Journal of Inequalities in Pure and
Applied Mathematics
A SHARP INEQUALITY OF OSTROWSKI-GRÜSS TYPE
ZHENG LIU
Institute of Applied Mathematics
Faculty of Science
Anshan University of Science and Technology
Anshan 114044, Liaoning
People’s Republic of China.
volume 7, issue 5, article 192,
2006.
Received 9 February, 2006;
accepted 23 May, 2006.
Communicated by: J. Sándor
EMail: lewzheng@163.net
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
The main purpose of this paper is to use a Grüss type inequality for RiemannStieltjes integrals to obtain a sharp integral inequality of Ostrowski-Grüss type
for functions whose first derivative are functions of Lipschitizian type and precisely characterize the functions for which equality holds.
A Sharp Inequality of
Ostrowski-Grüss Type
2000 Mathematics Subject Classification: 26D15.
Key words: Ostrowski-Grüss type inequality, Grüss type inequality for RiemannStieltjes integrals, Lipschitzian type function, Sharp bound.
Zheng Liu
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
7
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1.
Introduction
In 1935, G. Grüss (see [4, p. 296]) proved the following integral inequality
which gives an approximation for the integral of a product of two functions in
terms of the product of integrals of the two functions.
Theorem A. Let h, g : [a, b] → R be two integrable functions such that φ ≤
h(x) ≤ Φ and γ ≤ g(x) ≤ Γ for all x ∈ [a, b], where φ, Φ, γ, Γ are real
numbers. Then we have
(1.1)
|T (h, g)|
Z b
Z b
Z b
1
1
1
:= h(x)g(x)dx −
h(x)dx ·
g(x)dx
b−a a
b−a a
b−a a
1
≤ (Φ − φ)(Γ − γ),
4
and the inequality is sharp, in the sense that the constant
by a smaller one.
It is clear that the constant
1
4
cannot be replaced
1
4
is achieved for
a+b
.
h(x) = g(x) = sgn x −
2
From then on, (1.1) has been known in the literature as the Grüss inequality.
In 1998, S.S. Dragomir and I. Fedotov [2] established the following Grüss
type inequality for Riemann-Stieltjes integrals:
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
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Theorem B. Let h, u : [a, b] → R be so that u is L-Lipschitzian on [a, b], i.e.,
|u(x) − u(y)| ≤ L |x − y|
for all x, y ∈ [a, b], h is Riemann integrable on [a, b] and there exists the real
numbers m, M so that m ≤ h(x) ≤ M for all x ∈ [a, b]. Then we have the
inequality
Z b
Z b
1
u(b)
−
u(a)
(1.2)
h(x)du(x) −
h(t)dt ≤ L(M − m)(b − a)
b−a
2
a
a
and the constant
1
2
A Sharp Inequality of
Ostrowski-Grüss Type
is sharp.
In a recent paper [3], the inequality (1.2) has been improved and refined as
follows:
Theorem C. Let h, u : [a, b] → R be so that u is L-Lipschitzian on [a, b], h
is Riemann integrable on [a, b] and there exist the real numbers m, M so that
m ≤ h(x) ≤ M for all x ∈ [a, b]. Then we have
Z b
Z
u(b) − u(a) b
(1.3)
h(x)du(x)
−
h(t)dt
b
−
a
a
a
Z b
Z b
1
h(x) −
dx
≤L
h(t)dt
b−a a
a
p
≤ L(b − a) T (h, h)
1
≤ L(M − m)(b − a).
2
Zheng Liu
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All the inequalities in (1.3) are sharp and the constant 12 is the best possible one.
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Theorem D. Let h, u : [a, b] → R be so that u is (l, L)-Lipschitzian on [a, b],
i.e., it satisfies the condition
l(x2 − x1 ) ≤ u(x2 ) − u(x1 ) ≤ L(x2 − x1 )
for a ≤ x1 ≤ x2 ≤ b with l < L, h is Riemann integral on [a, b] and there exist
the real numbers m, M so that m ≤ h(x) ≤ M for all x ∈ [a, b]. Then we have
the inequality
Z b
Z
u(b) − u(a) b
(1.4)
h(t)dt
h(x)du(x)
−
b
−
a
a
a
Z b
Z b
1
L−l
h(x) −
h(t)dtdx
≤
2
b−a a
a
p
L−l
≤
(b − a) T (h, h)
2
1
≤ (L − l)(M − m)(b − a).
4
All the inequalities in (1.4) are sharp and the constant 14 is the best possible one.
In [1], L.J. Dedić et al. have proved the following Ostrowski type inequality
as
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
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Theorem E. If u is L-Lipschitzian on [a, b], then for every x ∈ [a, b] we have
Z b
b
−
a
u(a)
+
u(b)
a
+
b
0
(1.5) u(t)dt −
u(x) +
+ x−
u (x) 2
2
2
a
!
3
a + b 1 (b − a)3
≤L
x−
+
.
3
2 48
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In this paper, we will use Theorem C and Theorem D to obtain some sharp
integral inequalities of Ostrowski-Grüss type for functions whose first derivative are functions of Lipschitzian type. Thus a further generalization of the
Ostrowski type inequality and a perturbed version of the inequality (1.5) is obtained.
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
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2.
The Results
Theorem 2.1. Let u : [a, b] → R be a differentiable function so that u0 is (l, L)Lipschitzian on [a, b], i.e., satisfies the condition
(2.1)
l(x2 − x1 ) ≤ u0 (x2 ) − u0 (x1 ) ≤ L(x2 − x1 )
for a ≤ x1 ≤ x2 ≤ b with l < L.Then for all x ∈ [a, b] we have
Z b
b
−
a
u(a)
+
u(b)
a
+
b
0
(u(x) +
+ x−
u (x)
(2.2) u(t)dt −
2
2
2
a
a + b 2 (b − a)2 L − l
u0 (b) − u0 (a)
x−
−
−
≤ 4 I(a, b, x),
4
2
12
Zheng Liu
where
(2.3)
A Sharp Inequality of
Ostrowski-Grüss Type
I(a, b, x)=


























a+3b
−x
4
[3(x−a)+(b−x)]
i 3 a ≤ x ≤ ξ,
(b−a)2 2
4 1
a+b 2
+ 3 2 x− 2
,
+ 48
a+b
[(x−a)+3(b−x)]
−x x− 3a+b
2
4
h
i3
ξ < x < ζ,
(b−a)2 2
1
a+b 2
+4 2 x− 2
+ 48
,
h
1
6
16
3
























a+b
−x
2
1
6
1
6
1
6
h
1
2
2
x− a+b
2
x− a+b
2
+
(b−a)2
48
a+3b
−x
4
i 32
,
ζ ≤ x ≤ θ,
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[3(x−a)+(b−x)]
h
2 (b−a)2 i 32
+4 12 x− a+b
+ 48
,
2
3a+b
x− a+b
x−
[(x−a)+3(b−x)]
2
h 4
2 (b−a)2 i 32
+ 34 12 x− a+b
+ 48
,
2
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η ≤x≤b
J. Ineq. Pure and Appl. Math. 7(5) Art. 192, 2006
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with
√
a+b
3(b − a)
ξ=
−
,
2√
6
6(b − a)
ζ =a+
,
6
and a < ξ <
3a+b
4
<ζ<
a+b
2
√
a+b
3(b − a)
η=
+
,
2√
6
6(b − a)
θ =b−
6
<θ<
a+3b
4
< η < b.
Proof. Integrating by parts produces the identity
Z
A Sharp Inequality of
Ostrowski-Grüss Type
b
Zheng Liu
K(x, t)du0 (t)
(2.4)
a
Z
=
a
b
1
u(a) + u(b)
a+b
0
u(t)dt − (b − a) u(x) +
+ x−
u (x) ,
2
2
2
where
(
(2.5)
K(x, t) =
1
(t
2
− a) t −
a+b
1
(t
2
− b) t −
a+b
2
2
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(2.6)
1
b−a
Z
b
K(x, t)dt =
a
1
4
"
x−
a+b
2
2
2
−
(b − a)
12
#
.
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J. Ineq. Pure and Appl. Math. 7(5) Art. 192, 2006
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Applying the Grüss type inequality (1.4) gives
Z b
Z
u0 (b) − u0 (a) b
0
K(x,
t)du
(t)
−
K(x,
t)dt
b
−
a
a
a
Z b
Z b
L−l
1
K(x, t) −
≤
K(x, s)dsdt.
2
b−a a
a
Then for any fixed x ∈ [a, b] we can derive from (2.4), (2.5) and (2.6) that
(2.7)
Z b
b
−
a
u(a)
+
u(b)
a
+
b
0
u(t)dt −
u(x) +
+ x−
u (x)
2
2
2
a
"
#
2
a+b
(b − a)2 L − l
u0 (b) − u0 (a)
−
x−
−
I(a, b, x),
≤
4
2
12
4
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
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where
x
a+b
1
a + b 2 (b − a)2 −
x−
−
I(a, b, x) =
dt
(t − a) t− 2
2
2
12
a
Z b
2 a
+
b
1
a
+
b
(b
−
a)
2
(t − b) t−
dt.
+
−
x−
−
2
2
2
12
x
Z
The last two integrals can be calculated as follows:
For brevity, we put
"
#
2
1
a+b
(b − a)2
a+b
p1 (t) := (t − a) t −
−
x−
−
,
2
2
2
12
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t ∈ [a, x],
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a+b
p2 (t) := (t − b) t −
2
1
−
2
"
a+b
x−
2
2
#
(b − a)2
−
,
12
t ∈ [x, b].
Then we have
"
2 #
1 (b − a)2
a+b
p1 (a) = p2 (b) =
− x−
;
2
12
2
a+b
(b − a)2
x−
+
,
2
24
1
b−a
a+b
(b − a)2
p2 (x) =
x−
x−
+
.
2
2
2
24
1
p1 (x) =
2
Set
√
b−a
x+
2
√
a+b
3(b − a)
a+b
3(b − a)
−
,
η=
+
,
2
6
2
6
√
√
6(b − a)
6(b − a)
ζ =a+
,
θ =b−
.
6
6
It is easy to find that p1 (a) = p2 (b) ≤ 0 for x ∈ [a, ξ] ∪ [η, b], p1 (a) = p2 (b) > 0
for x ∈ (ξ, η) and p1 (x) ≤ 0 for x ∈ [a, ζ], p1 (x) > 0 for x ∈ (ζ, b], p2 (x) > 0
for x ∈ [a, θ), p2 (x) ≤ 0 for x ∈ [θ, b]. Notice that
ξ=
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
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3a + b
a+b
a + 3b
a<ξ<
<ζ<
<θ<
< η < b,
4
2
4
we see that there are five possible cases to be determined.
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(i) In case x ∈ [ζ, θ]. p1 (a) = p2 (b) > 0, p1 (x) ≥ 0, p2 (x) ≥ 0 and it is
easy to find by elementary calculus that the function
p1 (t) is strictly decreasing
3a+b
and
strictly
increasing
in
,
x
,
also,
as the function
p2 (t) is
in a, 3a+b
4
4
a+3b
a+3b
strictly decreasing
in x, 4 and strictly increasing in 4 , b . Moreover,
3a+b
a+3b
p1 4 = p2 4 < 0. So, p1 (t) has two zeros in (a, x) at the points
" # 12
2
3a + b
(b − a)2
1
a+b
t1 =
+
−
x−
4
2
2
48
and
" # 12
2
(b − a)2
3a + b
1
a+b
.
t2 =
+
x−
+
4
2
2
48
Also p2 (t) has two zeros in (x, b) at the points
" # 12
2
1
a+b
(b − a)2
a + 3b
+
t3 =
−
x−
4
2
2
48
and
" # 12
2
a + 3b
1
a+b
(b − a)2
t4 =
+
x−
+
.
4
2
2
48
Thus we have
(2.8)
I(a, b, x)
#
2
Z t1 "
a+b
1
a+b
(b − a)2
=
(t − a) t−
−
x−
+
dt
2
2
2
24
a
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
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" 2
#
(b − a)2
1
a+b
a+b
+
x−
−
− (t − a) t−
dt
2
2
24
2
t1
Z x
a+b
1
a + b 2 (b − a)2
+
(t − a) t−
−
x−
+
dt
2
2
2
24
t2
#
2
Z t3 "
(b − a)2
a+b
1
a+b
dt
+
(t − b) t−
−
x−
+
2
2
2
24
x
2
#
Z t4 " 1
a+b
(b − a)2
a+b
+
x−
−
− (t − b) t−
dt
2
2
24
2
t3
#
2
Z b"
a+b
1
a+b
(b − a)2
+
(t − b) t−
−
x−
+
dt
2
2
2
24
t4
" # 32
2
a+b
(b − a)2
16 1
+
.
=
x−
3 2
2
48
Z
t2
(ii) In case x ∈ [a, ξ], p1 (a) = p2 (b) ≤ 0, p1 (x) < 0,p2 (x) > 0 and p1 (t) is
)
strictly decreasing in (a, x) as well as p2 (t) is strictly decreasing in (x, a+3b
4
a+3b
and strictly increasing in ( a+3b
,
b)
with
t
∈
(x,
)
such
that
p
(t
)
=
0.
3
2 3
4
4
Thus we have
(2.9)
I(a, b, x)
2
#
Z x" a+b
(b − a)2
a+b
1
=
x−
−
− (t − a) t−
dt
2
2
24
2
a
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
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"
#
2
a+b
(b − a)2
1
a+b
+
(t − b) t−
−
x−
+
dt
2
2
2
24
x
2
#
Z b" 1
a+b
(b − a)2
a+b
+
x−
−
− (t − b) t−
dt
2
2
24
2
t3
1 a+b
a + 3b
=
−x
− x [3(x − a) + (b − x)]
6
2
4
" # 32
2
(b − a)2
4 1
a+b
.
+
x−
+
3 2
2
48
Z
t3
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
(iii) In case (ξ, ζ), p1 (a) = p2 (b) > 0, p1 (x) < 0,p2 (x) > 0 and p1 (t) has a
unique zero t1 ∈ (a, x), p2 (t) has two zeros t3 , t4 ∈ (x, b). Thus we have
(2.10)
I(a, b, x)
#
2
Z t1 "
a+b
1
a+b
(b − a)2
=
(t − a) t−
−
x−
+
dt
2
2
2
24
a
Z x 1
a + b 2 (b − a)2
a+b
+
x−
−
− (t − a) t−
dt
2
2
24
2
t1
#
2
Z t3 "
a+b
1
a+b
(b − a)2
+
(t − b) t−
−
x−
+
dt
2
2
2
24
x
2
#
Z t4 " 1
a+b
(b − a)2
a+b
+
x−
−
− (t − b) t−
dt
2
2
24
2
t3
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Z b
a+b
1
a + b 2 (b − a)2
+
(t − b) t−
−
x−
+
dt
2
2
2
24
t4
1 a+b
3a + b
−x
x−
[(x − a) + 3(b − x)]
=
6
2
4
#3
" 2
2 2
a+b
(b − a)
1
x−
+
.
+4
2
2
48
(iv) In case x ∈ (θ, η), p1 (a) = p2 (b) > 0, p1 (x) > 0, p2 (x) < 0 and p1 (t) has
two zeros t1 , t2 ∈ (a, x), p2 (t) has a unique zero t4 ∈ (x, b). Thus we have
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
(2.11)
I(a, b, x)
Z t1 1
a + b 2 (b − a)2
a+b
=
(t − a) t−
−
x−
+
dt
2
2
2
24
a
2
#
Z t2 " a+b
(b − a)2
a+b
1
+
x−
−
− (t − a) t−
dt
2
2
24
2
t1
#
2
Z x"
a+b
1
a+b
(b − a)2
+
(t − a) t−
−
x−
+
dt
2
2
2
24
t2
2
#
Z t4 " 2
1
a+b
(b − a)
a+b
+
x−
−
− (t − b) t−
dt
2
2
24
2
x
#
2
Z b"
2
a+b
1
a+b
(b − a)
+
(t − b) t−
−
x−
+
dt
2
2
2
24
t4
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a+b
a + 3b
x−
− x [3(x − a) + (b − x)]
2
4
# 32
" 2
1
a+b
(b − a)2
+4
x−
+
.
2
2
48
1
=
6
(v) In case x ∈ [η, b], p1 (a) = p2 (b) ≤ 0, p1 (x) > 0, p2 (x) < 0 and p1 (t) has a
unique zero t2 ∈ (a, x), p2 (t) ≤ 0 for t ∈ [x, b]. Thus we have
(2.12)
I(a, b, x)
2
#
Z t2 " 1
(b − a)2
a+b
a+b
=
x−
−
− (t − a) t−
dt
2
2
24
2
a
#
2
Z x"
a+b
(b − a)2
1
a+b
+
(t − a) t−
−
x−
+
dt
2
2
2
24
t2
2
#
Z b" 1
a+b
(b − a)2
a+b
+
x−
−
− (t − b) t −
dt
2
2
24
2
x
1
a+b
3a + b
=
x−
x−
[(x − a) + 3(b − x)]
6
2
4
" # 32
2
a+b
(b − a)2
4 1
x−
+
+
.
3 2
2
48
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
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Consequently, the inequality (2.2) with (2.3) follows from (2.7), (2.8), (2.9),
(2.10), (2.11) and (2.12).
The proof is completed.
J. Ineq. Pure and Appl. Math. 7(5) Art. 192, 2006
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Remark 1. It is not difficult to prove that the inequality (2.2) with (2.3) is sharp
in the sense that we can construct the function u to attain the equality in (2.2)
with (2.3). Indeed, we may choose u such that
 1
(t − a)2 ,
a ≤ t < x,

2


 L
(t − x)2 + 2l [2(x − a)t − (x2 − a2 )], x ≤ t < t3 ,
2
u(t) =

 2l [(t − t3 )2 + 2(x − a)t − (x2 − a2 )]


+ L2 [2(t3 − x)t − (t23 − x2 )],
t3 ≤ t ≤ b,
which follows
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu

a ≤ t < x,

 l(t − a),
0
L(t − x) + (x − a)l,
x ≤ t < t3,
u (t) =


l(t − t3 + x − a) + (t3 − x)L, t3 ≤ t ≤ b,
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for any x ∈ [a, ξ], and
 L
(t − a)2 ,


2



l

(t − t1 )2 + L2 [2(t1 − a)t − (t21 − a2 )],

2




L

[(t − x)2 + 2(t1 − a)t − (t21 − a2 )]


 2
+ 2l [2(x − t1 )t − (x2 − t21 )],
u(t) =

l

[(t − t23 ) + 2(x − t1 )t − (x2 − t21 )]

2



+ L2 [2(t3 − x + t1 − a)t − (t23 − x2 + t21 − a2 )],





L

[(t − t4 )2 + 2(t3 − x + t1 − a)t − (t23 − x2 + t21 − a2 )]


 2
+ 2l [2(t4 − t3 + x − t1 )t − (t24 − t23 + x2 − t21 )],
a ≤ t < t1 ,
t1 ≤ t < x,
x ≤ t < t3 ,
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t4 ≤ t ≤ b,
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which follows

L(t − a),





 l(t − t1 ) + (t1 − a)L,



0
L(t − x + t1 − a) + (x − t1 )l,
u (t) =





l(t − t3 + x − t1 ) + (t3 − x + t1 − a)L,




L(t − t4 + t3 − x + t1 − a) + (t4 − t3 + x − t1 )l,
a ≤ t < t1 ,
t1 ≤ t < x,
x ≤ t < t3 ,
t3 ≤ t < t4 ,
t4 ≤ t ≤ b,
for any x ∈ (ξ, ζ), and

L
(t − a)2 ,


2




l

(t − t1 )2 + L2 [2(t1 − a)t − (t21 − a2 )],

2





L

[(t − t2 )2 + 2(t1 − a)t − (t21 − a2 )]

2




+ 2l [2(t2 − t1 )t − (t22 − t21 )],
u(t) =

l

[(t − t23 ) + 2(t2 − t1 )t − (t22 − t21 )]


2




+ L2 [2(t3 − t2 + t1 − a)t − (t23 − t22 + t21 − a2 )],





L


[(t − t4 )2 + 2(t3 − t2 + t1 − a)t − (t23 − t22 + t21 − a2 )]

2



+ l [2(t − t + t − t )t− (t2 − t2 + t2 − t2 ) ],
2
4
3
2
1
4
3
2
1
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
a ≤ t < t1 ,
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t1 ≤ t < t2 ,
t2 ≤ t < t3 ,
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which follows

L(t − a),






 l(t − t1 ) + (t1 − a)L,



0
L(t − t2 + t1 − a) + (t2 − t1 )l,
u (t) =





l(t − t3 + t2 − t1 ) + (t3 − t2 + t1 − a)L,





L(t − t4 + t3 − t2 + t1 − a) + (t4 − t3 + t2 − t1 )l,
a ≤ t < t1 ,
t1 ≤ t < t2 ,
t2 ≤ t < t3 ,
t3 ≤ t < t4 ,
t4 ≤ t ≤ b,
for any x ∈ (ξ, ζ), and

L
(t − a)2 ,


2




l

(t − t1 )2 + L2 [2(t1 − a)t − (t21 − a2 )],

2





L

[(t − t2 )2 + 2(t1 − a)t − (t21 − a2 )]

2




+ 2l [2(t2 − t1 )t − (t22 − t21 )],
u(t) =

l

[(t − x2 ) + 2(t2 − t1 )t − (t22 − t21 )]


2




+ L2 [2(x − t2 + t1 − a)t − (x2 − t22 + t21 − a2 )],





L


[(t − t4 )2 + 2(x − t2 + t1 − a)t − (x2 − t22 + t21 − a2 )]

2



+ l [2(t − x + t − t )t − (t2 − x2 + t2 − t2 )],
2
4
2
1
4
2
1
A Sharp Inequality of
Ostrowski-Grüss Type
Zheng Liu
a ≤ t < t1 ,
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which follows

L(t − a),







l(t − t1 ) + (t1 − a)L,



L(t − t2 + t1 − a) + (t2 − t1 )l,
u0 (t) =





l(t − x + t2 − t1 ) + (x − t2 + t1 − a)L,





L(t − t4 + x − t2 + t1 − a) + (t4 − x + t2 − t1 )l,
a ≤ t < t1 ,
t1 ≤ t < t2 ,
t2 ≤ t < x,
x ≤ t < t4 ,
t4 ≤ t ≤ b,
for any x ∈ (θ, η), and
 l
(t − a)2 ,
a ≤ t < t2 ,

2




 L (t − t2 )2 + l [2(t2 − a)t − (t2 − a2 )], t2 ≤ t < x,
2
2
2
u(t) =

l

[(t − x)2 + 2(t2 − a)t − (t22 − a2 )]


2


+ L2 [2(x − t2 )t − (x2 − t22 )],
x ≤ t ≤ b,
which follows


l(t − a),
a ≤ t < t2 ,



L(t − t2 ) + (t2 − a)l,
t2 ≤ t < x,
u0 (t) =



 l(t − x + t − a) + (x − t )L, x ≤ t ≤ b.
2
2
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Zheng Liu
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for any x ∈ [η, b].
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It is clear that all the above u0 (t) satisfy the condition (2.1) on [a, b].
Remark 2. For x =
a+b
,
2
we have
Z b
2
a
+
b
b
−
a
u(a)
+
u(b)
(b
−
a)
0
0
u(t)dt −
u
+
+
[u (b) − u a)]
2
2
2
48
a
(L − l)(b − a)3
√
≤
.
144 3
Corollary 2.2. If u0 is L-Lipschitzian on [a, b], then for all x ∈ [a, b] we have
(2.13)
Z b
u(a) + u(b)
a+b
b−a
0
u(x) +
+ x−
u (x)
u(t)dt −
2
2
2
a
u0 (b) − u0 (a)
a + b 2 (b − a)2 L
−
x−
−
≤ 2 I(a, b, x),
4
2
12
where I(a, b, x) is as defined in (2.3).
Proof. It is immediate by taking l = −L in the theorem.
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Zheng Liu
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References
[1] L.J. DEDIĆ, M. MATIĆ, J. PEČARIĆ AND A. VUKELIĆ, On generalizations of Ostrowski inequality via Euler harmonic identities, J. of Inequal. &
Appl., 7(6) (2002), 787–805.
[2] S.S. DRAGOMIR AND I. FEDOTOV, An inequality of Grüss type for
Riemann-Stieltjes integral and applications for special means, Tamkang J.
of Math., 29(4) (1998), 286–292.
[3] Z. LIU, Refinement of an inequality of Grüss type for Riemann-Stieltjes
integral, Soochow J. of Math., 30(4) (2004), 483–489.
[4] D.S. MITRINOVIĆ, J. PEČARIĆ AND A.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
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