volume 7, issue 2, article 78, 2006.
Received 01 September, 2005;
accepted 27 February, 2006.
Communicated by:S.S. Dragomir
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Journal of Inequalities in Pure and Applied Mathematics
A NOTE ON INTEGRAL INEQUALITIES INVOLVING THE PRODUCT OF TWO FUNCTIONS
B.G. PACHPATTE
57 Shri Niketan Colony Near Abhinay Talkies
Aurangabad 431 001 (Maharashtra) India.
EMail:bgpachpatte@gmail.com
c
2000Victoria University ISSN (electronic): 1443-5756 260-05
A Note on Integral Inequalities Involving the Product of Two
Functions B.G. Pachpatte
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Abstract
In this note, we establish new integral inequalities involving two functions and their derivatives. The discrete analogues of the main results are also given.
2000 Mathematics Subject Classification:26D10, 26D15, 26D99, 41A55.
Key words: Integral inequalities, Product of two functions, Discrete analogues, Ap- proximation formulae, Identities.
Contents
1 Introduction. . . 3
2 Statement of Results. . . 4
3 Proof of Theorem 2.1 . . . 6
4 Proof of Theorem 2.2 . . . 9 References
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1. Introduction
Inequalities have proved to be one of the most powerful and far-reaching tools for the development of many branches of mathematics. The monographs [1] – [3] contain an extensive number of surveys of inequalities up to the year of their publications. In the last few decades, much significant development in the clas- sical and new inequalities, particularly in analysis has been witnessed. The aim of the present note is to establish new integral inequalities, providing approxi- mation formulae which can be used to estimate the deviation of the product of two functions. The discrete versions of the main results are also given.
A Note on Integral Inequalities Involving the Product of Two
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2. Statement of Results
Our main results are given in the following theorem.
Theorem 2.1. Letf, g ∈C1([a, b],R),[a, b]⊂R, a < b. Then (2.1)
f(x)g(x)−1
2[g(x)F +f(x)G]
≤ 1 4
|g(x)|
Z b
a
|f0(t)|dt+|f(x)|
Z b
a
|g0(t)|dt
,
and
(2.2) |f(x)g(x)−[g(x)F +f(x)G] +F G|
≤ 1 4
Z b
a
|f0(t)|dt
Z b
a
|g0(t)|dt
,
for allx∈[a, b], where
(2.3) F = f(a) +f(b)
2 , G= g(a) +g(b)
2 .
The constant 14 in (2.1) and (2.2) is sharp.
Remark 1. If we take g(x) = 1and henceg0(x) = 0 in (2.1), then by simple calculation we get the inequality
(2.4) |f(x)−F| ≤ 1
2 Z b
a
|f0(t)|dt,
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which is established in [5, p.28]. We believe that the inequality established in (2.2) is new to the literature.
The discrete versions of the inequalities in Theorem2.1are embodied in the following theorem.
Theorem 2.2. Let{ui},{vi}fori= 0,1,2, . . . , n,n ∈Nbe sequences of real numbers. Then
(2.5)
uivi− 1
2[viU +uiV]
≤ 1 4
"
|vi|
n−1
X
j=0
|∆uj|+|ui|
n−1
X
j=0
|∆vj|
# ,
and
(2.6) |uivi−[viU +uiV] +U V| ≤ 1 4
n−1
X
j=0
|∆uj|
! n−1 X
j=0
|∆vj|
! ,
fori= 0,1,2, . . . , n, where
(2.7) U = u0+un
2 , V = v0+vn 2 ,
and ∆is the forward difference operator. The constant 14 in (2.5) and (2.6) is sharp.
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3. Proof of Theorem 2.1
From the hypotheses of Theorem2.1we have the following identities (see [5], [6, p. 267]):
(3.1) f(x)−F = 1
2 Z x
a
f0(t)dt− Z b
x
f0(t)dt
,
(3.2) g(x)−G= 1
2 Z x
a
g0(t)dt− Z b
x
g0(t)dt
.
Multiplying both sides of (3.1) and (3.2) byg(x)andf(x)respectively, adding the resulting identities and rewriting we have
(3.3) f(x)g(x)−1
2[g(x)F +f(x)G]
= 1 4
g(x)
Z x
a
f0(t)dt− Z b
x
f0(t)dt
+f(x) Z x
a
g0(t)dt− Z b
x
g0(t)dt
. From (3.3) and using the properties of modulus we have
f(x)g(x)− 1
2[g(x)F +f(x)G]
≤ 1 4
|g(x)|
Z b
a
|f0(t)|dt+|f(x)|
Z b
a
|g0(t)|dt
.
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This is the required inequality in (2.1).
Multiplying the left sides and right sides of (3.1) and (3.2) we get (3.4) f(x)g(x)−[g(x)F +f(x)G] +F G
= 1 4
Z x
a
f0(t)dt− Z b
x
f0(t)dt Z x
a
g0(t)dt− Z b
x
g0(t)dt
.
From (3.4) and using the properties of modulus we have
|f(x)g(x)−[g(x)F +f(x)G] +F G| ≤ 1 4
Z b
a
|f0(t)|dt Z b
a
|g0(t)|dt
. This proves the inequality in (2.2).
To prove the sharpness of the constant 14 in (2.1) and (2.2), assume that the inequalities (2.1) and (2.2) hold with constantsc > 0 andk > 0respectively.
That is, (3.5)
f(x)g(x)−1
2[|g(x)|F +|f(x)|G]
≤c
|g(x)|
Z b
a
|f0(t)|dt+|f(x)|
Z b
a
|g0(t)|dt
,
and
(3.6) |f(x)g(x)−[|g(x)|F +|f(x)|G] +F G|
≤k Z b
a
|f0(t)|dt
Z b
a
|g0(t)|dt
,
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forx ∈[a, b]. In (3.5) and (3.6), choosef(x) =g(x) = xand hencef0(x) = g0(x) = 1,F =G= a+b2 .Then by simple computation, we get
(3.7)
x−1
2(a+b)
≤2c(b−a), and
(3.8)
x(x−(a+b)) +
a+b 2
2
≤k(b−a)2.
By taking x = b, from (3.7) we observe that c ≥ 14 and from (3.8) it is easy to observe thatk ≥ 14, which proves the sharpness of the constants in (2.1) and (2.2). The proof is complete.
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4. Proof of Theorem 2.2
From the hypotheses of Theorem2.2we have the following identities (see [5], [6, p. 352]):
(4.1) ui−U = 1
2
"i−1 X
j=0
∆uj −
n−1
X
j=i
∆uj
#
and
(4.2) vi−V = 1
2
"i−1 X
j=0
∆vj−
n−1
X
j=i
∆vj
# .
Multiplying both sides of (4.1) and (4.2) by vi and ui (i = 0,1,2, . . . , n)re- spectively, adding the resulting identities and rewriting we get
(4.3) uivi− 1
2[viU +uiV]
= 1 4
"
vi
"i−1 X
j=0
∆uj −
n−1
X
j=i
∆uj
# +ui
"i−1 X
j=0
∆vj−
n−1
X
j=i
∆vj
##
.
Multiplying the left sides and right sides of (4.1) and (4.2) we have (4.4) uivi−[viU+uiV] +U V
= 1 4
"i−1 X
j=0
∆uj−
n−1
X
j=i
∆uj
# "i−1 X
j=0
∆vj−
n−1
X
j=i
∆vj
# .
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From (4.3) and (4.4) and following the proof of Theorem2.1, we get the desired inequalities in (2.5) and (2.6).
Assume that the inequalities (2.5) and (2.6) hold with constants α > 0 and β > 0 respectively. Taking {ui} = {vi} = {i} for i = 0,1,2, . . . , n and U = V = n2 and following similar arguments to those used in the last part of the proof of Theorem2.1, it is easy to observe thatα ≥ 14 andβ ≥ 14 and hence the constants in (2.5) and (2.6) are sharp. The proof is complete.
Remark 2. Dividing both sides of (3.3) and (3.4) by (b − a), then integrat- ing both sides with respect to x over [a, b] and closely looking at the proof of Theorem2.1we get
(4.5)
1 b−a
Z b
a
f(x)g(x)dx
− 1
2 (b−a)
F Z b
a
g(x)dx+G Z b
a
f(x)dx
≤ 1
4 (b−a)
Z b
a
|g(x)|dx
Z b
a
|f0(x)|dx
+ Z b
a
|f(x)|dx
Z b
a
|g0(x)|dx
, and
(4.6)
1 b−a
Z b
a
f(x)g(x)dx
− 1
(b−a)
F Z b
a
g(x)dx+G Z b
a
f(x)dx−F G
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≤ 1 4
Z b
a
|f0(x)|dx
Z b
a
|g0(x)|dx
. We note that the inequalities (4.5) and (4.6) are similar to those of the well known inequalities due to Grüss and ˇCebyšev, see [3,4].
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References
[1] E.F. BECKENBACH AND R. BELLMAN, Inequalities, Springer-Verlag, Berlin-New York, 1970.
[2] G.H. HARDY, J.E. LITTLEWOOD AND G. PÓLYA, Inequalities, Cam- bridge University Press, 1934.
[3] D.S. MITRINOVI ´C, Analytic Inequalities, Springer-Verlag, Berlin-New York, 1970.
[4] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C AND A.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.
[5] B.G. PACHPATTE, A note on Ostrowski type inequalities, Demonstratio Math., 35 (2002), 27–30.
[6] B.G. PACHPATTE, Mathematical Inequalities, North-Holland Mathemati- cal Library, Vol. 67, Elsevier Science, 2005.