2009; Accepted 23 July 2009

1. Introduction

If p>1 , 1/p+1/q=1 , _an ,_bn >0 such that ∞>^∑n=1∞anp >0 and ∞>^∑n=1∞bnq >0 , then the well-known Hardy-Hilbert's inequality and its equivalent form are given by [figure omitted; refer to PDF] [figure omitted; refer to PDF] where the constant factors are all the best possible [1]. It attracted some attention in the recent years. Actually, inequalities (1.1) and (1.2) have many generalizations and variants. Equation (1.1) has been strengthened by Yang and others ( including integral inequalities ) [2-11].

In 2006, Yang gave an extension of [2] as follows.

If p>1 , 1/p+1/q=1 , r>1,1/r+1/s=1,t∈[0,1],(2-min {r,s})t+min {r,s}≥λ>(2-min {r,s})t, such that ∞>^{∑n=1∞np(1-t+(2t-λ)/r)-1anp} >0 , ∞>^{∑n=1∞nq(1-t+(2t-λ)/s)-1bnq} >0 , then [figure omitted; refer to PDF] B(u,v) is the Beta function.

In 2007 Xie gave a new Hilbert-type Inequality [3] as follows.

If p>1,1/p+1/q=1,a,b,c>0,2/3≥μ>0 , and the right of the following inequalities converges to some positive numbers, then

[figure omitted; refer to PDF] The main objective of this paper is to build a new Hilbert's inequality with a best constant factor and some parameters.

In the following, we always suppose that

(1) 1/p+1/q=1, p>1 , a≥0 , -1<α<1 ,

(2) both functions u(x) and v(x) are differentiable and strict increasing in (_n0 -1,∞) and (_m0 -1,∞), respectively,

(3) ^{u[variant prime]} (x)/^uα (x),^{v[variant prime]} (x)/^vα (x) are strictly increasing in (_n0 -1,∞) and (_m0 -1,∞), respectively. {_{^{u[variant prime]} n^{v[variant prime]} m} /[(^un2 +2a_unvm +^vm2 )^unαvmα ]} is strict decreasing on n and m ,

(4) u(n)=_un ,u(_n0 )=_u0 ,u((_n0 -1⁾⁺ )=v((_m0 -1⁾⁺ )=0,u(∞)=∞,v(∞)=∞,^{u[variant prime]} (n)=_{^{u[variant prime]} n} ,v(m)=_vm ,v(_m0 )=_v0 ,^{v[variant prime]} (m)=^{vm[variant prime]} .

2. Some Lemmas

Lemma 2.1.

Define the weight coefficients as follows: [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] then [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

Let f(z)=1/[(1+2az+^z2 )^zα ]=1/[(z-_z1 )(z-_z2 )^zα ] then K=(2πi/(1-^e-2απi ))[Res(f,_z1 )+Res(f,_z2 )] if a>1 then _z1 =-a-^a2 -1,_z2 =-a+^a2 -1 [figure omitted; refer to PDF] if a=cos θ (0<θ<π/2 ), then _z1 =-^eiθ ,_z2 =-^e-iθ [figure omitted; refer to PDF] On the other hand, W(p,m)<ω(p,m) . Setting u(x)=_vm σ , then ω(p,m)=K^vmpα-2α-1 /(^{vm[variant prime])p-1} . Similarly, W...(q,n)<ω...(q,n)=K^unqα-2α-1 /(^{un[variant prime])q-1} .

Lemma 2.2.

For 0<[straight epsilon]<min {p,p(1-α)} one has [figure omitted; refer to PDF]

Proof.

[figure omitted; refer to PDF] The lemma is proved.

Lemma 2.3.

Setting _wn =_un (or _vm ) and _w0 =_n0 (or _m0 , resp.), then k>0. {^{τw[variant prime]} /^τwk } is strictly decreasing, then [figure omitted; refer to PDF] There A∈(0,^{τ_w0 [variant prime]} /^{τ_w0 k} ),( for any N ).

Proof.

We have [figure omitted; refer to PDF] Easily, A had up bounded when N[arrow right]∞.

3. Main Results

Theorem 3.1.

If _an >0, _bn >0 , 0<^{∑n=1∞vmpα-2α-1} /(^{vm[variant prime])p-1anp} <∞ , 0<^{∑n=_n0 ∞unqα-2α-1} /(^{un[variant prime] )q-1bnq} <∞ , then [figure omitted; refer to PDF] [figure omitted; refer to PDF] K is defined by Lemma 2.1.

Proof.

By Hölder's inequality [12] and (2.5), [figure omitted; refer to PDF] setting _bn =^{unpα-2α+p-1un[variant prime](∑m=_m0 ∞_am /(un2 +2a_unvm +vm2 ))p-1} >0. By(3.1) we have [figure omitted; refer to PDF] By 0<^{∑n=_n0 ∞} (^unqα-2α-1 /(^{un[variant prime])q-1} )^bnq <∞ and (3.4) taking the form of strict inequality, we have (3.1). By Hölder's inequality[12], we have [figure omitted; refer to PDF] as 0<^{{∑n=_n0 ∞ (unqα-2α-1 /(un[variant prime])q-1 )bnq }1/q} <∞ . By (3.2), (3.5) taking the form of strict inequality, we have (3.1).

Theorem 3.2.

If α=0 , then both constant factors, K and ^Kp of (3.1) and (3.2), are the best possible.

Proof.

We only prove that K is the best possible. If the constant factor K in (3.1) is not the best possible, then there exists a positive H (with H<K ), such that [figure omitted; refer to PDF] For 0<[straight epsilon]<min{p,q} , setting _a...m =^{vm-[straight epsilon]/pvm[variant prime]} ,_b...n =^{un-[straight epsilon]/qun[variant prime]} , then [figure omitted; refer to PDF] On the other hand (u(x)=σv(y) and v(y)=τ ), [figure omitted; refer to PDF] By (3.6), (3.7), (3.8), and Lemma 2.3, we have [figure omitted; refer to PDF] [figure omitted; refer to PDF] We have K≤H , ([straight epsilon][arrow right]⁰⁺ ) . This contracts the fact that H<K .

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