On Hilbert’s Inequality for Double Series and Its Applications

International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 165089,12pages

doi:10.1155/2008/165089

Research Article

On Hilbert’s Inequality for Double Series and Its Applications

Zhou Yu and Gao Mingzhe

Department of Mathematics and Computer Science, Normal College of Jishou University, Jishou, Hunan 416000, China

Correspondence should be addressed to Gao Mingzhe,[email protected] Received 2 September 2007; Accepted 26 March 2008

Recommended by Feng Qi

This study shows that a refinement of the Hilbert inequality for double series can be established by introducing a real functionuxand a parameterλ. In particular, some sharp results of the classical Hilbert inequality are obtained by means of a sharpening of the Cauchy inequality. As applications, some refinements of both the Fejer-Riesz inequality and Hardy inequality inHpfunction are given.

Copyrightq2008 Z. Yu and G. Mingzhe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let{an}and{bn}be two sequences of complex numbers. Ifλ0,1, then

∞ m1−λ

∞ n1−λ

ambn

mnλ ≤ π

_∞

n1−λ

|an|²

1/2 _∞

n1−λ

|bn|² 1/2

, 1.1

∞ m1−λ

∞ n1−λ

m /n

a_mb_n m−n

≤ π

_∞

m1−λ

|an|²

_{1/2 ∞}

n1−λ

|bn|² _1/2

, 1.2

where the constant factorπ is the best possible. It is all known that the inequalities1.1and 1.2are called the Hilbert theorem for double series. The two forms 1.1 and1.2on the Hilbert inequality were combined into one similar form in some papers e.g., 1, 2, etc.,

that is,

∞ m1−λ

∞ n1−λ

ambn

mnλ

2

∞ m1−λ

∞ n1−λ

m /n

ambn

m−n

2

≤ π² ∞ n1−λ

|an|^{2 ∞}

n1−λ

|bn|². 1.3

Recently, the various extensions on 1.1 appeared in some papers e.g., 3, 4, etc..They focalize on changing the denominator of the function of the left-hand side of1.1. Such as the denominator mnλ is replaced byαmβn^μ in3, and the denominatormnλ is replaced by mum nvn^μ in 4. Some new results in these papers were yielded.

The inequality 1.3 is obviously a significant refinement of1.1and 1.2. However, both extensions and refinements on1.2and1.3do not commonly appear in previous papers. The main purpose of the present paper is to establish both an extension and a significant refinement on1.3. Explicitly, letux>0x∈0, ∞be a real function and let limx→∞ux ∞. If the denominatormnλof the first term of the left-hand side of1.3is replaced byumun, and the denominatorm−nof the second term of the left-hand side of1.3is replaced by um−un, then a new inequality established is significant in theory and applications; and as applications, we will give both extensions and refinements on Fejer-Riesz’s inequality and Hardy’s inequality. For convenience, we introduce some notations and functions as follows:

Va, b ^∞

m0

∞ umn0/vn

ambn

um− un,

Uka, b ^∞

m0

∞ n0

ambn

um un^k, k1,2, x^2∞

n0

|xn|²,

Tkx ^∞

n0

|xn|

u^kn, k1,2, f, g

_2π

0

ftgtdt, α²

_2π

0

|α|²dt, where αf, g, h.

1.4

In particular, whenba, the notationsUka, aandVa, aare denoted, respectively, byUka andVa. Throughout this paper, we will frequently use these notations, and we stipulate that Zdenotes integer set and that un Z_nλ/2, whereZ_n ∈ Z,n ∈ N₀, λis an integer or 0< λ <1.

2. Lemmas

In order to prove our assertions, we need the following lemmas.

Lemma 2.1. If both_∞

n0anand_∞

n0bnare absolute convergent, then i_∞

m0_∞

n0ambnis absolute convergent, iia²andb²are convergent.

The proof of it has been given in the paper2. Hence, it is omitted here.

Lemma 2.2. Letf, g, h∈ L²0, 2π, and lethbe a variable unit-vector. Then, f, g² ≤ f²g²

1−

|f, h|

f −|g, h| g

2

. 2.1

In particular, whenhis orthogonal tof, we have

f, g² ≤ f²g²

1−|g, h| g

2

; 2.2

and the equality in2.2holds if and only iff, gandhare linearly dependent.

The proof of these results has been given in5,6.

Define a function by

rx πU₁xsin 2λπ−U₂xsin²λπ. 2.3

Lemma 2.3. Let{an}and{bn}be two sequences of complex numbers. Ifλis an integer or 1/2 ≤ λ <1, then

rarb≥0. 2.4

Proof. Whenλis an integer, it is clear thatrarb 0. So we consider only the case for 1/2 ≤ λ <1. It is easy to deduce that

U₁x ₁

0

^∞

m0

x_mt^{um−1/2 2}dt >0, U₂x

₁

0

ds s

_s

0

^∞

n0

x_nt^{un−1/2 2}dt >0.

2.5

When 1/2≤λ <1, it follows from2.3thatrx<0 for anyx∈C. Hence, we haverarb>0.

Lemma 2.4. Letfz _∞

n0a_nz^un−1/2.Iffzis analytic in the unit-disc|z|<1, then 1

2π _π

−πt|f−e^it|²dt

|Va|. 2.6

Proof.

_π

−πtf−e^it2dt

_π

−πt f

−e^it

f−e^itdt

_π

−πt ∞ m0

∞ n0

a_ma_ncosπt isinπt^um−1/2

cosπ−t isinπ−tun−1/2

dt 2π

⎛

⎜⎜

⎝ ∞ m0

∞ umn0/un

a_ma_n um−un

⎞

⎟⎟

⎠sinλπi

⎛

⎜⎜

⎝ ∞ m0

∞ umn0/un

a_ma_n um−un

⎞

⎟⎟

⎠cosλπ 2πVa.

2.7 Thereby, the relation2.6holds.

3. Theorems and their corollaries

In order to prove our assertions, we need also to introduce the following functions:

s1x

√2|T1xcosλπ−1/2πT2xsinλπ| π²x²− rx^1/2 , s2x

√2|T1xsinλπ−1/πT2xsin²λπ/2|

π²x² rx^1/2 .

3.1

Theorem 3.1. Letrxbe a function defined by2.3, let{an}and{bn}be two nonzero sequences of complex numbers, and let both_∞

n0anand_∞

n0bnbe absolute convergent. Then, iifλis an integer, then

U1a, b²Va, b²≤π²a²b²1−R², 3.2 where

R²

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ 1 π²

T1a

a −T2b πb

2

T1b

b −T2a πa

2

ifλis odd, 1

π²

T1a a

2 T1b

b

2

ifλis even;

3.3

iiif 0< λ <1, then U₁a, bcos 2λπ− 1

2πU₂a, bsin 2λπ

²Va, b²≤

π²a²b²− 1

π²rarb 1−R²

, 3.4

where R² min{s1a−s₂b², s1b−s₂a²}, s_ix i 1,2 is defined by 3.1. In particular, when 1/2≤λ <1, we haverarb>0.

Proof. Define two functions by

fa, t ^∞

m0

a_m√

tsinumt,

gb, t ^∞

n0

bn

√tcosunt, t ∈ 0, 2π.

3.5

Since both _∞

n0an and _∞

n0bn are absolute convergent by Lemma 2.1, the double series _∞

m0_∞

n0ambn is absolute convergent. Accordingly, fa, tgb, t is uniformly convergent in the interval0, 2π. Thereby, the interchange in order of summation and integration can be made. In what follows, we stipulate that the interchanges in order of summation and integration are justified. It is easy to deduce that

f²π²a²−ra, g²π²b²rb, f, g

_2π

0

fa, tgb, tdt π

Va, b

U₁a, bcos 2λπ− 1

2πU₂a, bsin 2λπ ,

3.6

whererxis a function defined by2.3. ByLemma 2.2, we have Va, b

U₁a, bcos 2λπ− 1

2πU₂a, bsin 2λπ ² 1

π²f, g²≤ 1

π²f²g²1−r 1

π²

π²a²−ra

π²b²rb 1−r,

3.7

wherer |f, h|/f − |g, h|/g²,his a variable unit-vector, it can be properly chosen in accordance with our requirement.

iWhenλis an integer, it is known from2.3thatrx 0.

We selecth₁√

2t/2πit is easy to deduce thath11,and f, h1

√ 2

∞ m0

am

um . 3.8

Since the series_∞

n0anis absolute convergent, it is justified that the complex numberan

is replaced by|an|in3.8. Hence, we have f, h₁√

2T₁a. 3.9

Similarly

g, h₁

⎧⎪

⎨

⎪⎩

√2

πT₂b ifλis odd, 0 ifλis even.

3.10

We therefore obtain that

r₁f, h1

f −g, h1 g

2

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ 2 π²

T₁a

a −T₂b πb

2

ifλ is odd, 2

π² T₁a

a

2

ifλis even.

3.11

Hence, the inequality3.7can be reduced to

Va, b U1a, b²≤ {π²a²b²}1−r1. 3.12 Notice thatU₁b, a U₁a, bandVb, a −Va, b. If we still select the unit-vector h2√

2t/2π, then, interchangingaandbin3.12, we have −Va, b U1a, b²≤

π²a²b² 1−r2

, 3.13

wherer₂is defined by

r₂f, h₂

f −g, h₂ g

₂

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩ 2 π²

T1b

b − T2a

πa

2

ifλis odd, 2

π² T₁b

b

2

ifλis even.

3.14

Adding3.12and3.13, then the inequality3.2follows after simplifications.

iiWhen 0< λ <1, we firstly considerhin3.7. We still select unit-vectorh₁√ 2t/2π.

It is easy to deduce that f, h1√

2

_∞

m0

am

um

cosλπ− 1 2π

_∞

m0

am

um²

sinλπ , g, h₁√

2

_∞

n0

b_n un

sinλπ− 1 π

_∞

n0

b_n un²

sin²λπ

2 .

3.15

Since the series_∞

n0a_n and_∞

n0b_n are absolute convergent, it is justified that the complex numbers a_n and b_n are replaced, respectively, by |an| and |bn| in the above relations.

Let s1x f, h1/f, s2x g, h1/g. By using 3.1, we find s1a, s2b.

LetR²₁ s1a−s₂b². We obtain from3.7 Va, b

U1a, bcos 2λπ− 1

2πU2a, bsin 2λπ ²

≤

π²a²b²−b²ra− a²rb− 1

π²rarb 1−R²₁

.

3.16

Notice thatU1b, a U1a, b,U2b, a U2a, b,andVb, a −Va, b. If we still select the unit-vectorh₂√

2t/2π, then, interchangingaandbin3.16, we have −Va, b

U1a, bcos 2λπ− 1

2πU2a, bsin 2λπ ²

≤

π²a²b²−

a²rb− b²ra

− 1

π²rbra 1−R²₂

,

3.17

whereR²₂ s1b−s2a². LetR² min{R²₁, R²₂}. Adding3.16and3.17, the inequality 3.4can be gotten after simplifications. In particular, when 1/2 ≤ λ < 1, by Lemma 2.3, we haverarb≥0. The proof ofTheorem 3.1is completed.

Corollary 3.2. Let_∞

n0anbe absolute convergent. Then, iifλis an integer, then

U₁a²|Va|²≤π²a⁴1−R², 3.18 where

R²

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ 2 π²a²

T1a−T2a π

2

ifλis odd,

2

π²a²T1a² ifλis even;

3.19

iiif 0< λ <1, then U₁acos 2λπ− 1

2πU₂asin 2λπ

²|Va|²≤

π²a⁴− 1

π²r²a 1−R²

, 3.20

whereR² s1a−s2a², six i1,2is defined by3.1.

In particular, whenun nλ/2, according to3.2, one obtains a refinement of1.3immediately.

Corollary 3.3. Ifλ0,1, then ^∞

m1−λ

∞ n1−λ

ambn

mnλ ²

∞ m1−λ

∞ n1−λm /n

ambn

m−n

2

≤ π²

∞ n1−λ

|an|^{2 ∞}

n1−λ

|bn|² 1−R²

, 3.21

where

R²

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ 1 π²

T1a

a − T2b πb

2 T1b

b − T2a

πa

2

ifλ1, 1

π²

T₁a a

2 T₁b

b

2

ifλ0,

3.22

where

T_kx ^∞

n1−λ

|xn|

nλ/2^k, k1,2. 3.23 Corollary 3.4. Ifun n1/4, then

∞ m0

∞ n0

a_mb_n

mn1/2 ^{2 ∞}

m0

∞ n0,m /n

a_mb_n m−n

²≤

π²a²b²− 1

π²rarb 1−R²

, 3.24 where

rarb>0, R²min{s1a−s₂b²,s1b−s₂a²},

s₁x −1/√

2πT2x π²x²− rx^1/2, s2x

√2|T1x −1/2πT2x| π²x² rx^1/2 ,

Tkx ^∞

n0

|xn|

n 1/4^k, k1,2.

3.25

Sinceλ1/2, it is known fromLemma 2.3thatrarb>0.

IfrarbandRin3.24are replaced by zero, then the inequality3.24can be reduced to _∞

m0

∞ n0

ambn

mn1/2 ^{2 ∞}

m0

∞ n0,m /n

ambn

m−n

²< π²a²b². 3.26

The inequalities3.24and3.26are refinements of the Hilbert-Ingham inequality ^∞

m0

∞ n0

ambn

mn1/2

≤πab. 3.27 One has yet a new inequality according toTheorem 3.1ii.

Theorem 3.5. With the assumptions asTheorem 3.1, ifλ1/4, then 1

2πU2a, b

²Va, b²≤

π²a²b²− 1

π²rarb 1−R²

, 3.28

where

R²min{s1a−s2b²,s1b−s2a²}, s₁x |T₁x −1/2πT2x|

π²x²− rx^1/2 , s₂x |T₁x −1/2πT2x √

2−1| π²x² rx^1/2 , T_kx ^∞

n0

|xn|

n1/8^k, k1,2.

3.29

4. Applications toHP function

Letfzbe analytic in the unit-disc|z|<1. Iffz _∞

n0a_nzⁿ∈H_pwithp >0,then ₁

0

ft^pdt≤1 2

_2π

0

f

e^itpdt, 4.1

where the coefficient 1/2 is the best possible. It is called the Fejer-Riesz inequality in Hp

function7.

We will give both an extension and a refinement of4.1in what follows.

Theorem 4.1. Letfz _∞

n0a_nz^un−1/2∈H_p,wherep >0 andun Z_n λ/2Z_n∈Z, λ∈ N0. Iffzis analytic in the unit-disc|z|<1, then

₁

0

ft^pdt ²

1 2π

_π

−πtf

−e^itpdt ²≤

1 2

_2π

0

f e^itpdt

2 1−R²

, 4.2

whereR²>0.

Proof. At first, we prove the theorem for casep 2.Letfz _∞

m0amz^um−1/2. It is easy to deduce that

₁

0

ft²dt^∞

m0

∞ n0 um un/0

a_ma_n

um un U₁a, 1

2π _2π

0

f

e^it2dta².

4.3

ByLemma 2.4, we have 1

2π _π

−πtf

−e^it2dt

∞ m0

∞ um/n0un

a_ma_n um−un

Va. 4.4 Since the series_∞

n0a_nis absolutely convergent, it is justified that the complex numbera_nis replaced by|an|.

According3.18, we have ₁

0ft²dt ²

1 2π

_π

−πtf

−e^it2dt ²≤

1 2

_2π

0

f e^it2dt

21−R², 4.5

whereR²is defined by3.19. It is easy to deduce that 2

π²a² 4 π

2π 0

f e^it2dt

−1

,

T1a ₁

0

t^−1/2ftdt, T₂a

₁

0

ds s

_s

0

t^−1/2ftdt.

4.6

Becauseun Z_nλ/2/1/π,T₁a−T₂a/π /0. It shows thatR²>0. Hence, the inequality 4.2is valid whenp 2. Ifp /2, then by the Blaschke decomposition theorem, it holds that fz BzGz, whereBz is Blaschke function andGz/0,|Bz| ≤ 1 in |z| < 1 and

|Be^it|1.

LetFz Gz^p/2∈H₂. According to the above result forp2,we have ₁

0

ft^pdt ²

1 2π

_π

−πtf

−e^itpdt ²

₁

0

Ft²dt ²

1 2π

_π

−πtF

−e^it2dt ²

≤ 1 2

_2π

0

F e^it2dt

2

1−R²_F

1 2

_2π

0

G e^itpdt

₂

1−R²_G 1

2 _2π

0

f e^itpdt

2

1−R².

4.7

Based on the case for p 2, we haveR²_F > 0. Hence,R² > 0.The proof ofTheorem 4.1 is completed.

Let

fz ^∞

m0

cmz^m ∈ H1. 4.8

Then,

∞ n0

|cn| n1 ≤ 1

2 _π

−π

f

e^itdt. 4.9

It is called the Hardy inequality inHpfunction7.

We will give both an extension and a refinement of4.9as follows.

Theorem 4.2. Letfz _∞

m0cmz^umbe analytic in the unit-disc|z|<1, whereum Zmλ/2 (withZ_m∈Zandλ∈N₀) andf∈H₁. Then,

_∞

n0

|c_n| un

2

1 2π

_π

−πtf

−e^it2dt ²≤

1 2

_π

−π

f e^itdt

21−R², 4.10

where

R²

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩ 4 π

1

0t⁻¹ftdt− 1 π

₁

0

ds s

_s

0

t⁻¹ftdt

2 2π

0

fe^itdt

−1

ifλis odd,

4 π

1 0

t⁻¹ftdt

2 2π

0

f e^itdt

−1

ifλis even.

4.11 Proof. By Blaschke decomposition theorem, we have

fz BzGz BzG^1/2zG^1/2z f1zf2z, 4.12 whereBzis Blaschke function,f₁, f₂ ∈ H₂. Letf₁z BzG^1/2z _∞

m0a_mz^um, f₂z G^1/2z _∞

n0bnz^un.It is easy to deduce that a²b^2∞

m0

|am|^2∞

n0

|bn|² 1 2π

_π

−π|G^1/2e^it|²dt 1

2π _π

−π|Be^itG^1/2e^it|²dt 1 2π

_2π

0

f e^itdt.

4.13

Owing tofz f1zf2z, it holds that¹

0t⁻¹ftdt ₁

0t⁻¹f1t²dt. It is easy to deduce that

∞ n0

|c_n| un≤ ^∞

n0

rsn

|ar| |as|

ur us ^∞

m0

∞ n0

|am| |an|

um un. 4.14

ByLemma 2.4, we find that 1

2π _π

−πtf

−e^itdt

1 2π

_π

−πtf1

−e^it2dt

∞ m0

∞ umn0/un

|am| |an|

um −un

. 4.15

It follows from4.13,4.14,4.15, andCorollary 3.2that ∞

n0

|cn| un

2

1

2π _π

−πtf

−e^itdt ²≤

1 2

_2π

0

f e^itdt

2

1−R², 4.16

whereR²is defined by3.19. It is easy to deduce that 2

π²a² 4 π

2π 0

f e^it2dt

−1

,

T1a ₁

0

t⁻¹f1t²dt ₁

0

t⁻¹ftdt, T₂a

₁

0

ds s

_s

0

t⁻¹f₁t²dt ₁

0

ds s

_s

0

t⁻¹ftdt.

4.17

These show that the inequality4.10is valid.

Acknowledgment

The research is supported by the Scientific Research Fund of Hunan Provincial Education Departmentno. 06C657.

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