A sharpened version of Hardy’s inequality for parameter p=5/4

R E S E A R C H

Open Access

A sharpened version of Hardy’s inequality for

parameter

p

= /

Yongping Deng

_{, Shanhe Wu}

_{and Deng He}

*_{Correspondence:} [email protected] 1_{Department of Mathematics and} Computer Science, Longyan University, Longyan, Fujian 364012, P.R. China

Full list of author information is available at the end of the article

Abstract

The purpose of this paper is to investigate a sharpened version of Hardy’s inequality for parameterp= 5/4. By evaluating the weight coefficientW(k, 5/4), sharpened Hardy’s inequality that contains the best coefficient

η

Keywords: Hardy’s inequality; weight coefficient; sharpened inequality; exponential parameter; best coefficient

1 Introduction

Letp> , /p+ /q= ,an≥ (n= , , . . .),  <

∞

n=a p

n<∞. Then

∞

n=



n n

k= ak

p

<qp

∞

n=

ap_n, ()

whereqp_{= (} p p–)

p_{is the best coefficient. Inequality () is called Hardy’s inequality which is}

of great use in the field of modern mathematics (see [, ]). A special case of () yields the following inequalities:

∞

n=



n n

k= ak



< 

∞

n=

a_n, ()

∞

n=



n n

k= ak



< 

∞

n=

a_n. ()

In , Yang and Zhu [] evaluated the weight coefficientW(k,p),

W(k,p) =k–/p

∞

n=k



np

_n

j=



j/p

p–

, k= , , . . . , ()

and established an improved version of inequality () as follows:

∞

n=



n n

k= ak



< 

∞

n=

 –  √n+ 

a_n. ()

With the same approach, that is, evaluating the weight coefficientW(k,p), Huang [–] gave some improvements on Hardy’s inequality forp=  andp= /,i.e.,

∞

n=



n n

k= ak



< 

∞

n=

 –  n/

a_n, ()

∞

n=



n n

k= ak

/

≤√

∞

n=

 – ·





√

n+ 

a/_n . ()

Some further extensions of Hardy’s inequality related to the range of parameterpwere given in Huang [, ].

In , Yang [] proved an inequality for the weight coefficientW(k, )

W(k, ) =√k

∞

n=k



n

_n

j=



j

≤  –√

k

 – W(, )

and established the following inequality:

∞

n=



n n

k= ak



< 

∞

n=

 –√θ

n

a_n, ()

whereθ=  –_W(, ) = . . . . is the best coefficient under the weight coefficient W(k, ).

In , Zhang and Xu made use of the monotonicity theorem [–] and obtained an improvement of inequality ():

∞

n=



n n

k= ak

p

≤

p p– 

p∞

n=

 – cp (n– /)–/p

ap_n, ()

where

cp=

(p– )[ – /p_{( – /p)],  <p≤,}

 – –/p_{( – /p)}p–_{, p> .}

By evaluating the weight coefficientW(k,p), and with the help of an inequality-proving package called BOTTEMA [, ], He [] investigated a sharpened version of Hardy’s inequality forp∈Nand obtained the following improved version of inequality ():

∞

n=



n n

k= ak



≤



∞

n=

 – θ

n/

a_n, ()

whereθ=  – _W(, ) = . . . . is the best coefficient under the weight coefficient W(k, ).

numbers):

∞

n=



n n

k= ak

p

≤

p p– 

p∞

n=

 – θp()

n–/p

ap_n, ()

whereθp() =  – (p–_p )pW(,p) is the best coefficient of () under the weight coefficient W(k,p).

Recently, based on the program HDISCOVER  written by Deng, He and Wu [], an automated verification of inequality () is achieved forp∈Q(Qis the set of rational numbers).

For more detailed information of Hardy’s inequality, we refer the interested readers to relevant research papers [, , –].

In this paper, by evaluating the weight coefficientW(k, /), we establish an improve-ment of Hardy’s inequality for parameterp= / as follows:

∞

n=



n n

k= ak

/

≤/

∞

n=

 –  ·



n/_+η /

a/_n , ()

whereη/=  /

[/_{–W(,/)]}–  = . . . . is the best coefficient under the weight coefficient W(k, /).

2 Lemmas

To prove the main results in Section , we will use the following lemmas.

Lemma (see[]) If p> ,then for all integers n≥,it holds that p

p– n

–/p_– p p– +

 +

 n/p

≤

n

j=



j/p ≤ p p– n

–/p_– p p– +

 +

 n/p +

 p–



pn+/p. ()

Lemma (see[]) If p> ,then for all integers n≥k≥,it holds that

 (p– )kp–+

 kp <

∞

n=k



np <

 (p– )kp–+

 kp +

p

kp+.

Lemma  Let p> , /p+ /q= ,and let gr,glbe the functions defined by

gr(x) = –

p+ q–  pqx/q +

 qx–



pqx, gl(x) = – p+ q

pqx/q +



qx, x∈[, +∞). Then– <gr(x) < , – <gl(x) < .

Further, we have

g_r(x) = p+ q–  pq_x+/q –

 qx+

 pqx

≥p+ q– 

pq_x –

 qx +

 pqx

=(px+x+ p)(p– ) p_x > ,

and consequently,gris strictly increasing on [, +∞).

Now, fromgr() = –/p> – andlimx→+∞gr(x) = , it follows thatgr(x)≥gr() = –/p>

– andgr(x) < .

Similarly, from

g_l(x) = p+ q pq_x+/q –

 qx ≥

p+ q

pq_x –

 qx =

p–  p_x > , gl() = –/p> – and lim

x→+∞gl(x) = ,

we deduce that – <gl(x) < .

Lemma  is proved.

Lemma  Let– <g(x) < .Ifα∈(, ],then

 +g(x) + (α– )g(x) +(α– )(α– )  g

_(x)

≤ +g(x)α≤ +αg(x) +α(α– )  g

_(x).

Ifα∈[, ],then

 +g(x)α≥ +αg(x) +α(α– )  g

_(x).

Proof Whenα∈(, ]. By using the Maclaurin formula

 +g(x)α=  +αg(x) +α(α– )  g

_(x)

+α(α– )(α– )( +θg(x))

α–

 g

_{(x), θ∈(, ),}

and noticing – <g(x) < , we find

 +θg(x) >  +g(x) > ,

α(α– )(α– )( +θg(x))α–

 g

_(x)≤,

(α– )(α– )(α– )( +θg(x))α–

 g

Thus

 +g(x)α≤ +αg(x) +α(α– )  g

_(x),

 +g(x)α= +g(x) +g(x)α–

≥ +g(x) + (α– )g(x) +(α– )(α– )  g

_(x)

.

Whenα∈[, ]. We have

α(α– )(α– )( +θg(x))α–

 g

_(x)≥.

Thus

 +g(x)α≥ +αg(x) +α(α– )  g

_(x).

The proof of Lemma  is complete.

Lemma  Let p> , /p+ /q= ,n≥k≥,and let[x]denote the greatest integer less than or equal to the real number x.Then we have

W(k,p)≤qp–k/q

∞

n=k



n+/q

 +gr(n)

[p]–

×

 +p– [p]gr(n) +

(p– [p])(p– [p] – )  g

 r(n)

.

Proof By Lemma  and the identitypq=p+q,q(p– ) =p, it follows that

W(k,p) =k–/p

∞

n=k



np

_n

j=



j/p

p–

≤k–/p

∞

n=k



np

p p– n

–/p_– p p– +

 +

 n/p+

 p–

 pn+/p

p–

=k/q

∞

n=k



np

qn/q–p+ q–  p +

 n/p –

 pn+/p

p–

=k/q

∞

n=k



npq

p–_n(p–)/q

 –p+ q–  pqn/q +

 qn –

 pqn

p–

=qp–k/q

∞

n=k



n+/q

 +gr(n)

p–

.

Combining Lemmas  and , we obtain

W(k,p)

≤qp–k/q

∞

n=k



n+/q

 +gr(n)

[p]–

 +gr(n)

≤qp–k/q

∞

n=k



n+/q

 +gr(n)

[p]–

×

 +p– [p]gr(n) +

(p– [p])(p– [p] – )  g

 r(n)

.

This completes the proof of Lemma .

Lemma  Let/p+ /q= ,n≥k≥.If p∈(, ),then

W(k,p)≥qp–_k/q

∞

n=k



n+/q

 +gl(n)

[p]

×

 +p– [p] – gl(n) +

(p– [p] – )(p– [p] – )

 g

 l(n)

.

If p∈[, +∞),then

W(k,p)≥qp–k/q

∞

n=k



n+/q

 +gl(n)

[p]–

×

 +p– [p] + gl(n) +

(p– [p] + )(p– [p])  g

 l(n)

.

Proof Sincepq=p+q,q(p– ) =p, using Lemma  gives

W(k,p) =k–/p

∞

n=k



np

_n

j=



j/p

p–

≥k–/p

∞

n=k



np

p p– n

–/p_– p p– +

 +

 n/p

p–

=k/q

∞

n=k



np

qn/q–p+ q p +

 n/p

p–

=k/q

∞

n=k



npq

p–_n(p–)/q

 – p+ q pqn/q+

 qn

p–

=qp–k/q

∞

n=k



n+/q

 +gl(n)

p–

.

Whenp∈(, ). From Lemmas  and , we have

W(k,p)≥qp–k/q

∞

n=k



n+/q

 +gl(n)

[p]–

 +gl(n)

p–[p]

≥qp–k/q

∞

n=k



n+/q

 +gl(n)

[p]

×

 +p– [p] – gl(n) +

(p– [p] – )(p– [p] – )

 g

 l(n)

Whenp∈[, +∞). Using Lemmas  and , we obtain

W(k,p)≥qp–k/q

∞

n=k



n+/q

 +gl(n)

[p]–

 +gl(n)

p–[p]+

≥qp–k/q

∞

n=k



n+/q

 +gl(n)

[p]–

×

 +p– [p] + gl(n) +

(p– [p] + )(p– [p])  g

 l(n)

.

Lemma  is proved.

Lemma (see[]) Let p> ,an≥ (n= , , . . .),  <

_∞

n=a p

n<∞.Then

∞ n=  n n k= ak p ≤ ∞ k=

k–/p

∞

n=k

 np _n j= 

j/p

p– ap_k

=

∞

k=

W(k,p)ap_k.

3 Main results

Theorem  For an arbitrary natural number k,the following inequality holds true:

W(k, /) <R/(k),

where

R/(k) = /

 –  k/–

, ,k/ +

 k–

 k/ –

, ,k/ +

 ,k

+  ,k/ +

 ,k +

 ,k/+

 ,k+

 ,k

.

Proof Using Lemma  gives

W(k, /)≤/k/

∞

n=k



n/

 + gr(n) –

 g

 r(n)

= /k/

∞

n=k r/(n),

where

r/(n) =



n/ –

 n/ –

, ,n/+

 n/ +

 ,n/–

 ,n/

–  ,n/ +

 ,n/ –

 ,n/.

Hence

W(k, /)≤/k/

∞

n=k



n/ –

 n/–

, ,n/+

 n/ +

 ,n/

–  ,n/ –

 ,n/+

 ,n/–

 ,n/

Using Lemma  and takingp= /, /, /, /, /, /, /, /, / in the right-hand side of inequality (), respectively, we get

∞

n=k



n/ <



k/ +

 k/ +

 k/,

–

∞

n=k

 n/ < –

 k/ –

 ,k/,

. . . ,

∞

n=k

 ,n/ <

 ,k/ +

 ,k/ +

 ,k/,

–

∞

n=k



,n/< –

 ,k/ –

 ,k/.

Adding up the above inequalities, we obtain

W(k, /) <R/(k).

Theorem  is proved.

Theorem  For an arbitrary natural number k,the following inequality holds true:

W(k, /) >L/(k), where

L/(k) = /

 –  k/–

 k/ –

, ,k/+

 k–

 k/ +

, ,k/

– , ,k/+

 ,k –

 ,k/+

 ,k/–

, ,k/

–  ,k–

 ,k/+

 ,k–

, ,k/

.

Proof Utilizing Lemma  gives

W(k, /)≥/k/

∞

n=k



n/

 +gl(n)

 – gl(n) +

 g

 l(n)

= /k/

∞

n=k l/(n),

where

l/(n) =



n/ –

 n/–

 ,n/ –

, ,n/ +

 n/ +

 ,n/

+ , ,n/ –

 ,n/ –

 ,n/ +

Hence

W(k, /)≥/k/

∞

n=k



n/ –

 n/–

 ,n/ –

, ,n/ +

 n/

+  ,n/+

, ,n/ –

 ,n/

–  ,n/ +

 ,n/

.

Using Lemma  and takingp= /, /, /, /, /, /, /, /, /, / in the left-hand side of inequality (), respectively, we get

∞

n=k



n/ >



k/ +

 k/,

–

∞

n=k

 n/ > –

 k/–

 k/–

 k/,

. . . , –

∞

n=k

 ,n/> –

 ,k/–

 ,k/–

, ,k/,

∞

n=k

 ,n/ >

 ,k/ +

 ,k/.

Adding up the above inequalities, we obtain

W(k, /) >L/(k).

Theorem  is proved.

Theorem  Let an≥ (n= , , . . .),  <

∞

n=a/n <∞.Then

∞

n=



n n

k= ak

/

≤/

∞

n=

 –  ·



n/_+η /

a/

n , ()

whereη/=  /

[/_{–W(,/)]} –  = . . . .is the best possible under the weight coefficient W(k, /).

Proof By Lemma , we have

∞

n=



n n

k= ak

/

≤ ∞

k=

W(k, /)a/_k .

Therefore, to prove inequality (), it suffices to show that

W(k, /)≤/

 –  ·



k/_+η /

Obviously, inequality () becomes an equality fork= . In what follows, we will assume thatk≥.

By Theorem W(k, /) <R/(k), we need only to prove that

R/(k)≤/

 –  ·



k/_+η /

.

Note that

η/=

/

[/_{–W(, /)]}–  = . . . . >

 , it suffices to show

R/(k)≤/

 –  ·



k/_{+ /}

. ()

Substitutingk=xin (), inequality () becomes

R/

x≤/

 –  ·



x+ /

, wherex≥√,

which is equivalent to the following inequality:

/

 –  x–

, ,x +

 x –

 x–

, ,x +

 ,x+

 ,x

+  ,x +

 ,x+

 ,x+

 ,x

≤/

 –  ·



x+ /

()

⇔ – 

x–

, ,x +

 x –

 x–

, ,x +

 ,x+

 ,x

+  ,x +

 ,x+

 ,x+



,x ≤ –

 ·



x+ /

⇔– 

x–

, ,x +

 x –

 x–

, ,x +

 ,x+

 ,x

+  ,x +

 ,x+

 ,x+

 ,x+

 ·



x+ /≤

⇔– f(x)

,,x_{(x+ )}≤,

where

f(x) = ,,x+ ,,,x+ ,,x– ,,,x

– ,,x+ ,,x+ ,,x– ,,x

– ,,x– ,,x– ,,x– ,,x

From the hypothesisx≥√ > ., we have

,,x+ ,,,x+ ,,x– ,,,x

– ,,x+ ,,x+ ,,x

>,,×.+ ,,,×.+ ,,×. – ,,,x– ,,x+ ,,x+ ,,x

= ,,,x– ,,x+ ,,x+ ,,x

=(,,,x– ,,)x_{+ ,,x+ ,,x}

>(,,,×. – ,,)×. + ,,×. + ,,x

= ,,,x.

Further, we have

f(x) > ,,,x– ,,x– ,,x

– ,,x– ,,x– ,,x

– ,,x– ,x– ,x– ,x– , > ,,,x– ,,x– ,,x

– ,,x– ,,x

– ,,x– ,,x– ,x– ,x

– ,x– ,x

= ,,x> .

Consequently, inequality () holds true, and inequality () is proved. Let us now show thatη/= 

/

[/_{–W(,/)]} –  = . . . . is the best possible under the

weight coefficientW(k, /).

Consider inequality () in a general form as

W(k, /)≤/

 –  ·



k/_+η /

. ()

Puttingk=  in () yields

η/≥

/

[/_{–W(, /)]}.

Thus the best possible value forη/in () should beηmin=  / [/_{–W(,/)]}.

Remark  From the definition ofW(k,p) and in the same way as in [], we can establish the following accurate estimates ofW(, /):

. <W(, /) < .. () Further, the approximation ofη/can be derived as follows:

η/=

/

[/_{–W(, /)]}–  = . . . . .

Remark  Forp= /, inequality () can be written as

∞

n=



n n

k= ak

/

≤/

∞

n=

 – – (

 )

/_{W(, /)} n/

a/_n . ()

It is easy to observe that 

· 

n/_+η/>

 ·



n/_{+ ,/,}>

 n/

and

 –

 

/

W(, /) <  –

 

/

×. = . . . . <  , hence

 – (_)/_{W(, /)} n/ <

 ·



n/_+η /

.

This implies that inequality () is stronger than inequality ().

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YD finished the proof and the writing work. SW gave YD some advice on the proof and writing. DH gave YD lots of help in revising the paper. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics and Computer Science, Longyan University, Longyan, Fujian 364012, P.R. China.}2_Gangtou Middle School, Fuqing, Fujian 350317, P.R. China.

Acknowledgements

The present investigation was supported, in part, by the Natural Science Foundation of Fujian province of China under Grant 2012J01014 and, in part, by the Foundation of Scientific Research Project of Fujian Province Education Department of China under Grant JK2012049.

Received: 10 December 2012 Accepted: 1 February 2013 Published: 20 February 2013

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