GENERALIZATIONS OF HARDY’S INTEGRAL INEQUALITIES

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GENERALIZATIONS OF HARDY’S INTEGRAL INEQUALITIES

YANG BICHENG and LOKENATH DEBNATH (Received 25 January 1997and in revised form 15 May 1997)

Abstract.This paper deals with some new generalizations of Hardy’s integral inequal- ities. Some cases concerning whether the constant factors involved in these inequalities are best possible are discussed in some detail.

Keywords and phrases. Hardy’s integral inequalities, weight function, best possible con- stants.

1991 Mathematics Subject Classification. 26D15.

1. Introduction. Hardy et al. ([2, Ch. 9]) proved the following integral inequalities:

if

p >1, 1 p+1

q=1, f (x)≥0, and 0<

_∞

0 f^p(x)dx <∞, (1.1)

then _∞

0

1 x

_x

0 f (t)dt p

dx < q^p _∞

0 f^p(t)dt, (1.2)

where the constantq^pin (1.2) is best possible.

The dual form of (1.2) is as follows:

if

0<

_∞

0

xf (x)_p

dx <∞, (1.3)

then _∞

0

∞

x f (t)dtp

dx < p^p _∞

0

tf (t)pdt, (1.4)

where the constantp^pin (1.4) is still best possible.

Both inequalities (1.2) and (1.4) are known as Hardy’s integral inequalities. They play an important role in mathematical analysis and its applications. Recently, Yang et al.

[1] gave some new generalizations of (1.2) which can be stated as follows:

For anyaandb,(0< a < b <∞), the following inequalities hold:

_b

a

1 x

_x

a f (t)dt _p

dx < q^p

1−

a b

_1/qpb

af^p(t)dt; (1.5)

_∞

a

1 x

_x

af (t)dt _p

dx < q^p _∞

a

1−θp(t) f^p(t)dt

0< θp(t) <1

; (1.6) _b

0

1 x

_x

0 f (t)dtp

dx < q^p _b

0

1−t

b 1/q

f^p(t)dt. (1.7)

The main objective of this paper is to consider some cases whether the constant factors involved in (1.5), (1.6), and (1.7) are best possible. This is followed by some more new generalizations of (1.4), (1.5), (1.6), and (1.7).

2. Best possible constant factors. This section deals with calculations of the best possible constant factors involved in (1.5), (1.6), and (1.7).

Theorem2.1. Ifb > a >0,p >1,(1/p)+(1/q)=1,f (x)≥0, and0<_b

af^p(x)dx

<∞, then

_b

a

1 x

_x

af (t)dtp

dx < q^pηp(a,b) _b

af^p(t)dt, (2.1)

where the constant

ηp(a,b)= max

a≤t≤b

1 qt^1/q

_b

t x^−1−1/q 1−a

x

1/qp−1

dx

(2.2) and

1 p

1−a b

1/qp

< ηp(a,b) <

1−a b

1/qp

. (2.3)

Proof. In view of the proof given in [1, Thm. 2.1], we have _b

a

1 x

_x

a f (t)dt p

dx≤q^p−1 _b

a

_b

t x^−1−1/q

1−

a x

1/qp−1

dx

t^1/qf^p(t)dt

=q^p _b

agp(t)f^p(t)dt,

(2.4)

where the weight functiongp(t)is defined by gp(t):=1

qt^1/q _b

t x^−1−1/q 1−a

x

1/qp−1

dx, t∈[a,b]. (2.5) Settingηp(a,b):=maxa≤t≤bgp(t), sincegp(t)is a nonconstant continuous function, then by (2.4), we have (2.1). Sincegp(b)=0, and for anyt∈[a,b),

gp(t) <1 qt^1/q

_b

t x^−1−1/q 1−a

b

1/qp−1

dx

= −t^1/q 1−a

b

1/qp−1

[b^−1/q−t^−1/q]

= 1−a

b

1/qp−1 1−t

b 1/q

≤ 1−a

b 1/qp

,

(2.6)

thenηp(a,b) < [1−(a/b)^1/q]^p. Since gp(t)=

t a

1/q_b

t

1−

a x

1/qp−1

d

1− a

x 1/q

=1 p

t a

_1/q 1−

a b

_1/qp

−

1−

a t

_1/qp ,

(2.7)

andgp(a) >0, then we haveηp(a,b) > gp(a)=1/p[1−(a/b)^1/q]^p. This completes the proof.

Remark. This theorem implies that the constant factorq^p[1−(a/b)^1/q]^pin (1.5) is not best possible, and the best value ofkp(a,b)for which (1.5) exists is bounded.

More precisely,

0< kp(a,b)≤q^pηp(a,b) < q^p

1−

a b

_1/qp

. (2.8)

Theorem2.2. For anya,b >0, the same constant factorq^pin (1.6) and (1.7) is best possible.

Proof. If the constantq^pin (1.6) is not best possible, then there existsK(0< K <

q^p), such that _∞

a

1 x

_x

a f (t)dt _p

dx < K _∞

a

1−θp(t) f^p(t)dt < K _∞

a f^p(t)dt. (2.9) Since{q/[1−(q−1)]}^p(1−pq/(1+))→q^p(→0⁺), then there exists a small number(0< < p−1), such that{q/[1−(q−1)]}^p[1−(pq/(1+))] > K. Setting f(t)=t^−(1+)/p,t∈[a,∞), we obtain

_∞

a f^p(t)dt=1

a⁻, (2.10)

and by Bernoulli’s inequality (see [4, Ch. 2.4]), _∞

a

1 x

_x

af(t)dt _p

dx=

1−1+ p

_−p∞

a x⁻⁽¹⁺⁾

1−

a x

_1−(1+)/pp dx

>

1−1+ p

_−p∞

a x⁻⁽¹⁺⁾

1−p a

x

_1−(1+)/p dx

=

q 1−(q−1)

_p 1−pq

1+

1 a⁻> K

_∞

a f^p(t)dt.

(2.11)

This is a contradiction, and hence the constant factorq^pin (1.6) is best possible.

If the constant factorq^pin (1.7) is not best possible, then there existsK(0< K < q^p), such that

_b

0

1 x

_x

0 f(t)dt _p

dx < K _b

0f^p(t)dt. (2.12)

There exists a number(>0), such that[q/(1+q−)]^p> K. Settingf(t)=t^−(1−)/p, t∈(0,b], then we obtain_b

0f^p(t)dt=(1/)b, and _b

0

1 x

_x

0f(t)dtp

dx= q

1+q− p1

b> K _b

0f^p(t)dt. (2.13) This is a contradiction, and the constant factorq^pin (1.7) is best possible. The theorem is proved.

3. Some new general inequalities. We first prove two lemmas.

Lemma3.1. Letb >0,p >1,(1/p)+(1/q)=1,f (x)≥0, and0<_b

0(xf (x))^pdx <

∞. Then there exists a numberx0∈(0,b), such that for anyx∈(0,x0), the following

inequality is true:

_b

xf (t)dt _p

< p^(p−1)

x^−1/p−b^−1/p_p−1b

xt^p−1/pf^p(t)dt. (3.1) Proof. For anyx∈(0,b), by Holder’s inequality, we have

p

xf (t)dtp

=b

xt^(p+1)/pqf (t)t^−(p+1)/pqdtp

≤ _b

xt^(p+1)/qf^p(t)dt _b

xt^−(p+1)/pdt _p/q

=p^p−1

x^−1/p−b^−1/p_p−1b

xt^p−1/pf^p(t)dt.

(3.2)

We have to show that there existsx0∈(0,b)such that for anyxin 0< x < x0, the equality in (3.2) does not hold. Otherwise, there existsx=xn∈(0,b),n=1,2,3,...

and the sequence xn

decreases to zero such that (3.2) becomes an equality. More- over, there existcnanddnwhich are not always zero such that (see [3, p. 29])

cn

t^(p+1)/pqf (t) ^p=dn

t^−(p+1)/pqq, a.e. in

xn,b . (3.3)

Butf (t)≠0, a.e. in[0,b], then there exists an integerN, such that for anyn > N, f (t)≠0, a.e. in[xn,b]. Hence bothcn=c≠0, anddn=d≠0, for anyn > N, and

then _b

0

tf (t)_p

dt=_n→∞lim _b

xn

tf (t)_p dt=d

c_n→∞lim _b

xnt⁻¹dt= ∞. (3.4) This contradicts the fact that_b

0(xf (x))^pdx <∞. Thus, (3.1) is valid. The lemma is proved.

Lemma3.2. Leta >0,p >1,(1/p)+(1/q)=1,f (x)≥0, and0<_∞

a

xf (x)pdx <

∞. Then there existsx0∈(a,∞), such that for anyx∈(a,x0), ∞

x f (t)dtp

< p^p−1x^−1/q _∞

x t^p−1/pf^p(t)dt. (3.5) Proof. We have, by Holder’s inequality as in Lemma 3.1, and for anyx∈(a,∞),

_∞

x f (t)dt _p

≤p^p−1x^−1/q _∞

x t^p−1/pf^p(t)dt. (3.6) We show that there existsx0∈(a,∞), such that (3.6) does not assume equality for any x∈(a,x0). Otherwise, there existsx=xn∈(a,∞)(n=1,2,...), xn↓a, such that (3.6) becomes an equality. By the same argument as in Lemma 3.1, there exists c >0 andN, such that for anyn > N,[t^(p+1)/pqf (t)]^p=c(t^−(p+1)/pq))^q, a.e. in[xn,∞), and hence_∞

a(tf (t))^pdt=climn→∞_∞

xn(1/t)dt= ∞. This is a contradiction. Inequality (3.5) is true. The lemma is proved.

Theorem3.1. Letb>a>0,p>1,(1/p)+(1/q)=1,f (x)≥0, and0<_b

a(xf (x))^pdx<

∞.

Then _b

a

b

xf (t)dtp

dx < p^pµp(a,b) _b

a

tf (t)pdt, (3.7)

where

µp(a,b)= max

a≤t≤b

1 pt^−1/p

_t

a

x^−1/p−b^−1/pp−1dx

(3.8) and

1 p

1−

a b

_1/pp

≤µp(a,b) <

1−

a b

_1/pp

. (3.9)

Proof. Using inequality (3.2), we obtain _b

a

b

xf (t)dtp

dx≤p^p−1 _b

a

x^−1/p−b^−1/pp−1_b

xt^p−1/pf^p(t)dt dx

=p^p−1 _b

a

t^−1/p

_t

a

x^−1/p−b^−1/p_p−1

dx

tf (t)_p dt

=p^p _b

ahp(t) tf (t)_p

dt,

(3.10)

where the weight functionhp(t)is defined by hp(t):= 1

pt^−1/p _t

a

x^−1/p−b^−1/pp−1dx, t∈[a,b]. (3.11)

Settingµp(a,b):=maxa≤t≤bhp(t), by (3.10), we have (3.7). Sincehp(a)=0, and for anyt∈(a,b],

hp(t)= 1 pt^−1/p

_t

ax^−1+1/p 1−x

b

1/pp−1

dx

< 1 pt^−1/p

_t

ax^−1+1/p

1− a

b

_1/pp−1 dx

=

1−

a b

_1/pp−1 1−

a t

_1/p

≤

1−

a b

_1/pp ,

(3.12)

then we haveµp(a,b) < [1−(a/b)^1/p]^p. Since hp(t)= −

t b

_1/pt

a

1−

x b

_1/pp−1 d

1−

x b

_1/p

= 1 p

t b

1/p 1−

a b

1/pp

−

1− t

b

1/pp ,

(3.13)

then we have

µp(a,b)= max

a≤t≤bhp(t)≥hp(b)= 1 p

1−a b

1/pp

. (3.14)

This completes the proof.

Remark. It follows from this theorem that the best valueλp(a,b)for which in- equality (3.7) exists is bounded, that is,

0< λp(a,b)≤p^pµp(a,b) < p^p

1−

a b

1/pp

. (3.15)

Theorem3.2. Leta>0,p >1,(1/p)+(1/q)=1,f (x)≥0, and0<_∞

a(xf (x))^pdx <

∞. Then

_∞

a

_∞

x f (t)dt p

dx < p^p _∞

a

1−

a t

1/p tf (t)_p

dt, (3.16)

where the constant factorp^pin inequality (3.16) is best possible.

Proof. By inequality (3.5), we have _∞

a

_∞

x f (t)dt p

dx < p^p−1 _∞

a x^−1/q _∞

x t^p−1/pf^p(t)dt dx

=p^p−1 _∞

a

_t

ax^1/qdx

t^p−1/pf^p(t)dt

=p^p _∞

a

1−a

t 1/p

tf (t)pdt.

(3.17)

Inequality (3.16) is true. Ifp^pin (3.16) is not possible, then there existsK(0< K < p^p), such that

_∞

a

_∞

x f (t)dt _p

dx < K _∞

a

1−

a t

_1/p tf (t)_p

dt < K _∞

a

tf (t)_p

dt. (3.18) There exists a small number >0, such that p^p[1/(1+)^p] > K. Setting f(t)= t^{−1−(1+)/p},t∈[a,∞), then we have_∞

a (tf(t))^pdt=(1/)a⁻, and _∞

a

∞

x f(t)dtp

dx=p^p 1 (1+)^p·1

a⁻> K _∞

a

tf(t)pdt. (3.19) This is a contradiction, and the constant factorp^p in (3.16) is best possible. This proves the theorem.

Theorem3.3. Letb >0,p >1,(1/p)+(1/q)=1,f (x)≥0, and0<_b

0(xf (x))^pdx <

∞. Then

_b

0

_b

xf (t)dt _p

dx < p^p _b

0µp(t) tf (t)_p

dt, (3.20)

where the weight functionµp(t)=(1/p){1−[1−(t/b)^1/p]^p}(b/t)^1/p,t∈(0,b], and 0< µp(t) <1; the constantp^pin inequality (3.20) is best possible. Whenp=2, inequal- ity reduces to the form

_b

0

b

xf (t)dt2

dx <4 _b

0

1−1

2 t

b

tf (t)2dt. (3.21) Proof. In view of (3.1), we find

_b

0

_b

xf (t)dt p

dx < p^p−1 _b

0

x^−1/p−b^−1/p_p−1b

xt^p−1/pf^p(t)dt dx

=p^p−1 _b

0

t 0

x^−1/p−b^−1/pp−1dx

t^p−1/pf^p(t)dt

=p^p−1 _b

0

_t

0x^−1+1/p

1− x

b

_1/pp−1 dx

1 t

_1/p tf (t)_p

dt

=p^p _b

0µp(t) tf (t)_p

dt,

(3.22)

where

µp(t):= 1 p

1−

1− t

b

1/ppb t

1/p

, t∈(0,b]. (3.23)

By Bernoulli’s inequality, we have

0< µp(t) <1 p

1−

1−pt b

1/pb t

1/p

=1. (3.24)

Inequality (3.20) is true. Sinceµ2(t)=1−(1/2)

t/b, inequality (3.21) is also true.

If the constant factorp^pin (3.20) is not best possible, then there existsK(0< K <

p^p), such that

_b

0

b

xf (t)dtp

dx < K _b

0µp(t)

tf (t)pdt < K _b

0

tf (t)pdt. (3.25)

There exists a small number,(0< <1), such thatp^p(1/(1−)^p){1−p/[+(1−

)/p]}> K. Settingf(t)=t^{−1−(1−)/p},t∈(0,b], we obtain_b

0(tf (t))^pdt=(1/)b, and by Bernoulli’s inequality,

_b

0

b

xf(t)dtp

dx=p^p 1 (1−)^p

_b

0x⁻¹⁺

1−x b

(1−)/pp

dx

> p^p 1 (1−)^p

_b

0x⁻¹⁺

1−p

x b

(1−)/p dx

=p^p 1 (1−)^p



1− p +⁽¹⁻⁾_p



1 b

> K _b

0

tf(t)pdt.

(3.26)

This is a contradiction, and the constant factorp^p in (3.20) is best possible. This completes the proof.

Remark. (1) In the limitsa→0,b→ ∞, inequalities (3.7), (3.10), (3.16), and (3.20) reduce to (1.4).

(2) Inequalities (3.7), (3.10), (3.16), and (3.20) are new generalizations of (1.4).

(3) Inequalities (1.4), (3.7), (3.10), (3.16), and (3.20) represent a class, whereas (1.2), (1.6), (1.7), and (2.1) form another class. The constant factors involved in these two classes of inequalities are shown to be best possible except (2.1) and (3.7).

References

[1] Y. Bicheng, Z. Zhonhua, and L. Debnath,On New Generalizations of Hardy’s Integral In- equality, J. Math. Anal. Appl.217(1998), no. 1, 321–327. Zbl 893.26008.

[2] G. H. Hardy, J. E. Littlewood, and G. Polya,Inequalities, 2nd ed., Cambridge, at the University Press, 1952. MR 13,727e. Zbl 047.05302.

[3] J. C. Kuang,Applied Inequalities, 2nd ed., Hunan Jiaoyu Chubanshe, Changsha, 1993 (Chi- nese). MR 95j:26001.

[4] D. S. Mitrinovic,Analytic Inequalities, vol. 165, Springer-Verlag, New York, Berlin, 1970.

MR 43#448. Zbl 199.38101.

Bicheng: Department of Mathematics, Guangdong Education College, Guangzhou, Guangdong,510303China

Debnath: Department of Mathematics, Universityof Central Florida, Orlando, Florida32816, USA

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