General Mathematics Vol. 15, No. 1 (2007), 93-100
An inequality of Ostrowski type via a mean
value theorem 1
Emil C. Popa
Dedicated to Professor Ph.D. Alexandru Lupaş on his 65th anniversary
Abstract
The main of this paper is to establish an Ostrowski type inequality
by using a mean value theorem.
2000 Mathematics Subject Classification: 26D15, 26D20
Keywords: Ostrowski type inequalities. mean value theorem, diferentiable and integrable functions.
1
Introduction
The following result is known in the literature as Ostrowski’s inequality [3]:

µ
¶2 
¯
¯
a
+
b
¯ 
¯
Zb
x−

¯ 1
¯
1
2

¯
¯
f (t)dt¯ ≤  +
(1) ¯f (x) −
 (b − a) · kf ′ k∞
2
b−a
(b − a)

¯ 4
¯
a
1
Received 2 October, 2006
Accepted for publication (in revised form) 15 December, 2006
93
94
Emil C. Popa
where f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) and
whose derivative f ′
:
(a, b)
→
R is bounded on (a, b), i. e.
kf ′ k = sup |f ′ (t)| < ∞.
t∈(a,b)
Many researchers have given considerable attention to the inequality (1)
and several generalizations, extensions and related results have appeared in
the literature.
The main aim of this paper is to establish an Ostrowski type inequality
by using a mean value theorem (see [2] and [5]).
2
Statement and results
In [5] the author has proved the following mean value theorem:
Theorem 2.1 Let f : [a, b] → R be continuous on [a, b] and differentiable
on (a, b) and ξ ∈ (a, b) with
f ′ (ξ) =
f (b) − f (a)
.
b−a
For every M (α, β), α ∈
/ [a, b],
min{y1 , y2 } ≤ β ≤ max{y1 , y2 },
there exists a point c in (a, b) such that
f ′ (c) =
β − f (c)
,
α−c
where y1 = f (a) + (α − a)f ′ (ξ), y2 = f (ξ) + (α − ξ)f ′ (ξ).
In the proofs of our results we make use this mean value theorem.
The following result holds.
An inequality of Ostrowski type via a mean value theorem
95
Theorem 2.2 Let f : [a, b] → R be continuous on [a, b] and differentiable
on (a, b). Then for any x ∈ [a, b] we have the inequality
¯
¯
¯·
¯
¸
Zb
¯ a+b
¯
α−x
¯
(2)
f (t)dt¯¯ ≤
¯ 2 − α f (x) + b − a
¯
¯
a



a+b 2
1 x − 2  
′

≤
 4 +  b − a   (b − a) · kf + lf k∞ ,
where α ∈
/ [a, b] and l(t) = α − t, t ∈ [a, b].
Proof. Applying the theorem 2.1 for β = y1 , we obtain that for any
x, t ∈ [a, b], there is a c between x and t such that
f (x) + (α − x)
f (t) − f (x)
= y1 = β = f (c) + (α − c)f ′ (c).
t−x
We have
|f (c) + (α − c)f ′ (c)| ≤ sup |f (t) + (α − t)f ′ (t)|
t∈[a,b]
= kf + lf ′ k∞
where l(t) = α − t, t ∈ [a, b].
Hence
¯
¯
¯
¯
¯f (x) + (α − x) f (t) − f (x) ¯ ≤ kf + lf ′ k ,
∞
¯
t−x ¯
|(t − x)f (x) + (α − x)[f (t) − f (x)]| ≤ |t − x| · kf + lf ′ k∞ ,
|(t − α)f (x) + (α − x)f (t)| ≤ |t − x| · kf + lf ′ k∞ .
Integrating over t ∈ [a, b], we get
¯
¯
¯
¯
Zb
Zb
Zb
¯
¯
′
(3) ¯¯f (x) (t − α)dt + (α − x) f (t)dt¯¯ ≤ kf + lf k∞ · |t − x|dt.
¯
¯
a
a
a
96
Emil C. Popa
We observe that
Zb
a
Next
 2
b − a2


− (b − a)x,
t≥x

2
|t − x|dt =

 b 2 − a2

−
+ (b − a)x, t < x.
2
(x − b)2 − (x − a)2
(x − a)2 + (x − b)2
b 2 − a2
− (b − a)x =
≤
,
2
2
2
−
(x − a)2 − (x − b)2
(x − a)2 + (x − b)2
b 2 − a2
+ (b − a)x =
≤
.
2
2
2
Hence
Zb
a
¶2
µ
1
a+b
(x − a)2 + (x − b)2
2
= (b − a) + x −
.
|t − x|dt ≤
2
4
2
Of (3) we deduce
¯
¯
¯· 2
¯
¸
Zb
2
¯ b −a
¯
¯
− α(b − a) f (x) + (α − x) f (t)dt¯¯ ≤
¯
2
¯
¯
a
"
µ
¶2 #
1
a
+
b
≤
(b − a)2 + x −
kf + lf ′ k∞ ,
4
2
and finally
¯
¯
¯
¯·
¸
Zb
¯
¯ a+b
α
−
x
¯
f (t)dt¯¯ ≤
¯ 2 − α f (x) + b − a
¯
¯
a



a+b 2
1 x − 2  
′

≤
 4 +  b − a   (b − a) · kf + lf k∞ .
An inequality of Ostrowski type via a mean value theorem
97
Remark 2.1 If 0 ∈
/ [a, b], then for α = 0 we obtain the Dragomir’s inequality (see [1]):
¯
¯
Z b
¯ a + b f (x)
¯
1
¯
¯≤
f
(t)dt
·
−
¯ 2
¯
x
b−a a



a+b 2
x−

b−a
2 
1 + 
 · kf + lf ′ k∞
≤



|x|  4
b−a
where l(t) = −t, t ∈ [a, b].
The following interesting particular case holds.
Corollary 2.1 With the assumptions in Theorem 2.2, we have
¯
¯
¯ µ
¯
¶
Zb
¯
¯
a+b
1
¯
¯≤
f
(4)
−
f
(t)dt
¯
¯
2
b
−
a
¯
¯
a
≤
for any α 6∈ [a, b].
3
b−a
¯ kf + lf ′ k∞
¯
¯
¯a + b
− α¯¯
4 ¯¯
2
AN APPLICATION
Using an idea of [1] we consider the division of the interval [a, b] given by
∆ : a = x0 < x1 < . . . < xn−1 < xn = b.
Let ξi be a sequence of intermediate points, ξi ∈ [xi , xi+1 ], i = 0, n − 1. We
define the quadrature
(5)
¸
·
n−1
X
f (ξi ) x2i+1 − x2i
S∆ (f, ξi ) =
− α (xi+1 − xi ) =
ξ −α
2
i=0 i
98
Emil C. Popa
µ
¶
n−1
X
f (ξi ) xi+1 + xi
− α hi
=
ξ
−
α
2
i
i=0
where hi = xi+1 − xi .
We denote
Z
(6)
b
f (t)dt = S∆ (f, ξi ) + R∆ (f, ξi )
a
where S∆ (f, ξi ) is as defined in (5). In the following result we obtain an
estimate for the remainder R∆ (f, ξi ).
Theorem 3.1 Assume that f : [a, b] → R is continuous on [a, b] and differentiable on (a, b). Then we have
n−1
|R∆ (f, ξi )| ≤
(7)
X
1
h2i
kf + lf ′ k∞ ·
2h
i=0
where h = min{|a − α|, |b − α|}.
Proof. Applying the theorem 2.2 on the interval [xi , xi+1 ] for the intermediate points ξi , to obtain
¯
¯x
¯ Zi+1
¶ ¯
µ
¯
¯
f (ξi ) xi+1 + xi
¯≤
¯
−
α
h
f
(t)dt
−
i
¯
¯
ξ
−
α
2
i
¯
¯
xi
≤
h2i


1 
 +
|ξi − α| 4
ξi −

xi+1 + xi 2

′
2
  · kf + lf k∞ .
hi
¯
¯
¯
xi+1 + xi ¯¯ xi+1 − xi
¯
But ¯ξi −
and
¯≤
2
2
|ξi − α| ≥ min{|a − α|, |b − α|}.
An inequality of Ostrowski type via a mean value theorem
Hence
99
¯x
¯
¯ Zi+1
¶ ¯
µ
¯
¯
f (ξi ) xi+1 + xi
¯
¯≤
f
(t)dt
−
−
α
h
i
¯
¯
ξ
−
α
2
i
¯
¯
xi
≤
1
kf + lf ′ k∞ · h2i .
2h
Summer over i from 0 to n − 1 we deduce
¯
¯ b
¯Z
µ
¶ ¯
n−1
X
¯
¯
f (ξi ) xi+1 + xi
¯ f (t)dt −
− α hi ¯¯ ≤
¯
ξ −α
2
¯
¯
i=0 i
a
n−1
X
1
≤
h2i
· kf + lf ′ k∞ ·
2h
i=0
so
n−1
|R∆ (f, ξi )| ≤
Remark 3.1 For hi =
X
1
h2i .
kf + lf ′ k∞ ·
2h
i=0
b−a
, i = 0, n − 1 we get
n
|R∆ (f, ξi )| ≤
(b − a)2
· kf + lf ′ k∞ .
2hn
References
[1] S. S. Dragomir, An inequality of Ostrowski type via Pompeiu’s mean
value theorem, J. Inequal. Pure and Appl. Math, 6(3), art. 83, 2003.
[2] I. Muntean, Extensions of some mean value theorems, ”Babes-Bolyai
University, Faculty of Mathematics, Research Seminars on Mathematical Analysis, Preprint Nr. 7, 1991, 7-24.
100
Emil C. Popa
[3] A. Ostrowski, Über die Absolutabweichung einer differentierbaren
Funktionen von ihrem Integralmittelwert, Comment. Math. Helv.,
10(1938), 226-227.
[4] B. G. Pachpatte , New Ostrowski type inequalities via mean value theorems, J. Inequal. Pure and Appl. Math, 7(4), art. 137, 2006.
[5] E. C. Popa, Asupra unei teoreme de medie, Gaz. Mat., anul XCIV, nr
4,1989, 113-114.
Department of Mathematics
”Lucian Blaga” University of Sibiu
Faculty of Sciences
Str. Dr. I. Raţiu, no. 5-7,
550012 - Sibiu, ROMANIA
E-mail address: emil.popa@ulbsibiu.ro