Weighted Ostrowski Inequality N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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ON THE WEIGHTED OSTROWSKI INEQUALITY
N.S. BARNETT AND S.S. DRAGOMIR
School of Computer Science and Mathematics Victoria University, PO Box 14428
Melbourne City, VIC 8001, Australia.
EMail:{neil.barnett,sever.dragomir}@vu.edu.au
Received: 14 May, 2007
Accepted: 30 September, 2007 Communicated by: B.G. Pachpatte 2000 AMS Sub. Class.: 26D15, 26D10.
Key words: Ostrowski inequality, Integral inequalities, Absolutely continuous functions.
Abstract: On utilising an identity from [5], some weighted Ostrowski type inequalities are established.
Weighted Ostrowski Inequality N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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Contents
1 Introduction 3
2 Ostrowski Type Inequalities 6
3 Some Examples 17
Weighted Ostrowski Inequality N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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1. Introduction
In [5], the authors obtained the following generalisation of the weighted Montgomery identity:
(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,
where f : [a, b] → R is an absolutely continuous function, ϕ : [0,1] → R is a differentiable function with ϕ(0) = 0, ϕ(1) 6= 0 and w : [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,
where the weighted Peano kernelPw is
(1.4) Pw(x, t) :=
( Rt
aw(s)ds, a≤t≤x,
−Rb
t w(s)ds, x < t≤b.
Weighted Ostrowski Inequality N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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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
Weighted Ostrowski Inequality N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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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,
whereH(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 Os- trowski 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,
whenf is absolutely continuous, of bounded variation or Lipschitzian on the interval [a, b]. Some particular cases of interest are also given.
Weighted Ostrowski Inequality N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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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,
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vol. 8, iss. 4, art. 96, 2007
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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 (2.4)
f(x)− 1 ϕ(1)
Z b a
w(t)ϕ0 Z t
a
w(s)ds
f(t)dt
Weighted Ostrowski Inequality N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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=
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 1. 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 2.2. Letϕ and wbe as in Theorem 2.1. Ifw is continuous on[a, b] and f : [a, b]→Ris 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)
Weighted Ostrowski Inequality N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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≤ 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 α
p(t)dv(t)
≤ sup
t∈[α,β]
|p(t)|
β
_
α
(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),
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vol. 8, iss. 4, art. 96, 2007
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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)
Weighted Ostrowski Inequality N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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− 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.
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vol. 8, iss. 4, art. 96, 2007
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The following particular case is of interest for applications.
Corollary 2.3. Assume thatf, ϕ, ware as in Theorem2.2. In addition, ifϕis mono- tonic 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 Theorem2.2on observing that, ifϕis 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
.
Weighted Ostrowski Inequality N.S. Barnett and S.S. Dragomir
vol. 8, iss. 4, art. 96, 2007
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Corollary 2.4. With the assumptions of Theorem2.2and 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. 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 forL−Lipschitzian functions, can be stated as well.
Theorem 2.5. Letϕ and wbe as in Theorem 2.1. Ifw is continuous on[a, b] and f : [a, b] → R is an L1−Lipschitzian function on [a, x] and L2−Lipschitzian on
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vol. 8, iss. 4, art. 96, 2007
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[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, ifp : [α, β] → R isL−Lipschitzian andv is Riemann inte- grable, 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
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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 2.6. With the assumptions of Theorem2.5and 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 3. 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
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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
ϕ(1)max{L1, L2} Z b
a
|t−x|w(t)dt.
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3. Some Examples
The inequality (2.12) is a source of numerous particular inequalities that can be ob- tained by specifying the functionϕ : [0,1]→Rwhich 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 thatfis of bounded variation on[a, b], w(s)≥0for any s∈[a, b]and the involved integrals exist.
Another simple example can be given by choosing ϕ(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),
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vol. 8, iss. 4, art. 96, 2007
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for anyx∈[a, b]provided thatfis of bounded variation on[a, b], w(s)≥0for any s∈[a, b]and the involved integrals exist.
Finally, by choosing the functionϕ(t) = exp(t)−1,we obtain, from the inequal- ity (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 any x ∈ [a, b], provided f is of bounded variation on [a, b] and the involved integrals exist.
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References
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