Internat. J. Math. & Math. Sci.
(1985) 165-171
165
Vol. 8 No.
ON SOME EXTENSIONS OF HARDY’S INEQUALITY
CHRISTOPHER O. IMORU
Department of Mathematics
University of Ife
Ile-Ife
Nigeria
(Received March 24, 1983 and in revised form February 28, 1984)
ABSTRACT.
to
We present in this paper some new integral inequalities which are related
Hardy’s inequality, thus bringing
into
sharp focus some of the earlier results of
the author.
KEY WORDS AND PHRASES. f neua li ties
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
1.
YD15
INTRODUCTION.
The present note is a sequel to the author’s paper [i] in which the following
theorem that generalizes Shum’s result [2] was proved.
THEORF, M
(x)
0
for
I.I.
x
Let
0
be continuous and non-decreasing on [o,=] with g(O)
O,
1 and f(x) be non-negative and
=. Let p
i, r
#
g(=)
and
g(x)
Lebesgue-Stieltjes integrable with respect to
r
1
or
r < i, where
a
0
b
and
or
[a, o] according
as
Suppose
0.
f f(t)dg(t)
f f(t)dg(t)
F(x)
[0,b]
on
(r
i)
(r
1).
(I.i)
Then
0
g(x)
-r
F(x)Pdg(x) + [p/(r-l)]Pg(b)1-r
F(b)
p
(1.2)
p (r-l)
]P
fb
g(x) -r [g(x)
0
f(x)]Pdg(x)
(r
i)
and
-r
F(x)Pdg(x) + [p/(1-r) ]Pg(a)l-r
F(a)
p
(1.3)
[p/("l-r")]Pf
with both inequalities reversed in
g(x)-r[g(x)f(x)]Pdg(x)
0
(r
i),
p _< I.
Equality holds in either inequality, when either p
1 or f
O. The constant
p
or [p/(i-r)]
is the best possible when the left side of (1.2) or (1.3)
[p/(r-l)
JP
is unchanged, respectively.
C.O. IMORU
166
The objective of this paper is to obtain a new integral inequality which is an
Indeed, we shall show that Theorem I.i in its modified
extension of Theorem i.i.
form leads us to some extensions, variants and a new generalization of a class of
inequalities which are related to Hardy’s integral inequality. Moreover, we shall examine
the case when
[I]. In fact,
a case which was not discussed in
I.I,
in Theorem
1
r
Hardy’s inequality due to Ling-Yau Chan [3], which seemed new, are
certain extensions of
shown to be immediate consequences of the modified form of Theorem i.I.
An immediate corollary of this result shows
In Section 2 we state our main result.
.
that the inequalities (1.2) and (1.3) continue to persist except for an added constant
b
b
b <0 <- a
are replaced by
when
in (1.2) and
a’ where
a
0
In Section 3, we prove an elementary lemma which is then used to give the proof of
In Section 4, our result is applied to give some useful variants and
our main result.
extensions of
Hardy’s inequality.
Throughout what follows, unless otherwise stated, we shall assume
0 and g(d)
non-negative and non-decreasing on [c,dJ with g(c)
.
g
is continuous,
Furthermore,
f
is assumed to be a non-negative Lebesgue-Stieltjes integrable function with respect to
g
on
Our notations and terminologies are similar to the ones in [2].
[c,d].
real numbers
p
#
0
#
and
Q(.,,F)
f(t)Pdg(t)
g(t) (p-l)
(p-l) (6+1) f(t)Pdg(t)
g(t)
(6
0)
(6
0),
f(t) dg(t)
(6
0)
f(t) dg(t)
(6
0)
6
and
as follows:
8, F
0, the functions
6+1
8(x)
f..
Thus, for
[c,d] are defined
on
(1.4)
(1.5)
F(x)
’__
jx
and
2.
P
d
0
c
0
6
according as
0, where
6
or
g(x) 6p-i [g(x)f(x)]
p
c
for every
p e 1
b <-d. Suppose for
d
6p-i
p
a
dg(x)
L((x,d),dg)
or f(t)
L((c,x),dg)
f(t)
Let
(1.6)
Our main result is the following.
STATEMENT OF IAIN RESULT.
THEOREM 2.1.
p
g(x) 6p V(x)
161 p-1
8(x)
g(x)
Q(x,6,F)
and
F (x)
g(x)
C
dg(x)
x
(a,b)
0
p -< i.
or
if
then we have
a
g(x)
6p-i
161- p
+
whenever
p -> 1
or
0.
p
strict unless either
when the term
F(x) p dg(x)
1
p
6
6
-i
[g(b)Pv(b) P
fb
a
g(x)
(2.1)
0
The inequality is reversed if
or f
0.
g(a)
(l-r)/p, r
The constant factor
p F(a) p]
1
and
results.
a
6PF(a) P
g (a)
6-I
g(x) p [g(x)f(x)]Pdg(x)
-l[g(b)PF(b)P
REMARK 2.2. For
_<
F(x)Pdg(x)
_<
-
II
i.
p
-p
The inequality is
is the best possible
remains unchanged.
p
1
[p/(l-r)]g(b)l-rF(b)P
or
p
0, we obtain the following
g(a)l-rF(a)P]
(2.2)
+ [p/(r-l)
pfb g(x)
a
-r
g(x) (x)
Pd g(x)
(r
i)
SOME EXTENSIONS OF HARDY’S INEQUALITY
167
and
b
g(x)-rF(x)Pdg(x)
g(a)l-rF(a)P]
[p/(1-r)][g(b)l-rF(b)P
<-
(2.3)
+ [p/(l-r)]
p -_ 1
0
when
pfb g(x)-r[g(x)f(x)]Pdg(x)
0 and
Suppose the hypotheses g(c)
[o’
o
where
and
oog
f by
(g)
REMARK 2.3.
g(d)]
[g(c)
and non-decreasing on
(oog)(c)
with
[I], namely
g(d)
]-if,
by
g
i);
the inequalities are reversed.
These inequalities generalize our earlier results in
replacing
(r
a
is
Theorem I.i.
are relaxed.
and (oog)(d)
0
On
differentiable, non-negative
oo, we obtain
from inequality (2.1) the following form
b
o’(g(x))o(g(x))SP-i
4-110,(g(b)) ?’p
18I-P
+
where
o(g(a))SP
F(b) p
o’ (g(x)) l-po(g(x)
b
a
F(a) p
(2.4)
6P-l[o(g(x))f (x) ]Pdg(x)
p > 1
is, however, still defined by (1.5) and
F
or
O.
p
Inequality (2.4)
p -< i.
0
is reversed if
3.
f
G(x) p dg(x)
PROOF OF THEOREM:
The proof of our main result will depend essentially on the
following lemma.
LEMMA:
For
_>
p
1
or
O, the
p
ever
0
p
!
I, the function
[c,d] according
non-decreasing on
PROOF:
according as
#
0
as
O,
p -> 1
For
or
0 <_ O(x)
c
and
0
p
8
0
O,
is positive and
O.
or
O.
0(x) exist uncer the hypotheses
O,
8+1) f(t)Pdg(t)
g(t) (p-l)
g(t) 8p-I [g(t) f(t) ]p dg(t)
-< g(x)
-< g(x) -S
f,
c
g(t) 8p-1 [g(t)f(t) ]p dg(t)
which is obtained from the non-decreasing property of
g
and the hypothesis of the
theorem,
For
p ->
or
0
p
0
e(x)
and
8
O, we have by the same token
=fx g(t)(P-l)(6+l)f(t)P
d
dg(t)
g(x)-Sfx g(t) p-l[ g(t)f(t) ]p dg(t)
d
-< g(x)
If how-
is negative and non-increasing or
or
We shall show that the integrals defining
of the theorem.
#
Q(.,6,F), 8
function
c,d]
Q(.,6,F),
non-decreasing or non-increasing on
-6fd
c
g(t) 8p-1 [g(t)f(t) ]p dg(t)
=.
C.O. IMORU
168
For
and let
O,
p
[c,d],
(t.,t)z tl’t2
t
g(tl)-l-pg(t2)-(6$1)dg(t!
(d,d)
d*
and
or
i
p
dX(t)
where
[c,d].
x
fX*c, h(t)d(t)
I
fd*
f x:
h(t)dX(t)
x*
g(t2)P(6+l)f(t2)P
h(t)
let
dg(tg). Suppose x* (x,x),
-I g(x) -(p+I) Q(x,6,F)
(161p)
Then,
x*
dX_t.
c*
dX(t)
jl-P[
I
x*
h(t)
c*
jl-P[
f
d*
Q(x,,F),
Hence, the non-negative property of
i/Pd% (t) ]P
(6
O)
h(t)l/Pdx(t)]P
(6
O)
[c,d],
x
O,
(c,c)
c*
is a direct
consequence of Jensen’s inequality for convex functions.
Also from the Jensen-Steffensen inequality for sums, we have for
p
p
"
1
or
0,
-
(Zi)l
where
_EI
i
2,
n
CrY*
I
(3.1),
c*
-[
(]{p)-i
p
y, dX(t),
q2
fY*
c*
dX(t)
JP
and
x*
d%(t)nl -pfy*
[Y,
[c*
.7 c
]P
f:y*, h(t) i/p dX(t) jp
f:: dX(t)]l-p[/x* h(t) 1/p dX(t) ]p
c*
g(x)
-(p+l)
Q(x,6,F)
0
to the second
[Q(x,6,F)
x
is zero for
p
h(t) i/Pd X(t)
dX(t)] l-p
Q(y,,F)]
summand, we obtain
O.
is positive and non-decreasing on
c
and non-decreasing for
x >- c
is non-negative and non-decreasing on
the assertion of the lemma is valid for
or
O.
qi
1-p(qlZl i/p + q2 z 2 I/p
h(t) i/Pd X(t) /
(Iglp) -I g(.) -6 (p+l) Q(.,,F)
p >- 1
E
we have
Jensen’s inequality
On the application of
that for
with
ify, h(t)_ i/Pd X(t)_ fy,,y, dX(t)_ ]p
,
+
O,
satisfies the conditions
we get
0 -<
whence for
0
i g n
i=1
c* dX(t),
z2
(161p)-ig(x) -(p+I)
)P
Making use of the substitutions
d.
x
y
z
Hence
i/p
qi zi
(qi)l
n)
u
(ql +q2
ql
in inequality
qi (I
qlZl + q2z2
c
n
(i.IE
n
qi -<
i=u
In particular, for
Now suppose
qi
is non-decreasing and
n
Y.
0 -<
l-p
(i I
i__E1 qizi
i -< n
0 g
n
n
Pn(Z’q’P)
and
6
p
OQ(d.,F)
1
or
p
0.
[c,d].
and
[c,d].
But
Q(c,,F)
O;
Consequently,
Similar argument also shows
is non-negative and non-decreasing
on
SOME EXTENSIONS OF HARDY’S INEQUALITY
[c,d].
6
+
0
6
o8
c <_ x
<- d
We note in passing that Holder’s inequality yields
F(x)
where
s
when
p >_ i
example, for
p
and
This completes the proof of the lemma.
0.
REMARK 3.1.
for
Q(.,6,F) for 0
p <- i
[c,d] according as
Also, it can be proved in the same manner that
is negative and non-increasing or non-decreasing on
0
6
169
=fc
x
p
i
or
and
6
0
f(t)g(t)
s
g(t)
-s
(p + i)(6 + l)/p.
0, 6 < 0,6
p
p <- i, 6
0
and
0
Q(x,e,F) -> 0
0. For
0,6
we have
I/p
dg(t) <- 0(x)
x
g(t)
-6-1
dg(t)] l-(i/p)
The result follows after raising both sides to the power
and simplifying.
The method by which Theorem i.i is obtained in [I] will be used to prove our main
Suppose
result.
or
i
p
negative property of
the monotonicity of
0, 6
p
P dg (x,
I,
6-1[g(b) 60(b)
<_
#
0,
and
b -<. d.
a
c
Using first the non-
Q(.,6,F), next an application of integration by parts, and finally
Q(.,6,F) (cf. Lemma), we obtain
g(a)
6
/’
16 I-/_
+
B(a)
-</ab g(x )8-1
0(x) dg (x,
g(x) 6p-1 [g(x) f(x) ]Pdg(x)
-1 161p-l[g(b)6PF(b)p g(a)6PF(a)P]
+ j6J
g(x) 6p-I [g (x) f (x) ]P dg(x).
-lfb
a
Hence
b
l,l- /a g(x)6P-l[g(x)f(x)]
+
This proves the assertion of the theorem when
when
p <- I
0
is similar;
g(a)6PF(a)P]
6-1[g(b)6PF(b)P
g(x) 6P-iF(x)Pdg(x)
p >- i
hence it is omitted.
p
0.
or p
dg(x).
The proof of the theorem
We also omit the proof of the ex-
actness of constant and the condit*ons for equality since the proof is similar to the
.
one given in [2].
Thus the proof of the theorem is complete.
APPLICATIONS. We examine here some of the consequences of Theorem 2.1. The
conditions for equality and the exactness of constant as stated in Theorem 2.1 will be
The next two results yield the case
tacitly assumed in all our results.
r
I of
Theorem I.I.
COROLLARY 4.1.
f(t)
L((c,x)),dg)
or r
i,
where
p ->- i
Let
or
g(c)
i
F(x)
or
p
0
.
L((x,d),dg)
f(t)
and
g(d)
and
For
i.
r
for every
x
<-
a
b
: d,
according as
let
r
I
Suppose
fx
f(t) dg(t)
(r
I)
=
]
f(t) dg(t)
(r
1).
c
c
(a,b)
(4.1)
C.O. IMORU
170
Then we have
g(x) -i (log g(x)) -r/p F(x) ]p dg(x)
b
a
g(b))-l-rF(b) p
[p/ (l-r) ][ (log
/b g(x)
p
+ [p/ii_rl
with the inequality reversed if
PROOF.
-I
a
0
g(a))l-rF(a)PJ
(log
[g(x)(log
g(x))l (r/p)f(x)]p
(4.2)
dg(x);
i.
p
The assertion of the Corollary follows on taking
log u, i -< u -<
(u)
=,
(l-r)/p, r
and
i
in inequality
(2.4).
0 and r
REMARK. For g(x)
p we obtain as special cases of the Corollary,
x, r
Chan’s results [3, Theorem i] and [3, Theorem 3] respectively.
COROLLARY 4.3. Let p ->
b _< d, let
0 and r # i. For c -< a
or p
f(t)
L((c,x),dg) or f(t)
L((x,d),dg)
0 and g(d)
i. Suppose F is
g(c)
fb
0
when
-l[
log g(x)
Then
I(r-2)/PF(x)] p
dg(x)
_<
[p/ (l-r) ][ log g(b) r-I F(b) p
+
[p/ii_rl]p
/b
g(x)
a
-i
Take
i
(4.1).
[g(x) Ilog g(x)
or
r-I
flog g(a)
i, the inequality is reversed.
o(u)
flog ul -I 0 u i and
p
PROOF.
g (x)
according as r
as in
1
r
where
F(a) p]
x e
(a,b),
(4.3)
ll+((r-2/P)f(x)]Pdg(x);
(l-r)/p
6
r
i
in inequality
(2.4) and the conclusion of the Corollary follows at once.
REMARK 4.4. Corollary 4.3 is a generalization of Chan’s results, [3, Theorems 2 and 4].
i in the Corollary, we obtain
2 and r
2-p, p
Indeed, if we take g(x)
x, r
Theorems 2 and 4 in
[I], respectively.
Finally, we note that many interesting inequalities may be obtained by specializing
0. For example, f(x)
f(x), g(x) and the real number 6
f(x)(g’(x)) -I in inequalities (2.2), (2.3), (4.2) and (4.3). Of
interest are inequalities (2.2) and (2.3) which for p > i or p < 0 become
any or all of the functions
may be replaced by
particular
a
Go(X)
F(x) p dx
_<
[p/(l-r)][o2(b)F(b)P o2(a
+ [p/il-rl3
here
o(X) g(x)-rg’(x), l(X)
02(x)
G
x
[43.
e
8(x)
e
d, we obtain an
where
8
a
o (x) f (x) Pdx
1
g(x) p-r (g’(x)) I-p
o(x) l-I/po l(x)I/p
Inequality (4.4) is reversed when
g(x)
fb
0 < p
F(a)P]
and
g(x)l-r
i.
For
r
p, a
c
is non-negative and non-decreasing with
and
8(x)
as
extension of certain inequalities considered by Kufner and Triebel
SOME EXTENSIONS OF HARDY’S INEQUALITY
171
RE FE REN CE S
I.
IMORI T, C.O., On some integral inequalities related to.Hardy’s, Can. Math. Bull. 20
(3) (1977), 307-312.
2.
[% D.T., On integral inequalities related to Hardy’s, Can. Math. Bull. 14 (2)(1971),
295
230.
3.
CHAN, LING-YAU, Some extensions of Hardy’s inequality, Can. Math. Bull. 22 (2) (1979),
165-169.
KUFNFR, A. and TRIEBEL, H., Ceneralizations of Hardy’s inequality. Confer. Sere. Mat.
Univ. Bari No. 156 (1978), 21pp. (1979).
4.