Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 343191, 12 pages
doi:10.1155/2012/343191
Research Article
Sharp Integral Inequalities Based on a
General Four-Point Quadrature Formula via a
Generalization of the Montgomery Identity
J. Pečarić1 and M. Ribičić Penava2
1
2
Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia
Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31 000 Osijek, Croatia
Correspondence should be addressed to M. Ribičić Penava, mihaela@mathos.hr
Received 28 March 2012; Accepted 10 July 2012
Academic Editor: Marianna Shubov
Copyright q 2012 J. Pečarić and M. Ribičić Penava. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We consider families of general four-point quadrature formulae using a generalization of the Montgomery identity via Taylor’s formula. The results are applied to obtain some sharp inequalities for
functions whose derivatives belong to Lp spaces. Generalizations of Simpson’s 3/8 formula and
the Lobatto four-point formula with related inequalities are considered as special cases.
1. Introduction
The most elementary quadrature rules in four nodes are Simpson’s 3/8 rule based on the
following four point formula
b
b−a
2a b
a 2b
b − a5 4
f ξ,
ftdt fa 3f
3f
fb −
8
3
3
6480
a
1.1
where ξ ∈ a, b, and Lobatto rule based on the following four point formula
√ √ 2
5
5
1
f−1 5f −
5f
f1 −
f 6 η ,
ftdt 6
5
5
23625
−1
1
1.2
2
International Journal of Mathematics and Mathematical Sciences
where η ∈ −1, 1. Formula 1.1 is valid for any function f with a continuous fourth
derivative f 4 on a, b and formula 1.2 is valid for any function f with a continuous sixth
derivative f 6 on −1, 1.
Let f : a, b → R be differentiable on a, b and f : a, b → R integrable on a, b.
Then the Montgomery identity holds see 1
fx 1
b−a
b
ftdt b
a
P x, tf tdt,
1.3
a
where the Peano kernel is
P x, t ⎧
t−a
⎪
⎪
⎨ b − a , a ≤ t ≤ x,
⎪
⎪
⎩ t − b , x < t ≤ b.
b−a
1.4
In 2, Pečarić proved the following weighted Montgomery identity
fx b
wtftdt b
a
Pw x, tf tdt,
1.5
a
where w : a, b → 0, ∞ is some probability density function, that is, integrable function,
b
t
satisfying a wtdt 1, and Wt a wxdx for t ∈ a, b, Wt 0 for t < a and Wt 1
for t > b and Pw x, t is the weighted Peano kernel defined by
Pw x, t a ≤ t ≤ x,
x < t ≤ b.
Wt,
Wt − 1,
1.6
Now, let us suppose that I is an open interval in R, a, b ⊂ I, f : I → R is such that f n−1 is
absolutely continuous for some n ≥ 2, w : a, b → 0, ∞ is a probability density function.
Then the following generalization of the weighted Montgomery identity via Taylor’s formula
states given by Aglić Aljinović and Pečarić in 3
fx b
wtftdt −
a
1
n − 1!
n−2 i1
f
x b
i0
b
i 1!
Tw,n x, sf
n
wss − xi1 ds
a
1.7
sds,
a
where x ∈ a, b and
⎧ s
n−1
⎪
⎪
a ≤ s ≤ x,
⎨ wuu − s du,
a
Tw,n x, s b
⎪
⎪
⎩− wuu − sn−1 du, x < s ≤ b.
s
1.8
International Journal of Mathematics and Mathematical Sciences
3
If we take wt 1/b − a, t ∈ a, b, equality 1.7 reduces to
fx 1
b−a
b
ftdt −
a
1
n − 1!
b
n−2
b − xi2 − a − xi2
f i1 x
i 2!b − a
i0
1.9
Tn x, sf n sds,
a
where x ∈ a, b and
Tn x, s ⎧
a − sn
⎪
⎪
⎪
,
−
⎪
⎪
⎨ nb − a
a ≤ s ≤ x,
⎪
⎪
⎪
b − sn
⎪
⎪
,
⎩−
nb − a
x < s ≤ b.
1.10
For n 1, 1.9 reduces to the Montgomery identity 1.3.
In this paper, we generalize the results from 4. Namely, we use identities 1.7 and
1.9 to establish for each number x ∈ a, a b/2 a general four-point quadrature formula
of the type
b
1
− Ax fa fb
wtftdt 2
a
Ax fx fa b − x R f, w; x ,
1.11
where Rf, w; x is the remainder and A : a, a b/2 → R is a real function. The obtained
formula is used to prove a number of inequalities which give error estimates for the general
four-point formula for functions whose derivatives are from Lp -spaces. These inequalities are
generally sharp. As special cases of the general non-weighted four-point quadrature formula,
we obtain generalizations of the well-known Simpson’s 3/8 formula and Lobatto four-point
formula with related inequalities.
2. General Weighted Four-Point Formula
Let f : a, b → R be such that f n−1 exists on a, b for some n ≥ 2. We introduce the
following notation for each x ∈ a, a b/2:
Dx 1
− Ax fa fb Ax fx fa b − x ,
2
4
International Journal of Mathematics and Mathematical Sciences
n−2 i1
f
x b
wss − xi1 ds
tw,n x Ax
i 1! a
i0
n−2 i1
f
a b − x b
i1
wss − a − b x ds
i 1!
a
i0
n−2 i1
f
a b
1
− Ax
wss − ai1 ds
2
1!
i
a
i0
n−2 i1
f
b b
i1
wss − b ds ,
i 1! a
i0
1
− Ax Tw,n a, s Tw,n b, s − AxTw,n x, s Tw,n a b − x, s
Tw,n x, s −
2
s
⎧ 1
⎪
⎪
Ax
wuu − sn−1 du
−
⎪
⎪
2
⎪
a
⎪
⎪
⎪
b
⎪
⎪
1
⎪
⎪
⎪
− Ax
wuu − sn−1 du,
a ≤ s ≤ x,
⎪
⎪
2
⎪
s
⎪
⎪
⎪
b
⎨ s
1
n−1
n−1
−
wuu − s du − wuu − s du , x < s ≤ a b − x,
⎪
2 a
⎪
s
⎪
⎪
s
⎪
⎪
⎪
1
⎪
⎪
−
− Ax
wuu − sn−1 du
⎪
⎪
2
⎪
a
⎪
⎪
⎪
b
⎪
⎪
1
⎪
⎪
⎩ Ax
wuu − sn−1 du,
a b − x < s ≤ b.
2
s
2.1
In the next theorem we establish the general weighted four-point formula.
Theorem 2.1. Let I be an open interval in R, a, b ⊂ I, and let w : a, b → 0, ∞ be some
probability density function. Let f : I → R be such that f n−1 is absolutely continuous for some
n ≥ 2. Then for each x ∈ a, a b/2 the following identity holds
b
wtftdt Dx tw,n x a
1
n − 1!
b
Tw,n x, sf n sds.
2.2
a
Proof. We put x ≡ a, x ≡ x, x ≡ a b − x and x ≡ b in 1.7 to obtain four new formulae. After
multiplying these four formulae by 1/2 − Ax, Ax, Ax and 1/2 − Ax, respectively, and
adding, we get 2.2.
Remark 2.2. Identity 2.2 holds true in the case n 1. It can also be obtained by taking
x ≡ a, x ≡ x, x ≡ a b − x and x ≡ b in 1.5, multiplying these four formulae by 1/2 −
Ax, Ax, Ax and 1/2 − Ax, respectively, and adding. In this special case we have
b
a
wtftdt Dx b
a
Tw,1 x, sf sds,
2.3
International Journal of Mathematics and Mathematical Sciences
5
where
1
− Ax Tw,1 a, s Tw,1 b, s − AxTw,1 x, s Tw,1 a b − x, s
2
1
− Ax Pw a, s Pw b, s − AxPw x, s Pw a b − x, s
−
2
⎧
1
⎪
⎪
⎪ 2 − Ax − Ws, a ≤ s ≤ x,
⎪
⎪
⎪
⎪
⎨
1
− Ws,
x < s ≤ a b − x,
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎩ 1 Ax − Ws, a b − x < s ≤ b.
2
2.4
Tw,1 x, s −
Theorem 2.3. Suppose that all assumptions of Theorem 2.1 hold. Additionally, assume that p, q is
a pair of conjugate exponents, that is, 1 ≤ p, q ≤ ∞, 1/p 1/q 1, let f n ∈ Lp a, b for some
n ≥ 1. Then for each x ∈ a, a b/2 we have
b
1 Tw,n x, · f n .
wtftdt − Dx − tw,n x ≤
n − 1!
a
q
p
2.5
Inequality 2.5 is sharp for 1 < p ≤ ∞.
Proof. By applying the Hölder inequality we have
b
1
1 n
Tw,n x, sf sds ≤
Tw,n x, · f n .
n − 1!
n − 1! a
q
p
2.6
By using the above inequality from 2.2 we obtain estimate 2.5. Let us denote Unx s Tw,n x, s. For the proof of sharpness, we will find a function f such that
b
x
n
Un sf sds Unx q f n .
a
p
2.7
For 1 < p < ∞, take f to be such that
f n s sign Unx s · |Unx s|1/p−1 ,
2.8
f n s sign Unx s.
2.9
where for p ∞ we put
Remark 2.4. Inequality 2.5 for Ax 1/4 was proved by Aglić Aljinović et al. in 4.
6
International Journal of Mathematics and Mathematical Sciences
3. Non-Weighted Four-Point Formula and Applications
Here we define
n−2 b − xi2 − a − xi2
tn x Ax
f i1 x −1i1 f i1 a b − x
i 2!b − a
i0
n−2 b − ai1
1
f i1 a −1i1 f i1 b
− Ax
,
2
i 2!
i0
1
− Ax Tn a, s Tn b, s AxTn x, s Tn a b − x, s
Tn x, s − n
2
⎧
1
1
a − sn
b − sn
⎪
⎪
Ax
−
Ax
, a ≤ s ≤ x,
⎪
⎪
⎪
2
2
b − a
b − a
⎪
⎪
⎪
⎨ a − sn b − sn
,
x < s ≤ a b − x,
⎪
2b − a
⎪
⎪
⎪
⎪
⎪
1
1
a − sn
b − sn
⎪
⎪
⎩
− Ax
Ax
, a b − x < s ≤ b.
2
2
b − a
b − a
3.1
3.2
Theorem 3.1. Let I be an open interval in R, a, b ⊂ I, and let f : I → R be such that f n−1 is
absolutely continuous for some n ≥ 1. Then for each x ∈ a, a b/2 the following identity holds
1
b−a
b
1
ftdt Dx tn x n!
a
b
Tn x, sf n sds.
3.3
a
Proof. We take wt 1/b − a, t ∈ a, b in 2.2.
Theorem 3.2. Suppose that all assumptions of Theorem 3.1 hold. Additionally, assume that p, q is
a pair of conjugate exponents, that is, 1 ≤ p, q ≤ ∞, 1/p 1/q 1 and f n ∈ Lp a, b for some
n ≥ 1. Then for each x ∈ a, a b/2 we have
1
1 b
ftdt − Dx − tn x ≤ Tn x, · f n .
n!
b − a a
q
p
3.4
Inequality 3.4 is sharp for 1 < p ≤ ∞.
Proof. We take wt 1/b − a, t ∈ a, b in 2.5.
Now, we set
Ax b − a2
,
12x − ab − x
x∈
a,
ab
.
2
3.5
This special choice of the function A enables us to consider generalizations of the well-known
Simpson’s 3/8 formula 1.1 and Lobatto formula 1.2
International Journal of Mathematics and Mathematical Sciences
7
3.1. x 2a b/3
Suppose that all assumptions of Theorem 3.1 hold. Then the following generalization of
Simpson’s 3/8 formula reads
1
b−a
b
ftdt D
a
2a b
3
tn
2a b
3
1
n!
b
2a b
, s f n sds,
Tn
3
a
3.6
where
1
2a b
a 2b
fa 3f
3f
fb ,
8
3
3
2i2 −1i1 b − ai1
n−2 1
2a
b
2a
b
a
2b
tn
−1i1 f i1
f i1
3
8 i0
3
3
3i1 i 2!
D
2a b
3
n−2 b − ai1
1
f i1 a −1i1 f i1 b
,
8 i0
i 2!
n
2a b
2a b
a 2b
,s −
Tn a, s 3Tn
, s 3Tn
, s Tn b, s
Tn
3
8
3
3
⎧
n
n
2a b
7a − s b − s
⎪
⎪
a≤s≤
,
⎪
⎪
⎪
8b − a
3
⎪
⎪
⎪
⎨
2a b
a 2b
a − sn b − sn
,
<s≤
,
⎪
2b
−
a
3
3
⎪
⎪
⎪
⎪
⎪
a − sn 7b − sn a 2b
⎪
⎪
⎩
< s ≤ b.
8b − a
3
3.7
In the next corollaries we will use the beta function and the incomplete beta function
of Euler type defined by
B x, y 1
t
x−1
1 − t
0
y−1
dt,
Br x, y r
tx−1 1 − ty−1 dt,
x, y > 0.
3.8
0
Corollary 3.3. Suppose that all assumptions of Theorem 3.1 hold. Additionally, assume that p, q is
a pair of conjugate exponents and n ∈ N.
a If f n ∈ L∞ a, b, then
1 b
25
2a b ftdt − D
b − af ∞ ,
≤
b − a a
3
288
8
International Journal of Mathematics and Mathematical Sciences
1 b
2a b
2a
b
ftdt − D
− tn
b − a a
3
3
1
≤
n 1!
3n1 3 · 2n1 3−1n b − an
4 · 3n1
b−a
−
2
n −1n1 1
2
n f ,
∞
n ≥ 2.
3.9
b If f n ∈ L2 a, b, then
1 b
2a b
2a
b
− tn
ftdt − D
b − a a
3
3
1
≤
n!
32n 5 · 22n1 11 b − a2n−1 −1n b − a2n−1
32
32 · 32n 2n 1
1/2 n ×7Bn 1, n 1 9B2/3 n 1, n 1 − 9B1/3 n 1, n 1
f .
2
3.10
c If f n ∈ L1 a, b, then
1 b
2a b
2a
b
2a b 1
n − tn
ftdt − D
≤ Kn
f ,
n!
b − a a
1
3
3
3
3.11
where K1 2a b/3 5/24, K2 2a b/3 5/18 b − a, K3 2a b/3 7/54b − a2
and Kn 2a b/3 1/8b − an−1 , for n ≥ 4.
The first and the second inequality are sharp.
Proof. We apply 3.4 with x 2a b/3 and p ∞
2ab/3 b 7a − sn b − sn ds
Tn 2a b , s ds 3
8b − a
a
a
a2b/3 b
a − sn b − sn a − sn 7b − sn ds ds
2b − a
8b − a
2ab/3
a2b/3
3n1 − 2n1 7 · −1n b − an
2
8 · 3n1 n 1
International Journal of Mathematics and Mathematical Sciences
2n1 −1n1 b − an
1 −1n1 b − an
−
3n1 n 1
2n1 n 1
n1
3 3 · 2n1 3−1n b − an
b − a n −1n1 1
,
−
2
2n 1
4 · 3n1 n 1
9
3.12
for n ≥ 2 and
b T1 2a b , s ds 25 b − a.
3
288
a
3.13
To obtain the second inequality we take p 2
b 2ab/3 2
7a − sn b − sn 2
ds
Tn 2a b , s ds 3
8b − a
a
a
a2b/3 b
a − sn b − sn 2
a − sn 7b − sn 2
ds
ds
2b − a
8b − a
2ab/3
a2b/3
2n
3 5 · 22n1 11 b − a2n−1 −1n b − a2n−1
32
32 · 32n 2n 1
× 7Bn 1, n 1 9B2/3 n 1, n 1 − 9B1/3 n 1, n 1.
3.14
If p 1, we have
2a b
, s max
sup Tn
3
s∈a,b
7a − sn b − sn ,
sup
8b − a
s∈a,2ab/3
a − sn b − sn sup
2b − a
s∈2ab/3,a2b/3
a − sn 7b − sn .
sup
8b − a
s∈a2b/3,b
By an elementary calculation we get
7a − s b − s sup
8b − a
s∈a,2ab/3
a − s 7b − s 5 b − a,
sup
24
8b − a
s∈a2b/3,b
3.15
10
International Journal of Mathematics and Mathematical Sciences
7a − s2 b − s2 a − s2 7b − s2 11
sup
sup
b − a,
72
8b
−
a
8b
−
a
s∈a,2ab/3
s∈a2b/3,b
7a − sn b − sn 8b − a
s∈a,2ab/3
sup
a − sn 7b − sn b − an−1
,
8b − a
8
s∈a2b/3,b
sup
3.16
for n ≥ 3. The function y : a, b → R, yx a − xn b − xn , is decreasing on a, ab/2
and increasing on a b/2, b if n is even, and decreasing on a, b if n is odd. Thus
a − sn b − sn −1n 2n b − an−1
sup
.
2b − a
2 · 3n
s∈2ab/3,a2b/3
3.17
5
2a b
, s sup T1
3
24
s∈a,b
3.18
1 2n −1n
2a b
sup Tn
, s b − an−1 max ,
.
3
8
2 · 3n
3.19
Finally,
and for n ≥ 2
s∈a,b
√
3.2. a, b −1, 1, x − 5/5
Suppose that all assumptions of Theorem 3.1 hold. Then the following generalization of
Lobatto formula reads
1
2
√ √
√ 1
1
5
5
5
tn −
, s f n sds,
ftdt D −
Tn −
5
5
n! −1
5
−1
1
where
√ √ √ 1
5
5
5
f−1 5f −
5f
f1 ,
D −
5
12
5
5
√ √ √ n−2
5
5
5
5
i1
i1
i1
tn −
−
f
−1 f
5
12 i0
5
5
×
5
√ i2
√ i2
5
−1i1 5 − 5
2 · 5i2 i 2!
n−2 2i1
1
f i1 −1 −1i1 f i1 1
,
12 i0
i 2!
3.20
International Journal of Mathematics and Mathematical Sciences
√
√
√
n
5
5
5
Tn −
,s −
Tn −1, s 5Tn −
, s 5Tn
, s Tn 1, s
5
12
5
5
√
⎧
5
11−1 − sn 1 − sn
⎪
⎪
−1 ≤ s ≤ −
,
⎪
⎪
⎪
24
5
⎪
√
√
⎨
n
n
5
5
−1 − s 1 − s
−
<s≤
,
⎪
4
5
5
⎪
⎪
√
⎪
n
n
⎪
⎪
5
⎩ −1 − s 111 − s
< s ≤ 1.
24
5
11
3.21
Corollary 3.4. Suppose that all assumptions of Theorem 3.1 hold. Additionally, assume that p, q is
a pair of conjugate exponents and n ∈ N.
a if f n ∈ L∞ −1, 1, then
√ √ 1 1
101
5 5 f ,
−
ftdt − D −
≤
∞
2 −1
5 180
6
√ √ 1 1
5
5 − tn −
ftdt − D −
2 −1
5
5 ⎛
1 ⎜
≤
⎝
n 1!
√ n1 √ n1
− −5 5
2n1 · 5n 5 5
3.22
12 · 5n
1 −1n1
−
2
n f ,
∞
n ≥ 2.
b if f n ∈ L2 −1, 1, then
√ √ 1 1
5
5 − tn −
ftdt − D −
2 −1
5
5 ⎛ √ 2n1
√ 2n1
85 5 − 5
102n1
1 2n−2 ⎜ 35 5 5
≤
·
⎝
n!
3
102n1 2n 1
−1n 11Bn 1, n 1 25B5√5/10 n 1, n 1
⎞1/2
⎟ −25B5−√5/10 n 1, n 1 ⎠ f n .
2
3.23
12
International Journal of Mathematics and Mathematical Sciences
c if f n ∈ L1 −1, 1, then
√ √ √ 1 1
5
5 1
5 n ftdt − D −
− tn −
≤ Kn −
f ,
2 −1
1
5
5 n!
5
3.24
√
√
√
√
√
√
where√K1 − 5/5 1/2
5, K√
2 − 5/5 3/5, K3 − 5/5 8/5 5, K4 − 5/5 28/25,
√
K5 − 5/5 88/25 5, Kn − 5/5 2n−3 /3, for n ≥ 6.
The first and the second inequality are sharp.
√
Proof. Applying 3.4 with a, b −1, 1, x − 5/5 and p ∞, p 2, p 1 and carrying
out the same analysis as in Corollay 3.3 we obtain the above inequalities.
References
1 D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities Involving Functions and Their Integrals
and Derivatives, vol. 53 of Mathematics and its Applications (East European Series), Kluwer Academic
Publishers, Dordrecht, The Netherlands, 1991.
2 J. E. Pečarić, “On the Čebyšev inequality,” Buletinul Ştiinţific al Universităţii Politehnica din Timişoara.
Seria Matematică-Fizică, vol. 25, no. 39, pp. 10–11, 1980.
3 A. Aglić Aljinović and J. Pečarić, “On some Ostrowski type inequalities via Montgomery identity and
Taylor’s formula,” Tamkang Journal of Mathematics, vol. 36, no. 3, pp. 199–218, 2005.
4 A. Aglić Aljinović, J. Pečarić, and M. Ribičić Penava, “Sharp integral inequalities based on general
two-point formulae via an extension of Montgomery’s identity,” The ANZIAM Journal, vol. 51, no. 1,
pp. 67–101, 2009.
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International Journal of
Differential Equations
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International Journal of
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