A REFINEMENT OF HÖLDER’S INEQUALITY AND APPLICATIONS

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Hölder’s Inequality and Applications Xuemei Gao, Mingzhe Gao and

Xiaozhou Shang vol. 8, iss. 2, art. 44, 2007

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A REFINEMENT OF HÖLDER’S INEQUALITY AND APPLICATIONS

XUEMEI GAO, MINGZHE GAO AND XIAOZHOU SHANG

Department of Mathematics and Computer Science Normal College, Jishou University

Jishou Hunan 416000, People’s Republic of China EMail:mingzhegao@163.com Received: 10 February, 2007

Accepted: 10 June, 2007

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 26D15, 46C99.

Key words: Inner product space, Gram matrix, Variable unit-vector, Minkowski’s inequality, Fan Ky’s inequality, Hardy’s inequality.

Abstract: In this paper, it is shown that a refinement of Hölder’s inequality can be established using the positive definiteness of the Gram matrix. As applications, some improvements on Minkowski’s inequality, Fan Ky’s inequality and Hardy’s inequality are given.

Acknowledgements: The authors would like to thank the anonymous referee for valuable comments that have been implemented in the final version of this paper.

A Project Supported by scientific Research Fund of Hunan Provincial Education Department (06C657).

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1. Introduction

For convenience, we need to introduce the following notations which will be fre- quently used throughout the paper:

(a^r, b^s) =

∞

X

n=1

a^r_nb^s_n, kak_r =

∞

X

n=1

a^r_n

!¹_r

, kak₂ =kak,

(f^r, g^s) = Z ∞

0

f^r(x)g^s(x)dx, kfk_r= Z ∞

0

f^r(x)dx ¹_r

, kfk₂ =kfk, and

S_r(α, y) = α^r/2, y

kαk^−r/2_r ,

where a = (a₁, a₂, . . .) are sequences of real numbers, f : [0,∞) → [0,∞) are measurable functions andαandy are elements of an inner product spaceE of real sequences.

Let a = (a₁, a₂, . . .) andb = (b₁, b₂, . . .)be sequences of real numbers in Rⁿ. Then Hölder’s inequality can be written in the form:

(1.1) (a, b)≤ kak_pkbk_q.

The equality in (1.1) holds if and only if a^p_i = kb^q_i, i = 1,2, . . ., where k is a constant.

This inequality is important in function theory, functional analysis, Fourier analysis and analytic number theory, etc. However, there are drawbacks in this inequality.

For example, let

a= (a₁, a₂, . . . , a_n,0, . . . ,0), b = (0,0, . . . , b_n+1, b_n+2, . . . , b_2n), a, b∈R²ⁿ.

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If we leta_i =b_j = 1, i = 1,2, . . . , n;j =n+ 1, n+ 2, . . . ,2n, and substitute them into (1.1), then we have0≤n. In this case, Hölder’s inequality is meaningless.

In the present paper we establish a new inequality that improves Hölder’s inequality and remedies the defect pointed out above. At the same time, some significant refinements for a number of the classical inequalities can be established. As space is limited, only several applications of the new inequality are given.

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2. Main Results

Letαand β be elements of an inner product spaceE.Then the inner product ofα andβ is denoted by(α, β) and the norm of α is given by kαk = p

(α, α).In our previous papers ([1], [2]), the following result has been obtained by means of the positive definiteness of the Gram matrix.

Lemma 2.1. Letα, βandγbe three arbitrary vectors ofE. Ifkγk= 1, then (2.1) |(α, β)|² ≤ kαk²kβk²−(kαk |x| − kβk |y|)²,

wherex= (β, γ), y = (α, γ). The equality in (2.1) holds if and only ifαandβare linearly dependent, orγis a linear combination ofαandβ, andxy = 0butxandy are not simultaneously equal to zero.

For the sake of completeness, we give here a short proof of (2.1), which can also be found in [2].

Proof of Lemma2.1. Consider the Gram determinant constructed by the vectorsα, β andγ:

G(α, β, γ) =

(α, α) (α, β) (α, γ) (β, α) (β, β) (β, γ) (γ, α) (γ, β) (γ, γ)

.

According to the positive definiteness of Gram matrix we haveG(α, β, γ)≥ 0, and G(α, β, γ) = 0if and only if the vectorsα, β andγ are linearly dependent.

Expanding this determinant and using the conditionkγk= 1we obtain G(α, β, γ) =kαk²kβk²−(α, β)²−

kαk²x²−2(α, β)xy+kβk²y²

≤ kαk²kβk²−(α, β)²−

kαk²x²−2|(α, β)xy|+kβk²y²

≤ kαk²kβk²−(α, β)²− {kαk |x| − kβk |y|}²

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wherex= (β, γ)andy= (α, γ). It follows that the equality holds if and only if the vectorsαandβare linearly dependent; or the vectorγis a linear combination of the vectorαandβ, andxy= 0butxandyare not simultaneously equal to zero.

Applying Lemma2.1, we can now establish the following refinement of Hölder’s inequality.

Theorem 2.2. Leta_n,b_n ≥ 0, (n = 1,2, . . .), ¹_p + ¹_q = 1andp > 1. If0 <kak_p <

+∞and0<kbk_q <+∞, then

(2.2) (a, b)≤ kak_pkbk_q(1−r)^m, where

r = (S_p(a, c)−S_q(b, c))², m= min 1

p,1 q

, kck= 1

and

a^p/2, c

b^q/2, c

≥0.

The equality in (2.2) holds if and only if a^p/2and b^q/2 are linearly dependent; or if the vectorcis a linear combination ofa^p/2andb^q/2, and a^p/2, c

b^q/2, c

= 0, but the vectorcis not simultaneously orthogonal toa^p/2andb^q/2.

Proof. Firstly, we consider the casep 6= q. Without loss of generality, we suppose that p > q > 1. Since ¹_p + ¹_q = 1, we havep > 2. Let R = ^p₂, Q = _p−2^p , then

1

R+_Q¹ = 1. By Hölder’s inequality we obtain (a, b) =

∞

X

k=1

akbk =

∞

X

k=1

akb^q/p_k

b^1−q/p_k (2.3)

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≤

∞

X

k=1

a_kb^q/p_k R!_R¹ _∞ X

k=1

b^1−q/p_k Q!_Q¹

= a^p/2, b^q/22/p

kbk^q(1−2/p)_q .

The equality in (2.3) holds if and only ifa^p/2andb^q/2are linearly dependent. In fact, the equality in (2.3) holds if and only if for anyk, there existsc0(c0 6= 0)such that

a_kb^q/p_k R

=c₀

b^1−q/p_k Q

. It is easy to deduce thata^p/2_k =c0b^q/2_k .

Ifα, βandγin (2.1) are replaced bya^p/2, b^q/2 andcrespectively, then we have

(2.4) a^p/2, b^q/2²

≤ kak^p_pkbk^q_q(1−r),

wherer = (S_p(a, c)−S_q(b, c))². Substituting (2.4) into (2.3), we obtain after sim- plifications

(2.5) (a, b)≤ kak_pkbk_q(1−r)¹^p.

It is known from Lemma2.1 that the equality in (2.5) holds if and only ifa^p/2 and b^q/2are linearly dependent; or if the vectorcis a linear combination ofa^p/2andb^q/2, and a^p/2, c

b^q/2, c

= 0,but the vectorcis not simultaneously orthogonal to a^p/2 andb^q/2.

Note the symmetry ofpandq. The inequality (2.2) follows from (2.5).

Secondly, consider the casep= 2. By Lemma2.1, we obtain:

(2.6) (a, b)≤ kak kbk(1−r)¹² , wherer=

(a,c)

kak −^(b,c)_kbk 2

,kck= 1and(a, c) (b, c)≥0.The equality in (2.6) holds if and only ifaandb are linearly dependent, or the vectorcis a linear combination

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ofaandb, and (a, c)(b, c) = 0, but(a, c)and(b, c)are not simultaneously equal to zero.

The proof of the theorem is thus completed.

Consider the example given in the Introduction. Let c = (c₁, c₂, . . . , c_2n), c ∈ R²ⁿ, wherec_i = ^√¹_n, i= 1,2, . . . , nandc_j = 0, j =n+ 1, n+ 2, . . . ,2n.It is easy to deduce thatkck = 1andr = 1. Substituting them into (2.2), it follows that the equality is valid.

The following theorem provides a similar result to Theorem2.2.

Theorem 2.3. Let f(x), g(x) ≥ 0 (x ∈ (0,+∞)), ¹_p + ¹_q = 1 and p > 1. If 0<kfk_p <+∞and0<kgk_q <+∞, then

(2.7) (f, g)≤ kfk_pkgk_q(1−r)^m, where

r = (S_p(f, h)−S_q(g, h))², m = min 1

p,1 q

, khk= 1, i.e. khk=

Z ∞ 0

h²(x)dx ¹₂

= 1 and

f^p/2, h

g^q/2, h

≥0.

The equality in (2.3) holds if and only iff^p/2andg^q/2 are linearly dependent; or the vectorhis a linear combination off^p/2andg^q/2, and f^p/2, h

g^q/2, h

= 0, but the vectorhis not simultaneously orthogonal tof^p/2 andg^q/2.

Its proof is similar to that of Theorem2.2. Hence it is omitted.

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3. Applications

3.1. A Refinement of Minkowski’s Inequality

We firstly give a refinement of Minkowski’s inequality for the discrete form.

Theorem 3.1. Leta_k, b_k ≥0,p >1. If0<kak_p <+∞and0<kbk_p <+∞, then

(3.1) ka+bk_p <

kak_p+kbk_p

(1−r)^m, where

ka+bk_p =

∞

X

k=1

(a_k+b_k)^p

!_p¹ , r = min{r(a), r(b)}, m= min

1

p,1−1 p

, r(x) =

( x^p/2, c

kxk^p/2_p − (a+b)^p/2, c ka+bk^p/2_p

)2

, x=a, b;

(a+b)^p/2, c

=

∞

X

k=1

(a_k+b_k)^p/2c_k, andcis a variable unit-vector.

Proof. Letm= minn

1

p,1−¹_po ,

ka+bk_p =

∞

X

k=1

(a_k+b_k)^p

!_p¹ .

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By Theorem2.2, we have (3.2)

∞

X

k=1

a_k(a_k+b_k)^p−1 ≤ kak_p

∞

X

k=1

(a_k+b_k)^p

!1−¹_p

(1−r(a))^m and

(3.3)

∞

X

k=1

bk(ak+bk)^p−1 ≤ kbk_p

∞

X

k=1

(ak+bk)^p

!¹⁻¹_p

(1−r(b))^m, where

r(x) =

( x^p/2, c

kxk^p/2_p − (a+b)^p/2, c ka+bk^p/2_p

)2

, x=a, b,

ka+bk^p/2_p =

∞

X

k=1

(ak+bk)^p

!¹₂ , (a+b)^p/2, c

=

∞

X

k=1

(a_k+b_k)^p/2c_k, andcis a variable unit-vector.

Adding (3.5) and (3.3) we obtain, after simplifying:

(3.4) ka+bk_p ≤ kak_p(1−r(a))^m+kbk_p(1−r(b))^m.

Let r = min{r(a), r(b)}, then the inequality (3.1) follows. This completes the proof of Theorem3.1.

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If we choose a unit-vectorcsuch that itsith component is 1 and the rest is zero, i.e.c= (0,0, . . . ,0,1

(i)

,0, . . .), then

r(x) =

( x^p/2_i

kxk^p/2_p − (a_i+b_i)^p/2 ka+bk^p/2_p

)2

x=a, b.

Similarly, we can establish a refinement of Minkowski’s integral inequality.

Theorem 3.2. Letf(x), g(x) ≥ 0, p > 1. If0 < kfk_p < +∞ and0 < kgk_p <

+∞, then

(3.5) kf+gk_p <

kfk_p+kgk_p

(1−r)^m, where

kf +gk_p = Z ∞

0

(f(x) +g(x))^pdx ¹_p

, r= min{r(f), r(g)}, m = min

1

p,1− 1 p

, r(t) =

( t^p/2, h

ktk^p/2_p − (f+g)^p/2, h kf+gk^p/2_p

)2

, t=f, g,

(f +g)^p/2, h

= Z ∞

0

(f(x) +g(x))^p/2h(x)dx, andhis a variable unit-vector, i.e.

khk= Z ∞

0

h²(x)dx ¹₂

= 1.

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Its proof is similar to that of Theorem3.1. Hence it is omitted.

Remark 1. The variable unit-vectorhcan be chosen in accordance with our requirements. For example, we may choosehsuch that

h(x) =

s 2 π(1 +x²). 3.2. A Strengthening of Fan Ky’s Inequality

Theorem 3.3. Let A, B and C be three positive definite matrices of order n, 0 ≤ λ≤1. Then

(3.6) |A|^λ|B|^1−λ

≤ |λA+ (1−λ)B|



1−





|AC|¹⁴

¹₂(A+C)

1 2

− |BC|¹⁴

¹₂(B +C)

1 2





2



m

, where|C|=πⁿ, m= min{λ,1−λ}.

Proof. Whenλ = 0,1, the inequality (3.3) is obviously valid. Hence we need only consider the case0< λ <1.

IfDis a positive definite matrix of ordern, then it is known from [4] that

(3.7) J_n =

Z +∞

−∞

· · · Z +∞

−∞

e^−(x,Dx)dx= π^n/2

|D|¹² , wherex= (x₁, x₂, . . . , x_n), anddx =dx₁dx₂· · ·dx_n.

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LetF (x) = e^−λ(x,Ax)andG(x) = e−(1−λ)(x,Bx). Ifp= ¹_λ andq = _1−λ¹ , according to (3.4) and (2.7) we have

π^n/2

|λA+ (1−λ)B|¹² (3.8)

= Z +∞

−∞

· · · Z +∞

−∞

F (x)G(x)dx

≤

Z +∞

−∞

· · · Z +∞

−∞

F^p(x)dx

¹_pZ +∞

−∞

· · · Z +∞

−∞

G^q(x)dx ¹_q

(1−r)^m

= π^n/2(1−r)^m |A|^λ|B|^1−λ¹₂, where

r= S¹

λ(F, H)−S ¹

1−λ(G, H)2

=

F^2λ¹ , H

kFk⁻1^2λ¹ λ

−

G^2(1−λ)¹ , H kGk⁻

1 2(1−λ) 1 1−λ

, whereH =e⁻¹²^(x,Cx),Cis a positive definite matrix of ordern, and

kHk=

Z +∞

−∞

· · · Z +∞

−∞

H²(x)dx ¹₂

= 1.

By the definition of the variable unit-vectorH, it is easy to deduce that|C| = πⁿ.

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Hence we have

F^2λ¹ , H

= Z +∞

−∞

· · · Z +∞

−∞

F^2λ¹ (x)H(x)dx

= π^n/2

¹₂ (A+C)

1 2

=

( |C|

¹₂(A+C)

)¹₂

and

kFk^1/2λ_1/λ =

Z +∞

−∞

· · · Z +∞

−∞

F^1/λ(x)dx ¹₂

=

( π^n/2

|A|^1/2 )¹₂

= |C|

|A|

¹₄ , whence

S_1/λ(F, H) = |AC|¹⁴

¹₂(A+C)

1 2

. Similarly,

S1/(1−λ)(G, H) = |BC|¹⁴

¹₂ (B +C)

1 2

, therefore we obtain

(3.9) r=





|AC|¹⁴

¹₂(A+C)

1 2

− |BC|¹⁴

¹₂(B+C)

1 2





2

. It follows from (3.8) and (3.9) that the inequality (3.3) is valid.

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3.3. An Improvement of Hardy’s Inequality

We give firstly a refinement of Hardy’s inequality for the discrete form.

Theorem 3.4. Leta_n ≥0, β_n = _n¹ Pn

k=1a_k, ¹_p +¹_q = 1andp >1. If0<kak_p <

+∞, then

(3.10) kβk_p ≤

p p−1

kak_p(1−r)^m, where

r= (a^p/2, c)

kak^p/2_p − (β^p/2, c) kβk^p/2_p

!2

, cis a variable unit-vector andm= minn

1 p,¹_qo

.

Proof. Firstly, we estimate the difference of the following two terms:

β_n^p − p

p−1β_n^p−1a_n =β_n^p − p

p−1(nβ_n−(n−1)β_n−1)β_n^p−1 (3.11)

=β_n^p

1− np p−1

+(n−1)p

p−1 (β_n^p)^p−1β_n−1^p ¹_p . Applying the arithmetic-geometric mean inequality to the second term on the right- hand side of (3.11) we get

(3.12) (β_n^p)^p−1β_n−1^p ¹_p

≤ 1

p (p−1)β_n^p +β_n−1^p .

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It follows from (3.11) and (3.12) that β_n^p − p

p−1β_n^p−1an ≤β_n^p

1− np p−1

+(n−1)

p−1 (p−1)β_n^p +β_n−1^p

= 1

p−1 (n−1)β_n−1^p −nβ_n^p . Summing the above inequality with respect ton, we have

N

X

n=1

β_n^p − p p−1

N

X

n=1

β_n^p−1a_n ≤ − 1

p−1(N β_N^p)≤0.

Hence

N

X

n=1

β_n^p ≤ p p−1

N

X

n=1

β_n^p−1an. LettingN → ∞, we get

(3.13)

∞

X

n=1

β_n^p ≤ p p−1

∞

X

n=1

β_n^p−1a_n.

Applying the inequality (2.2) to the right-hand side of (3.13) we obtain p

p−1

∞

X

n=1

a_nβ_n^p−1 ≤ p p−1

∞

X

n=1

a^p_n

!¹_p _∞ X

n=1

β_n^(p−1)q

!¹_q

(1−r)^m (3.14)

= p

p−1kak_p

kβk^p_p¹_q

(1−r)^m,

wherer= (S_p(a, c)−S_q(β^p−1, c))²,cis a variable unit-vector andm= minn

1 p,¹_qo

.

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We obtain from (3.13) and (3.14) after simplification

(3.15) kβk_p ≤

p p−1

kak_p(1−r)^m. It is easy to deduce that

S_p(a, c) = a^p/2, c

kak^p/2_p and S_q(β^p−1, c) = (β^(p−1)q/2, c)

kβ^p−1k^q/2_q = (β^p/2, c) kβk^p/2_p . Hence

r=

a^p/2, c

kak^−p/2_p − β^p/2, c

kβk^−p/2_p 2

,

wherecis a variable unit-vector. The proof of the theorem is completed.

A variable unit-vectorccan be chosen in accordance with our requirements. For example, we may choosec ∈ R^∞ such thatc = (1,0,0, . . .). Obviously, kck = 1 and

r=a^p₁

kak^−p/2_p − kβk^−p/2_p 2

.

Similarly, we can establish a refinement of Hardy’s integral inequality.

Theorem 3.5. Let f(x) ≥ 0, g(x) = ¹_xRx

0 f(t)dt, ¹_p + ¹_q = 1 and p > 1. If 0<R∞

0 f(t)dt <+∞,then

(3.16) kgk_p < p

p−1kfk_p(1−r)^m, where

r= (f^p/2, h)

kfk^p/2_p −(g^p/2, h) kgk^p/2_p

!2

,

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his a variable unit-vector, i.e.

khk= Z ∞

0

h²(t)dt ¹₂

= 1 and m= min 1

p,1 q

. Proof. Using integration by parts and then applying (2.2) we obtain that

kgk^p_p = Z ∞

0

g^p(t)dt= p

p−1 f, g^p−1 (3.17)

≤ p

p−1kfk_p g^p−1

q(1−r)^m

= p

p−1kfk_pkgk^p−1_p (1−r)^m, where r = (S_p(f, h)−S_q(g^p−1, h))², m = minn

1 p,¹_qo

and h is a variable unit- vector. It is easy to deduce that

S_p(f, h) = f^p/2, h

kfk^p/2_p and S_q g^p−1, h

= g^p/2, h kgk^p/2_p . It follows that the inequality (3.16) is valid. The theorem is thus proved.

A variable unit-vectorhcan be chosen in accordance with our requirements. For example, we may choosehsuch thath(x) =e^−x/2. Obviously, we then have

khk= Z ∞

0

h²(t)dt ¹₂

= 1.

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References

[1] MINGZHE GAO, On Heisenberg’s inequality, J. Math. Anal. Appl., 234(2) (1999), 727–734.

[2] TIAN-XIAO HE, J.S. SHIUE AND ZHONGKAI LI, Analysis, Combinatorics and Computing, Nova Science Publishers, Inc. New York, 2002, 197-204.

[3] JICHANG KUANG, Applied Inequalities, Hunan Education Press, 2nd ed.

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