ON NEW STRENGTHENED HARDY-HILBERT’S

Internat. Math. &

VOL. 21 NO. 2 (1998) 403-408

ON NEW STRENGTHENED HARDY-HILBERT’S

^INEQUAUTY

BICHENG YANG

Department

of Mathematics GuangdongEducationCollege Guangzhou,Guangdong510303,P.R.CHINA

and LOKENATHDEBNATH

Department

of Mathematics UniversityofCentral Florida Orlando,Florida32816, U.S.A.

(Received February27, 1997 andinrevisedform September 8,

1997)

ABSTRACT. Inthispaper,a newinequality for the weight coefficient

w(q, n)

intheform

sin(r/p) ^2mi/p

+

^{n_i/q q}

^{> I,-}

P

⁺

^{q I, 6N} isproved. This isfollowed byastrengthenedversionof the Hardy-I-lilbeninequality.

KEY

WORDS AND PHRASES: Hardy-Hilbert’s inequality, weight coefficient, Holder’s inequality.

1991AMS SUBCTCLASSIFICATION CODES: 26D15.

INTRODUC’HON

If

a ^_>

^{0, 0}

<

ⁿ

2a2. ^<

oo, then the Karlson’sinequalityis

o < a,,

ⁿ

.=1 .=1

where theconstant cannotbe made smaller.

However,

^{it canbe}strengthened

(see

^Mikhlin ],p. 7) as

2 2

an < n

^rt- n"

r=l n=l

(1.2)

In

recent years, considerable attention has been given to develop some types ofstrengthened inequality(see

[2]-[ 10])

byestimating the weight coefficientw(q,

n)

as

1

(mn--)l/q ^{q-1 N)}

^(1.3)

w(q,

n)----

(m + ⁿ⁾ (q > I’P- +

^{1, n}

Someimprovement ofHardy-Hilbert’s inequality

(see

Hardy^etal. 11

])

has been madeinthe form

(

a

^{bq, (I 4)}

m

+

ⁿ sin(Tr/p)

m=l n=l n=l

Intheir recentwork, Xu and Gau

[2]

considered thefollowing weight coefficient

(1.3)

^andproved the followinginequality

404 B.C.YANG ANDL.DEBNATH

co(el, n) <

sin(r/p) ^nt/p

+

^{n-a/q r;p---p-1. (1.$)}

ThenastrengthenedHardy-Hilbert’s inequality

wasproved. Thekeyis to estimatethecorresponding weightcoefficiemeffectively. Hsuand

Wang [3]

provedthe followinginequality

0 3

w(2,n)<r---_, 0=:-I----1.12132

⁺

^{(n6N). (1.7)}

vn

Then they gave a new strengthened Hilbert’s inequality which is the same as

(1.6)

with p 2.

Since 0 in

(1.7)

is not the best possible, Gau

[5]

^{obtained the} best possible value of

0=- k=1

^1.2811+ Subsequently, C-au

[6]

^consideredthe general caseandproved a newinequality for the weight coefficientco(q,

n)

^as

co(q, n) <

sin(m/p) nl/; q

>

^1,

P- ^{+ -q}

^1,n6.N

^(1.8)

where

0p

(p-

1).

Recently, C-au

[7]

replaced(p-

1)

by

0, O,(n) >

^{0 in}(1.8). Buttheproblemis that

O,(n)

^{dependsonbothpandq.} Simultaneously,

Yang

[8]foundthat

0p

^{0 0.341295}

+,

but the constant

0p

0 isnotthebest possiblevalue. Finally,

Yang

andOau

[9]

found the bestpossible^value for

0,

^{0 1 C} 0.42278433

+,

whereCis aEulerconstant. Theyalsoprovedthefollowingnew Hardy-Hilbert’s inequality

m+n

sin(r/p)

z-i’/ an

_sin(Tr/p)

_z’i’/

^bq

^(1.9)

m=l n=l n=l n=l

Itisimportanttopointoutthat

(1.5)

^and

(1.8)

^aredifferent,and theconstant

rt

ⁱⁿ

(1.5)

dependson p.

Themainobjective of thispaperistoprovean improved versionof

(1.5)

^as

(q, ) <

sinQr/p) ^2n]/

+

^{r-l/q q}

^>

^1,

1 1

-+-=l,n6N

P ^q

(1.10)

and thenproveastrengthenedversion ofHardy-Hilbert’s inequalityasfollows:

EE m+n ^bn sin(/p)

^r 2rtl/P-t-rt-1/q¹

sin(r/p)

^{r 2nl/q}

n-1/’

^{1 bq}

m=ln=l n=l n=l

1/q

(1.11) Forthis,weneed thefollowinginequality

(see Yang [8]

Lemma

1):

If

f(m) >

^0,

/(2"-l) (x) < 0,/(2")(m) ^>

O,z6

[1, oo)(r 1,2), f(’)(oo) ---0(r--

O,

1,2,3,4),

and

fo ^{f(z)dz <}

^{oo,then rtt=l}

_ ^{f(m) <_ f(z)dz}

^-I-

_- ^{f(1) f’(:).}

^(1.12)

STRENGTHENEDI-IARDY-I-IILBERT’S INEQUALITY 405 2. SOME LEMMAS

LEMMA2.1. Ifq

>

1,

p-z ₊ q-Z

1,nEN, then

r 1

w(q,n) <

sin(vr/p)

nVn ^[fn(p) + g"(P)]’

where

w(q, n)

is definedby

(1.5),

^and

1 1

Y"(P)

^:=_P

+ + ₍₁ +

-1 1

g.)^:=

12pn

2(1 +

2p)n²

1 1

12n ^{+ 3(1 +}

^3p)n3’

7 1 1

12 2n 12n² 12n^3"

PROOF.

Let

f(w)

zE

[I, oo)(q > I,

nE

N).

By

(1.12),we obtainthat

1

/’

¹

wz--l_

dz

+

12 1

’1 +

ⁿ

+

1

12(1 + ⁿ⁾

^{2" (2.2)}

Since

Puttingz n3/,wefindthat 1

wethenfindthat

and

l+n ¹

( ^I+ _ln)-I ^<- 1{

^1--+

1

1(nl_.)

^-2

(1 + n)

^{2 n}

-’"

¹

⁺ _<- 1( ^I----+

²

Substitutingthe above resultsin(2.2),by(1.5),^wehave(2.1). Thisproves the lemma.

406 B.C. YANG AND L.DEBNATH

LEMMA

2.2. Ifp

>

^1,n E

N,

then

f. ^(,) + o ^{(,) >}

₂¹ _12n¹ _2n¹_s"

(2.3)

and

PROOF. Since

f’(,) x

>1 11

1

+n

1 1

2n (+)n (+

1

+n

1 1

12n²

(1 + 1)2n (1 + 3)2n

^s

1 1 1

12n 4n 16ns

>

0,

1 1

a’.()

2, +

( + 2), ^> o,

then

fn(P) +

gn(P)is strictly increasingfor pE

(1, co),

^and

’.) + .() > i.(/.O,) ^{+ .0,))}

21

1 1

12n 2n^s"

Thus the lemmaisproved.

LEMMA

2.3. Ifq

>

1,

p-1 + ^q-1

^{1,n E}

^N,

^theninequality

(1.10)

^{is valid. Soisthe}following inequality:

7r 1

w(p,

n) < sin(r/p--’---

^2nUq

+

^n-Up"

^(2.4)

PROOF. Sinceforn

>_

3,

2 12n 2n 1

+ +-

_{n 6 24n 2n2 4ns}

>

2’

then

1 1 1 1

2 12n 2n³

>

2+n^-1

(n>3).

By

(2.1)and(2.3),wehave

r 1

(1

¹

,(,n) <

sin(/,)

^,/

\2

¹²ⁿ

r 1

<

sin(r/p) ^2n/p

+

^n-Uq

2n³

(n 3).

Taking

0,

¹

^C,

^by

(1.8) (see Yang

andGau

[9]),

^{we find}that

r 1-U r

w(q, 1) < <

sin(/p) 1 sin(r/p) 1

2xl+l

(2.5)

SinceC

< 3/5

0.6, then we have

1 1-C

2x^2a/p

+

^2-Uq

^<

^21/p

and

r I-C r 1

w(q, 2) <

.(mQr/p) "2V’ ^< sin(r/p’-

2 2/P

+

^2-x/q"

^(2.6)

Itfollows that forn 1, 2,

(1.10)

also holds. Then

(1.10)

isvalidfor anyn

N.

Interchanging p,q in (l.10),since

si,(,]/j;) in(}q),

^wehave

^(2.4).

^{The lemmaisproved.}

NEW STRENGHARDY-HILBERT’SINEQUALITY &07 3. MAIN RESULTS

THEOREM3.1. Ifp>l,

p-I+q-l=l,

an_0, b,_0, and

0<a, <oo,

n=l

0

< b, <

oo, theninequality (l.

1)

^isvalid. Wealso have

m=l n=l n=l

When p q 2, this inequality reducestotheform

/

^<Tr

,._ _.=

^m

+

ⁿ _.--x

2v ^{+ (3.2)}

PROOF.

By

Holder’s inequality,wehave

Hence,

by

(1.10)

and

(2.4),

inequality

(1.11)

holds.

Sinceby

(2.4), w(p, n) < ,

then by Holder’s inequality,weobtain

n=l ^U,

By

(1.10),wefind

a, r I ^/q

Thisproves result

(3.1).

ThustheproofofTheorem3.1 iscomplete.

08 B.C.YANGAND L.DEBNATH 4.

CONCLUDING

REMARKS

(a) Inequality

(1.11)

is a definiteimprovementover

(1.6).

"--’)

^then

(b)

^{Since, forn _)}3,C

>

_2.+

gin(Trip) ^2rd/n

+

^n-1/q

^<

Sin(Tr/p)

(1 -C)

t l/p

(n >_ 3). (4.1)

Inviewof(2.5),

(2.6)

and (3.3),^itfollows that

(1.9)

^and

(1.11)

^representtwo distinct versionsof strengthenedinequalities. But theyarenotcomparable.

(c)

Inequality

(3. l)

^reducesto

Thisisanequivalent formofHardy-Hilbert’s inequality

(1.4) (see

Hardyetal.

[11],

Chapter 9).

[l]

MIKtK,IN,

S.G., Constants

m Some

Inequalities

of

Analysis, John Wiley

& Sons,

New York, 1986.

[2] XU,

^{L.C. and GAU,}

Y.K.,

Note on Hardy-Riesz’s extension ofI-Iilben’s inequality, Chinese QuarterlyJournal

of

Mathematfcs,6, (1991),^75-77.

[3]

HSU, L.C. andWANG,

Y.J.,

Arefinement ofHilbert’sdouble seriestheorem, J.Math.Res.

Exp.

11, (1991), 143-144.

[4] ZHAO, D.J.,

^{Onarefinement}of Hilbert’s double seriestheorem,Math. Practiceand Theory,

(1993),

85-90.

[5]

^GAU,

M.Z.,

A note onHilbertdouble seriestheorem, HunanAnnals

of

Mathematics, 12, 1-2 (1992), 142-147.

[6] GAU,

M.Z.,

An improvement ofHardy-Riesz’sextensionoftheI-filbertinequality, J.Math.Res.

Exp.

I4,2(1994),255-259.

[7]

GAU,

M.Z.,

AnoteontheHardy-Hbertinequality,J.Math. Aria.Appl.204

(1996),

^346-351.

[8] YANG,

^{B.C., A}refinement on thegeneralHilbert’sdoubleseriestheorem,

J.

Math.Study, 29, 2 (1996),^64-70.

[9] YANG,

B.C. and

GAU, M.Z.,

Onabest valueofHardy-Hilbert’sinequality,AdvancesinMath., 26, 2

(1997),

159-164.

[10] YANG, B.C.and

DEBNATI-I, L.,

Someinequalities involving theconstant e,andanapplicationto Carleman’s inequality,

J.

MarkAnal.andAppl.,toappear

(1997).

[11 HARDY, G.H., LITTLEWOOD, J.E.

and

POLYA, G.,

Inequalities, Ca/nbridgeUniversity

Press,

Cambridge, 1952.