ON THE WEIGHTED OSTROWSKI INEQUALITY
N.S. BARNETT AND S.S. DRAGOMIR SCHOOL OFCOMPUTERSCIENCE ANDMATHEMATICS
VICTORIAUNIVERSITY, PO BOX14428 MELBOURNECITY, VIC 8001, AUSTRALIA.
neil.barnett@vu.edu.au sever.dragomir@vu.edu.au
Received 14 May, 2007; accepted 30 September, 2007 Communicated by B.G. Pachpatte
ABSTRACT. On utilising an identity from [5], some weighted Ostrowski type inequalities are established.
Key words and phrases: Ostrowski inequality, Integral inequalities, Absolutely continuous functions.
2000 Mathematics Subject Classification. 26D15, 26D10.
1. INTRODUCTION
In [5], the authors obtained the following generalisation of the weighted Montgomery iden- tity:
(1.1) f(x) = 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt
+ 1
ϕ(1) Z b
a
Pw,ϕ(x, t)f0(t)dt,
wheref : [a, b] → R is an absolutely continuous function,ϕ : [0,1] → Ris a differentiable function with ϕ(0) = 0, ϕ(1) 6= 0 andw : [a, b] → [0,∞) is a probability density function such that the weighed Peano kernel
(1.2) Pw,ϕ(x, t) :=
ϕ
Rt
aw(s)ds
, a≤t≤x, ϕ
Rt
aw(s)ds
−ϕ(1), x < t≤b, is integrable for anyx∈[a, b].
Ifϕ(t) = t,then (1.1) reduces to the weighted Montgomery identity obtained by Peˇcari´c in [21]:
(1.3) f(x) =
Z b a
w(t)f(t)dt+ Z b
a
Pw(x, t)f0(t)dt,
160-07
where the weighted Peano kernelPwis
(1.4) Pw(x, t) :=
( Rt
aw(s)ds, a≤t≤x,
−Rb
t w(s)ds, x < t≤b.
Finally, the uniform distribution is used to provide the Montgomery identity [17, p. 565]:
(1.5) f(x) = 1
b−a Z b
a
f(t)dt+ Z b
a
P(x, t)f0(t)dt,
with
P(x, t) :=
( t−a
b−a if a≤t ≤x,
t−b
b−a if x < t≤b,
that has been extensively used to obtain Ostrowski type results, see for instance the research papers [3] – [6], [7] – [16], [19] – [20], [22] and the book [15].
In the same paper [5], on introducing the generalised ˇCebyšev functional, (1.6) Tϕ(w, f, g) :=
Z b a
w(x)ϕ0 Z x
a
w(t)dt
f(x)g(x)dx
− 1 ϕ(1)
Z b a
w(x)ϕ0 Z x
a
w(t)dt
f(x)dx
× Z b
a
w(x)ϕ0 Z x
a
w(t)dt
g(x)dx
,
the authors obtained the representation:
(1.7) Tϕ(w, f, g) = 1 ϕ2(1)
Z b a
w(x)ϕ0 Z x
a
w(t)dt
× Z b
a
Pw,ϕ(x, t)f0(t)dt Z b
a
Pw,ϕ(x, t)g0(t)dt
dx
and used it to obtain an upper bound for the absolute value of the ˇCebyšev functional in the case wheref0, g0, ϕ0 ∈L∞[a, b].This bound can be stated as:
(1.8) |Tϕ(w, f, g)| ≤ 1
ϕ2(1)kf0k∞kg0k∞kϕ0k∞ Z b
a
w(x)H2(x)dx,
where H(x) := Rb
a |Pw,ϕ(x, t)|dt. The inequality (1.8) provides a generalisation of a result obtained by Pachpatte in [18].
The main aim of this paper is to obtain some weighted inequalities of the Ostrowski type by providing various upper bounds for the deviation off(x), x∈[a, b],from the integral mean
1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt,
when f is absolutely continuous, of bounded variation or Lipschitzian on the interval [a, b].
2. OSTROWSKI TYPE INEQUALITIES
In order to state some Ostrowski type inequalities, we consider the Lebesgue norms kgk[α,β],∞:=ess sup
t∈[α,β]
|g(t)|
and
kgk[α,β],`:=
Z β α
|g(t)|`dt 1`
, ` ∈[1,∞);
provided that the integral and the supremum are finite.
Theorem 2.1. Let ϕ : [0,1] → R be continuous on [0,1], differentiable on (0,1) with the property thatϕ(0) = 0andϕ(1)6= 0.Ifw: [a, b]→R+is a probability density function, then for anyf : [a, b]→Ran absolutely continuous function, we have
(2.1)
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt
≤ Z x
a
ϕ
Z t a
w(s)ds
|f0(t)|dt+ Z b
x
ϕ
Z t a
w(s)ds
−ϕ(1)
|f0(t)|dt
for anyx∈[a, b]. If
H1(x) :=
Z x a
ϕ
Z t a
w(s)ds
|f0(t)|dt
and
H2(x) :=
Z b x
ϕ
Z t a
w(s)ds
−ϕ(1)
|f0(t)|dt,
then
(2.2) H1(x)≤
ϕ R·
aw(s)ds
[a,x],∞kf0k[a,x],1; ϕ R·
aw(s)ds
[a,x],pkf0k[a,x],q ifp >1,1p +1q = 1 andf0 ∈Lq[a, x] ; ϕ R·
aw(s)ds
[a,x],1kf0k[a,x],∞ iff0 ∈L∞[a, x] ; and
(2.3) H2(x)≤
ϕ R·
aw(s)ds
−ϕ(1)
[x,b],∞kf0k[x,b],1; ϕ R·
aw(s)ds
−ϕ(1)
[x,b],rkf0k[x,b],t ifr >1,1r +1t = 1 andf0 ∈Lt[x, b] ; ϕ R·
aw(s)ds
−ϕ(1)
[x,b],1kf0k[x,b],∞ iff0 ∈L∞[x, b]
for anyx∈[a, b].
Proof. Follows from the identity (1.1) on observing that
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt (2.4)
=
Z x a
ϕ Z t
a
w(s)ds
f0(t)dt+ Z b
x
ϕ
Z t a
w(s)ds
−ϕ(1)
f0(t)dt
≤
Z x a
ϕ Z t
a
w(s)ds
f0(t)dt
+
Z b x
ϕ
Z t a
w(s)ds
−ϕ(1)
f0(t)dt
≤ Z x
a
ϕ
Z t a
w(s)ds
|f0(t)|dt+ Z b
x
ϕ
Z t a
w(s)ds
−ϕ(1)
|f0(t)|dt
for anyx∈[a, b],and the first part of (2.1) is proved.
The bounds from (2.2) and (2.3) follow by the Hölder inequality.
Remark 2.2. It is obvious that, the above theorem provides 9 possible upper bounds for the absolute value of the deviation off(x)from the integral mean,
1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt
although they are not stated explicitly.
The above result, which provides an Ostrowski type inequality for the absolutely continuous functionf,can be extended to the larger class of functions of bounded variation as follows:
Theorem 2.3. Letϕandwbe as in Theorem 2.1. Ifwis continuous on[a, b]andf : [a, b]→R is a function of bounded variation on[a, b],then:
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt (2.5)
≤ 1 ϕ(1)
"
sup
t∈[a,x]
ϕ
Z t a
w(s)ds
·
x
_
a
(f)
+ sup
t∈[x,b]
ϕ
Z t a
w(s)ds
−ϕ(1)
·
b
_
x
(f)
#
≤ 1
ϕ(1) ·max (
sup
t∈[a,x]
ϕ
Z t a
w(s)ds
,
sup
t∈[x,b]
ϕ
Z t a
w(s)ds
−ϕ(1)
)
·
b
_
a
(f),
whereWb
a(f)denotes the total variation off on[a, b].
Proof. We recall that, ifp: [α, β]→Ris continuous on[α, β]andv : [α, β]→Ris of bounded variation, then the Riemann-Stieltjes integralRβ
α p(t)dv(t)exists and (2.6)
Z β
p(t)dv(t)
≤ sup |p(t)|
β
_(v).
Since the functions ϕ R·
aw(s)ds
and ϕ R·
aw(s)ds
−ϕ(1) are continuous on [a, x] and [x, b], respectively, the Riemann-Stieltjes integrals
Z x a
ϕ Z t
a
w(s)ds
df(t) and Z b
x
ϕ
Z t a
w(s)ds
−ϕ(1)
df(t)
exist and (2.7)
Z x a
ϕ Z t
a
w(s)ds
df(t)
≤ sup
t∈[a,x]
ϕ
Z t a
w(s)ds
·
x
_
a
(f),
while (2.8)
Z b x
ϕ
Z t a
w(s)ds
−ϕ(1)
df(t)
≤ sup
t∈[x,b]
ϕ
Z t a
w(s)ds
−ϕ(1)
·
b
_
x
(f).
Integrating by parts in the Riemann-Stieltjes integral, we have Z x
a
ϕ Z t
a
w(s)ds
df(t) (2.9)
= f(t)ϕ Z t
a
w(s)ds
x
a
− Z x
a
f(t)d
ϕ Z t
a
w(s)ds
=f(x)ϕ Z x
a
w(s)ds
− Z x
a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt
and
Z b x
ϕ
Z t a
w(s)ds
−ϕ(1)
df(t) (2.10)
=
ϕ Z t
a
w(s)ds
−ϕ(1)
f(t)
b
x
− Z b
x
f(t)d
ϕ Z t
a
w(s)ds
−ϕ(1)
=−
ϕ Z t
a
w(s)ds
−ϕ(1)
f(x)− Z b
x
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt.
If we add (2.9) and (2.10) we deduce the following identity of the Montgomery type for the Riemann-Stieltjes integral which is of interest in itself:
(2.11) f(x) = 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt
+ 1
ϕ(1) Z x
a
ϕ Z t
a
w(s)ds
df(t)
+ 1
ϕ(1) Z b
x
ϕ
Z t a
w(s)ds
−ϕ(1)
df(t),
for anyx∈[a, b].
Now, by (2.11) and (2.7) – (2.8) we obtain the estimate:
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt
≤ 1 ϕ(1)
Z x a
ϕ Z t
a
w(s)ds
df(t)
+ 1
ϕ(1)
Z b x
ϕ
Z t a
w(s)ds
−ϕ(1)
df(t)
≤ 1
ϕ(1) · sup
t∈[a,x]
ϕ
Z t a
w(s)ds
·
x
_
a
(f)
+ 1
ϕ(1) · sup
t∈[x,b]
ϕ
Z t a
w(s)ds
−ϕ(1)
·
b
_
x
(f), x∈[a, b]
which provides the first inequality in (2.5).
The last part of (2.5) is obvious.
The following particular case is of interest for applications.
Corollary 2.4. Assume that f, ϕ, w are as in Theorem 2.3. In addition, if ϕ is monotonic nondecreasing on[0,1],then
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt (2.12)
≤ ϕ Rx
a w(s)ds ϕ(1) ·
x
_
a
(f) +
"
1− ϕ Rx
a w(s)ds ϕ(1)
#
·
b
_
x
(f)
≤
"
1 2 +
ϕ Rx
a w(s)ds ϕ(1) − 1
2
# b _
a
(f).
Proof. Follows by Theorem 2.3 on observing that, ifϕ is monotonic nondecreasing on[a, b], then:
sup
t∈[a,x]
ϕ
Z t a
w(s)ds
= sup
t∈[a,x]
ϕ Z t
a
w(s)ds
=ϕ Z x
a
w(s)ds
and
sup
t∈[x,b]
ϕ
Z t a
w(s)ds
−ϕ(1)
= sup
t∈[x,b]
ϕ(1)−ϕ Z t
a
w(s)ds
=ϕ(1)− inf
t∈[x,b]ϕ Z t
a
w(s)ds
=ϕ(1)−ϕ Z x
a
w(s)ds
.
Corollary 2.5. With the assumptions of Theorem 2.3 and ifK := supt∈(0,1)|ϕ0(t)|< ∞,then we have the bounds:
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt (2.13)
≤ 1 ϕ(1) ·K
"
sup
t∈[a,x]
Z t a
w(s)ds
·
x
_
a
(f) + sup
t∈[x,b]
Z b t
w(s)ds
·
b
_
x
(f)
#
≤ K ϕ(1)max
( sup
t∈[a,x]
Z t a
w(s)ds
, sup
t∈[x,b]
Z b t
w(s)ds
) b _
a
(f).
Remark 2.6. Ifw(s)≥0fors∈[a, b],then from (2.13) we get
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt (2.14)
≤ K ϕ(1)
"
Z x a
w(s)ds·
x
_
a
(f) + Z b
x
w(s)ds·
b
_
x
(f)
#
≤ K ϕ(1)
1 2
Z b a
w(s)ds+1 2
Z x a
w(s)ds− Z b
x
w(s)ds
·
b
_
a
(f).
The following result, that provides an Ostrowski type inequality for L−Lipschitzian func- tions, can be stated as well.
Theorem 2.7. Letϕandwbe as in Theorem 2.1. Ifwis continuous on[a, b]andf : [a, b]→R is anL1−Lipschitzian function on[a, x]andL2−Lipschitzian on[x, b],withx∈[a, b],then
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt (2.15)
≤ 1 ϕ(1)
L1·
Z x a
ϕ
Z t a
w(s)ds
dt
+L2· Z b
x
ϕ
Z t a
w(s)ds
−ϕ(1)
dt
≤max{L1, L2} · 1 ϕ(1)
Z x a
ϕ
Z t a
w(s)ds
dt
+ Z b
x
ϕ
Z t a
w(s)ds
−ϕ(1)
dt
.
Proof. We recall that, if p : [α, β] → R isL−Lipschitzian and v is Riemann integrable, then the Riemann-Stieltjes integralRβ
α f(t)du(t)exists and (2.16)
Z β α
p(t)dv(t)
≤L Z β
α
|p(t)|dt.
Now, if we apply the above property to the integrals Z x
a
ϕ Z t
a
w(s)ds
df(t) and Z b
α
ϕ
Z t a
w(t)ds
−ϕ(1)
df(t),
then we can state that (2.17)
Z x a
ϕ Z t
a
w(s)ds
df(t)
≤L1· Z x
a
ϕ
Z t a
w(s)ds
dt
and (2.18)
Z b x
ϕ
Z t a
w(s)ds
−ϕ(1)
df(t)
≤L2· Z b
x
ϕ
Z t a
w(s)ds
−ϕ(1)
dt.
By making use of the identity (2.11), by (2.17) and (2.18) we deduce the first part of (2.15).
The last part is obvious.
The following particular case is of interest as well.
Corollary 2.8. With the assumptions of Theorem 2.7 and ifK := supt∈(0,1)|ϕ0(t)|<∞,then
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt (2.19)
≤ K ϕ(1)
L1·
Z x a
Z t a
w(s)ds
dt+L2· Z b
x
Z b t
w(s)ds
dt
≤ K
ϕ(1)max{L1, L2} Z x
a
Z t a
w(s)ds
dt+ Z b
x
Z b t
w(s)ds
dt
.
Remark 2.9. Ifw : [a, b] → Ris a nonnegative weight, thenRt
aw(s)ds,Rb
t w(s)ds ≥ 0for eacht ∈[a, b]and since
Z x a
Z t a
w(s)ds
dt = Z t
a
w(s)ds
·t
x
a
− Z x
a
w(t)dt
=x Z x
a
w(t)dt− Z x
a
tw(t)dt= Z x
a
(x−t)w(t)dt
and
Z b x
Z b t
w(s)ds
dt = t· Z b
t
w(s)ds
b
x
+ Z b
x
w(t)dt
=−x Z b
x
w(t)dt+ Z b
x
tw(t)dt= Z b
x
(t−x)w(t)dt,
then we get, from (2.19), the following result:
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt (2.20)
≤ K ϕ(1)
L1·
Z x a
(x−t)w(t)dt+L2 · Z b
x
(t−x)w(t)dt
≤ K
max{L , L } Z b
|t−x|w(t)dt.
3. SOMEEXAMPLES
The inequality (2.12) is a source of numerous particular inequalities that can be obtained by specifying the function ϕ : [0,1] → R which is continuous, differentiable and monotonic nondecreasing withϕ(0) = 0.
For instance, if we chooseϕ(t) =tα, α > 0,then we get the inequality:
f(x)−α Z b
a
w(t) Z t
a
w(s)ds α−1
f(t)dt (3.1)
≤ Z x
a
w(s)ds α
·
x
_
a
(f) +
1− Z x
a
w(s)ds α
·
b
_
x
(f)
≤ 1
2 +
Z x a
w(s)ds α
− 1 2
b
_
a
(f),
for anyx∈ [a, b]provided thatf is of bounded variation on[a, b], w(s)≥ 0for anys ∈[a, b]
and the involved integrals exist.
Another simple example can be given by choosingϕ(t) = ln (t+ 1). In this situation, we obtain the inequality:
f(x)− 1 ln 2
Z b a
"
w(t) Rt
aw(s)ds+ 1
#
f(t)dt (3.2)
≤ ln Rx
a w(s)ds+ 1
ln 2 ·
x
_
a
(f) +
"
1− ln Rx
a w(s)ds+ 1 ln 2
#
·
b
_
x
(f)
≤
"
1 2 +
ln Rx
a w(s)ds+ 1
ln 2 − 1
2
#
·
b
_
a
(f),
for anyx∈ [a, b]provided thatf is of bounded variation on[a, b], w(s)≥ 0for anys ∈[a, b]
and the involved integrals exist.
Finally, by choosing the functionϕ(t) = exp(t)−1,we obtain, from the inequality (2.12), the following result as well:
f(x)− 1 e−1
Z b a
w(t) exp Z t
a
w(s)ds
f(t)dt
≤ exp Rx
a w(s)ds
−1
e−1 ·
x
_
a
(f) + e−exp Rx
a w(s)ds
e−1 ·
b
_
x
(f)
≤
"
1 2 +
exp Rx
a w(s)ds
−1
e−1 − 1
2
#
·
b
_
a
(f),
for anyx∈[a, b],providedf is of bounded variation on[a, b]and the involved integrals exist.
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