DRAGOMIR-AGARWAL TYPE INEQUALITIES FOR
SEVERAL FAMILIES OF QUADRATURES
I. FRANJIĆ
J. PEČARIĆ
Faculty of Food Technology and Biotechnology
University of Zagreb
Pierottijeva 6,
10000 Zagreb, Croatia
EMail: ifranjic@pbf.hr
Faculty of Textile Technology
University of Zagreb
Prilaz baruna Filipovića 28a
10000 Zagreb, Croatia
EMail: pecaric@hazu.hr
Received:
05 May, 2009
Accepted:
15 September, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
26D15, 65D30, 65D32.
Key words:
Dragomir-Agarwal-type inequalities, k-convex functions, General 3-point, 4point and 5-point quadrature formulae, Corrected quadrature formulae.
Abstract:
Acknowledgements:
Inequalities estimating the absolute value of the difference between the integral
and the quadrature, i.e. the Dragomir-Agarwal-type inequalities, are given for
the general 3, 4 and 5-point quadrature formulae, both classical and corrected.
Beside values of the function in the chosen nodes, "corrected" quadrature formula
includes values of the first derivative at the end points of the interval and has a
higher accuracy than the adjoint classical quadrature formula.
The research of the authors was supported by the Croatian Ministry of Science,
Education and Sports, under the Research Grants 058-1170889-1050 (first author) and 117-1170889-0888 (second author).
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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Contents
1
Introduction
3
2
Preliminaries
5
3
Main Result
3.1 CASE α = Q3 and n = 3, 4 .
3.2 CASE α = CQ3 and n = 5, 6
3.3 CASE α = Q4 and n = 3, 4 .
3.4 CASE α = CQ4 and n = 5, 6
3.5 CASE α = Q5 and n = 5, 6 .
3.6 CASE α = CQ5 and n = 7, 8
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Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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1.
Introduction
The well-known Hermite-Hadamard inequality states that if f : [a, b] → R is a
convex function, then
Z b
a+b
1
f (a) + f (b)
(1.1)
f
≤
f (t)dt ≤
.
2
b−a a
2
Dragomir-Agarwal Type Inequalities
This pair of inequalities has been improved and extended in a number of ways.
One of the directions estimated the difference between the middle and rightmost
term in (1.1). For example, Dragomir and Agarwal presented the following result in
[2]: suppose f : I ⊆ R → R is differentiable on I and |f 0 |q is convex on [a, b] for
some q ≥ 1, where I is an open interval in R and a, b ∈ I (a < b). Then
1
Z b
b − a |f 0 (a)|q + |f 0 (b)|q q
f (a) + f (b)
1
.
−
f (t)dt ≤
2
b−a a
4
2
Generalizations to higher-order convexity for this type of inequality were given in
[3]. Related results for Euler-midpoint, Euler-twopoint, Euler-Simpson, dual EulerSimpson, Euler-Maclaurin, Euler-Simpson 3/8 and Euler-Boole formulae were given
in [11]. Furthermore, related results for the general Euler 2-point formulae were
given in [10], unifying the cases of Euler trapezoid, Euler midpoint and Eulertwopoint formulae.
The aim of this paper is to give related results for the general 3, 4 and 5-point
quadrature formulae, as well as for the corrected general 3, 4 and 5-point quadrature
formulae. In addition to values of the function at the chosen nodes, "corrected"
quadrature formulae include values of the first derivative at the end points of the
interval and also have higher accuracy than adjoint classical quadrature formulae.
They are sometimes called "quadratures with end corrections".
I. Franjić and J. Pečarić
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Our first course of action was to obtain the quadrature formulae. This was done
using the extended Euler formulae, in which Bernoulli polynomials play an important role. For the reader’s convenience, let us recall some basic properties of
Bernoulli polynomials. Bernoulli polynomials Bk (t) are uniquely determined by
Bk0 (x) = kBk−1 (x),
Bk (t + 1) − Bk (t) = ktk−1 ,
k ≥ 0, B0 (t) = 1.
For the kth Bernoulli polynomial we have Bk (1 − x) = (−1)k Bk (x), x ∈ R, k ≥ 1.
The kth Bernoulli number Bk is defined by Bk = Bk (0). For k ≥ 2, we have
Bk (1) = Bk (0) = Bk . Note that B2k−1 = 0, k ≥ 2 and B1 (1) = −B1 (0) = 1/2.
Bk∗ (x) are periodic functions of period 1 defined by Bk∗ (x + 1) = Bk∗ (x), x ∈ R,
and related to Bernoulli polynomials as Bk∗ (x) = Bk (x), 0 ≤ x < 1. For k ≥ 2,
Bk∗ (t) is a continuous function, while B1∗ (x) is a discontinuous function with a jump
of −1 at each integer. For further details on Bernoulli polynomials, see [1] and [9].
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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2.
Preliminaries
General 3-point quadrature formulas were obtained in [5] and general corrected 3point quadrature formulas in [6]; general closed 4-point quadrature formulas were
considered in [7] and finally, general closed 5-point quadrature formulas were derived in [8]. Namely, if f : [0, 1] → R is such that f (n−1) is continuous and of
bounded variation on [0, 1] for some n ≥ 1, then we have
Z
Z 1
1 1 α
α
(2.1)
f (t)dt − Qα (x) + Tn−1 (x) =
Fn (x, t)df (n−1) (t),
n!
0
0
for α = Q3, CQ3 and x ∈ [0, 1/2), for α = Q4, CQ4 and x ∈ (0, 1/2], and for
α = Q5, CQ5 and x ∈ (0, 1/2), where
f (x) + 24B2 (x)f 12 + f (1 − x)
1
,
QQ3 (x) := Q x, , 1 − x =
6(1 − 2x)2
2
7f (x) − 480B4 (x)f 12 + 7f (1 − x)
1
QCQ3 (x) := QC x, , 1 − x =
,
2
30(1 − 2x)2 (1 + 4x − 4x2 )
QQ4 (x) := Q(0, x, 1 − x, 1)
−6B2 (x)f (0) + f (x) + f (1 − x) − 6B2 (x)f (1)
,
=
12x(1 − x)
QCQ4 (x) := QC (0, x, 1 − x, 1)
30B4 (x)f (0) + f (x) + f (1 − x) + 30B4 (x)f (1)
,
=
60x2 (1 − x)2
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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1
QQ5 (x) := Q 0, x, , 1 − x, 1
2
h
1
=
f (x) + f (1 − x)
60x(1 − x)(1 − 2x)2
− (10x2 − 10x + 1)(1 − 2x)2 (f (0) + f (1))
1
2
+32x(1 − x)(5x − 5x + 1)f
,
2
1
QCQ5 (x) := QC 0, x, , 1 − x, 1
2
1
=
[f (x) + f (1 − x)
2
420x (1 − x)2 (1 − 2x)2
+ (98x4 − 196x3 + 102x2 − 4x − 1)(1 − 2x)2 (f (0) + f (1))
+ 64x2 (1 − x)2 (14x2 − 14x + 3)f (1/2)
and
b(n−1)/2c
α
Tn−1
(x)
(2.2)
=
Fnα (x, t) =
X
k=1
α
Gn (x, t)
(2.3)
(2.4)
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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1
Gα (x, 0) [f (2k−1) (1) − f (2k−1) (0)],
(2k)! 2k
− Gαn (x, 0),
and finally,
Bn∗ (x − t) + 24B2 (x) · Bn∗ 12 − t + Bn∗ (1 − x − t)
=
,
6(1 − 2x)2
∗
∗ 1
7B
(x
−
t)
−
480B
(x)
·
B
−
t
+ 7Bn∗ (1 − x − t)
4
n
n
2
GnCQ3 (x, t) =
,
30(1 − 2x)2 (1 + 4x − 4x2 )
GQ3
n (x, t)
Dragomir-Agarwal Type Inequalities
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(2.5)
(2.6)
Bn∗ (x − t) − 12B2 (x) · Bn∗ (1 − t) + Bn∗ (1 − x − t)
,
12x(1 − x)
60B4 (x) · Bn∗ (1 − t) + Bn∗ (x − t) + Bn∗ (1 − x − t)
,
GnCQ4 (x, t) =
60x2 (1 − x)2
GQ4
n (x, t) =
10x2 − 10x + 1 ∗
Bn (1 − t)
30x(x − 1)
Bn∗ (x − t) + Bn∗ (1 − x − t) 8(5x2 − 5x + 1) ∗ 1
+
+
Bn
−t ,
60x(1 − x)(1 − 2x)2
15(1 − 2x)2
2
(2.7) GQ5
n (x, t) =
98x4 − 196x3 + 102x2 − 4x − 1 ∗
Bn (1 − t)
210x2 (1 − x)2
Bn∗ (x − t) + Bn∗ (1 − x − t) 16(14x2 − 14x + 3) ∗ 1
+
+
Bn
−t .
420x2 (1 − x)2 (1 − 2x)2
105(1 − 2x)2
2
(2.8) GCQ5
(x, t) =
n
The following lemma was the key result for obtaining the results in [5], [6], [7]
and [8], and we shall need it here as well.
GQ3
2n−1 (x, t)
Lemma 2.1. For x ∈ {0} ∪ [1/6, 1/2) and n ≥ 2,
variable t on (0, 1/2). The sign of the function is determined by:
has no zeros in
(−1)n+1 GQ3
for x ∈ [1/6, 1/2) and (−1)n GQ3
2n−1 (x, t) > 0
2n−1 (0, t) > 0.
√
For x ∈ 0, 1/2 − 15/10 ∪ [1/6, 1/2) and n ≥ 3, GCQ3
2n−1 (x, t) has no zeros in
variable t on (0, 1/2). The sign of the function is determined by:
√
(−1)n GCQ3
for x ∈ [0, 1/2 − 15/10],
2n−1 (x, t) > 0
(−1)n+1 GCQ3
2n−1 (x, t) > 0
for x ∈ [1/6, 1/2).
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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√
For x ∈ (0, 1/2 − 3/6] ∪ [1/3, 1/2] and n ≥ 2, GQ4
2n−1 (x, t) has no zeros in
variable t on (0, 1/2). The sign of the function is determined by:
√
(−1)n+1 GQ4
for x ∈ (0, 1/2 − 3/6],
2n−1 (x, t) > 0
(−1)n GQ4
for x ∈ [1/3, 1/2].
2n−1 (x, t) > 0
√
CQ4
For x ∈ (0, 1/2 − 5/10] ∪ [1/3, 1/2] and n ≥ 3, G2n−1
(x, t) has no zeros in
variable t on (0, 1/2). The sign of the function is determined by:
√
(−1)n+1 GCQ4
for x ∈ (0, 1/2 − 5/10],
2n−1 (x, t) > 0
CQ4
(−1)n G2n−1
(x, t) > 0 for x ∈ [1/3, 1/2].
√
For x ∈ (0, 1/2 − 15/10] ∪ [1/5, 1/2) and n ≥ 3, GQ5
2n−1 (x, t) has no zeros in
variable t on (0, 1/2). The sign of the function is determined by:
i
√
(−1)n GQ5
(x,
t)
>
0
for
x
∈
0,
1/2
−
15/10
,
2n−1
n+1
GQ5
2n−1 (x, t)
> 0 for x ∈ [1/5, 1/2) .
√
√
For x ∈ (0, 1/2 − 21/14] ∪ [3/7 − 2/7, 1/2) and n ≥ 4, GCQ5
2n−1 (x, t) has no
zeros in variable t on (0, 1/2). The sign of the function is determined by:
i
√
(−1)n GCQ5
(x,
t)
>
0
for
x
∈
0,
1/2
−
21/14
,
2n−1
h
√
CQ5
(−1)n+1 G2n−1
(x, t) > 0 for x ∈ 3/7 − 2/7, 1/2 ,
(−1)
CQ3
Q4
CQ4
where GQ3
2n−1 is as in (2.3), G2n−1 as in (2.4), G2n−1 as in (2.5), G2n−1 as in (2.6),
CQ5
GQ5
2n−1 as in (2.7) and G2n−1 as in (2.8).
Dragomir-Agarwal Type Inequalities
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vol. 10, iss. 3, art. 65, 2009
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Applying properties of Bernoulli polynomials, it easily follows that functions Gαn
for α = Q3, CQ3, Q4, CQ4, Q5, CQ5 and n ≥ 1, have the following properties:
(2.9)
(2.10)
Gαn (x, 1 − t) = (−1)n Gαn (x, t), t ∈ [0, 1],
∂ j Gαn (x, t)
n!
= (−1)j
Gα (x, t), j = 1, 2, . . . , n,
j
∂t
(n − j)! n−j
α
also, that Gα2n−1 (x, 0) = 0 for n ≥ 1, and so F2n−1
(x, t) = Gα2n−1 (x, t).
α
These properties and Lemma 2.1 yield that functions F2n
, defined by (2.2), are
monotonous on (0, 1/2) and (1/2, 1), have constant sign on (0, 1), so the functions
α
|F2n
(t)| attain their maximal value at t = 1/2. Finally, using (2.9) and (2.10), it is
not hard to establish that under the assumptions of Lemma 2.1 we have:
Z 1
Z 1
α
α
(2.11)
|F2n (x, t)|dt = 2
t|F2n
(x, t)|dt = |Gα2n (x, 0)|,
0
Z0 1
Z 1
1 α
α
(2.12)
|G2n−1 (x, t)|dt = 2
t|Gα2n−1 (x, t)|dt = |F2n
(x, 1/2)| .
n
0
0
Dragomir-Agarwal Type Inequalities
Now that we have stated all the previously obtained results which form a basis
for the results of this paper, we proceed to the main result.
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3.
Main Result
To shorten notation, we denote the left-hand side of (2.1) by
R1
0
f (t)dt − ∆αn (x), i.e.
α
∆αn (x) := Qα (x) − Tn−1
(x)
for α = Q3, CQ3, Q4, CQ4, Q5, CQ5 and n ≥ 1.
Theorem 3.1. Let f : [0, 1] → R be n-times differentiable. If |f (n) |p is convex for
some p ≥ 1 and
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
• n ≥ 3 and
Title Page
1. α = Q3 and x ∈ {0} ∪ [1/6, 1/2),
√
2. α = Q4 and x ∈ (0, 1/2 − 3/6] ∪ [1/3, 1/2],
Contents
• n ≥ 5 and
√
1. α = CQ3 and x ∈ [0, 1/2 − 15/10] ∪ [1/6, 1/2),
√
2. α = CQ4 and x ∈ (0, 1/2 − 5/10] ∪ [1/3, 1/2],
√
3. α = Q5 and x ∈ (0, 1/2 − 15/10] ∪ [1/5, 1/2),
• n ≥ 7 and
1. α = CQ5 and x ∈ (0, 1/2 −
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√
21/14] ∪ [3/7 −
√
2/7, 1/2),
then we have
Z 1
(n)
1
p
(n)
p p
|f
(0)|
+
|f
(1)|
(3.1)
f (t)dt − ∆αn (x) ≤ Cα (n, x) ·
2
0
Close
while if, under same conditions, |f (n) | is concave, then
Z 1
(n) 1 α
,
(3.2)
f (t)dt − ∆n (x) ≤ Cα (n, x) · f
2 0
where
α
F2k x, 1 2 2
Cα (2k − 1, x) =
(2k)!
with functions
and (2.8).
α
F2k
and
defined as in (2.2) and
Gα2k
Cα (2k, x) =
1
|Gα (x, 0)|
(2k)! 2k
as in (2.3), (2.4), (2.5), (2.6), (2.7)
α
Proof. First, recall that F2k−1
(x, t) = Gα2k−1 (x, t). Now, starting from (2.1), we
apply Hölder’s and then Jensen’s inequality for the convex function |f (n) |p , to obtain
Z 1
α
n! f (t)dt − ∆n (x)
0
Z 1
≤
|Fnα (x, t)| · |f (n) (t)|dt
0
Z
1
1− p1
|Fnα (x, t)|dt
≤
≤
1
|f
0
Z
Z
(n)
p
((1 − t) · 0 + t · 1)| ·
1
p
|Fnα (x, t)|dt
0
1
1− p1
α
|Fn (x, t)|dt
× |f (n) (0)|p
Z
1
(1 − t)|Fnα (x, t)|dt + |f (n) (1)|p
0
Inequality (3.1) now follows from (2.11) and (2.12).
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vol. 10, iss. 3, art. 65, 2009
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0
Dragomir-Agarwal Type Inequalities
Z
0
1
p1
t|Fnα (x, t)|dt .
To prove (3.2), apply Jensen’s integral inequality to (2.1) to obtain
Z 1
α
n! f (t)dt − ∆n (x)
0
Z 1
≤
|Fnα (x, t)| · |f (n) ((1 − t) · 0 + t · 1)|dt
0
!
R1
Z 1
α
((1
−
t)
·
0
+
t
·
1)|F
(x,
t)|dt
n
α
(n)
0
≤
|Fn (x, t)|dt · f
.
R1
|Fnα (x, t)|dt
0
0
Theorem 3.1 provides numerous interesting special cases. Particular choices of
node will procure the Dragomir-Agarwal-type estimates for many classical quadrature formulas, as well as the adjoint corrected ones.
3.1.
CASE α = Q3 and n = 3, 4
For x = 0, Theorem 3.1 gives Dragomir-Agarwal-type estimates for Simpson’s formula; for x = 1/4 it provides the estimates for the dual Simpson formula and for
x = 1/6 for Maclaurin’s formula. These were already obtained in [11]; Simpson’s
formula was also considered
in [4].
√
For x = 1/2 − 3/6 (⇔ B2 (x) = 0), the following estimates are obtained for
the Gauss 2-point formula:
√ !
√ !
√ !
3
−
3
1
3
−
3
1
3
−
3
∆Q3
= f
+ f
n
6
2
6
2
6
Dragomir-Agarwal Type Inequalities
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vol. 10, iss. 3, art. 65, 2009
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and
CQ3
CQ3
√
√ !
9−4 3
3− 3
=
3,
≈ 1.2 · 10−3 ,
6
1728
√ !
3− 3
1
4,
=
≈ 2.3 · 10−4 .
6
4320
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
3.2.
CASE α = CQ3 and n = 5, 6
For x = 0, the following estimates for the corrected Simpson’s formula are produced:
1
1
1
CQ3
∆n (0) =
7f (0) + 16f
+ 7f (1) − [f 0 (1) − f 0 (0)]
30
2
60
and
1
CCQ3 (5, 0) =
≈ 8.68 · 10−6 ,
115200
1
CCQ3 (6, 0) =
≈ 1.65 · 10−6 .
604800
For x = 1/6, the following estimates for the corrected Maclaurin’s formula are
produced:
1
1
1
1
5
1 0
CQ3
∆n
=
27f
+ 26f
+ 27f
+
[f (1) − f 0 (0)]
6
80
6
2
6
240
vol. 10, iss. 3, art. 65, 2009
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and
1
1
CCQ3 5,
=
≈ 1.45 · 10−6 ,
6
691200
1
31
CCQ3 6,
=
≈ 3.56 · 10−7 .
6
87091200
For x = 1/4, the following estimates for the corrected dual Simpson’s formula
are produced:
1
1
3
1
1
1 0
CQ3
∆n
=
8f
−f
+ 8f
+
[f (1) − f 0 (0)]
4
15
4
2
4
120
Dragomir-Agarwal Type Inequalities
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vol. 10, iss. 3, art. 65, 2009
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and
Contents
1
1
CCQ3 5,
=
≈ 8.68 · 10−6 ,
4
115200
31
1
=
≈ 1.6 · 10−6 .
CCQ3 6,
4
19353600
√
(x, 0) = 0), the following estimates for the
For x = 1/2 − 15/10 (⇔ GCQ3
2
Gauss 3-point formula are produced:
"
√ !
√ !
√ !#
5
−
15
1
5
−
15
1
5
+
15
∆CQ3
=
5f
+ 8f
+ 5f
n
10
18
10
2
10
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and
√
25 − 6 15
5 − 15
=
CCQ3 5,
≈ 3.06 · 10−6 ,
10
576000
√ !
5 − 15
1
CCQ3 6,
=
≈ 4.96 · 10−7 .
10
2016000
p
√ .
For x = x0 := 1/2 − 225 − 30 30 30 (⇔ B4 (x) = 0) (the case when
the weight next to f (1/2) is annihilated), the following estimates for the corrected
Gauss 2-point formula are produced:
√
1
1
5 − 30 0
CQ3
∆n (x0 ) = f (x0 ) + f (1 − x0 ) −
[f (1) − f 0 (0)]
2
2
60
√
!
and
p
√ !
15 − 225 − 30 30
CCQ3 5,
30
p
p
√
√
√
46 225 − 30 30 − 120 30 − 4 30 + 150 30 − 825
=
≈ 7.86 · 10−6 ,
1728000
p
√ !
√
15 − 225 − 30 30
45 − 7 30
CCQ3 6,
=
≈ 1.47 · 10−6 .
30
4536000
3.3.
CASE α = Q4 and n = 3, 4
For x = 1/3, the estimates for the Simpson 3/8 formula from [11] are recaptured.
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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3.4.
CASE α = CQ4 and n = 5, 6
For x = 1/3, the following estimates for the corrected Simpson 3/8 formula are
produced:
1
1
2
1
1 0
CQ4
∆n
=
13f (0) + 27f
+ 27f
+ 13f (1) −
[f (1)−f 0 (0)]
3
80
3
3
120
Dragomir-Agarwal Type Inequalities
and
I. Franjić and J. Pečarić
1
1
CCQ4 5,
=
≈ 1.45 · 10−6 ,
3
691200
1
1
CCQ4 6,
=
≈ 3.67 · 10−7 .
3
2721600
√
(x,
0)
=
0
, the following estimates for the
For x = 1/2 − 5/10 ⇔ GCQ4
2
Lobatto 4-point formula are produced:
"
#
√ !
√ !
1
1
5
−
5
5
+
5
∆CQ4
=
f (0) + 5f
+ 5f
+ f (1)
n
3
12
10
10
and
CCQ4
CCQ4
√ !
√
5− 5
5
5,
=
≈ 3.88 · 10−6 ,
10
576000
√ !
5− 5
1
6,
=
≈ 6.61 · 10−7 .
10
1512000
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3.5.
CASE α = Q5 and n = 5, 6
For x = 1/4, the following estimates for Boole’s formula from [11] are recaptured.
CASE α = CQ5 and n = 7, 8
√
For x = 1/2 − 21/14 ⇔ GCQ5
(x,
0)
=
0
, the following estimates for the
2
Lobatto 5-point formula are produced:
1
CQ5
∆n
4
"
#
√ !
√ !
1
7 − 21
1
7 + 21
=
9f (0) + 49f
+ 64f
+ 49f
+ 9f (1)
180
14
2
14
3.6.
and
√
CCQ5
CCQ5
!
√
12 21 − 49
=
≈ 4.74 · 10−9 ,
1264435200
7 − 21
14
√ !
7 − 21
1
8,
=
≈ 7.03 · 10−10 .
14
1422489600
7,
For x = 1/4, the following estimates for the corrected Boole’s formula are produced:
1
1
1
1
CQ5
∆n
=
217f (0) + 512f
+ 432f
4
1890
4
2
3
1 0
+512f
+ 217f (1) −
[f (1) − f 0 (0)]
4
252
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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and
CCQ5
1
7,
4
=
17
≈ 3.49 · 10−9 ,
4877107200
1
1
=
CCQ5 8,
≈ 6.15 · 10−10 .
4
1625702400
√
Further, for x = 1/2 − 7/14 (⇔ 14x2 − 14x + 3 = 0), which is the case when
the weight next to f (1/2) is annihilated, the following estimates for the corrected
Lobatto 4-point formula are produced:
"
√ !
√ !
7− 7
1
7− 7
∆CQ5
=
37f (0) + 98f
n
14
14
270
#
√ !
1 0
7+ 7
+ 37f (1) −
[f (1) − f 0 (0)]
+98f
14
180
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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and
CCQ5
√ !
√
7− 7
343 − 16 7
=
≈ 8.81 · 10−9 ,
7,
14
34139750400
CCQ5
√ !
7− 7
1
=
≈ 1.41 · 10−9 .
8,
14
711244800
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Finally, for
1
x = x0 := −
2
p
√
45 − 2 102
14
⇔ 98x4 − 196x3 + 102x2 − 4x − 1
= 98
!
!
!
√
√
1
1
102
102
x2 − x +
=0 ,
x2 − x +
−
+
49
98
49
98
which is the case when the weight next to f (0) and f (1) is annihilated, the estimates
for the corrected Gauss 3-point formula are produced:
√
1977 + 16 102
CQ5
∆n (x0 ) =
[f (x0 ) + f (1 − x0 )]
6930
√
√
1
9 − 102 0
1488 − 16 102
+
f
−
[f (1) − f 0 (0)]
3465
2
420
and
p
√
√
24 60933 − 6014 102 − 49(87 − 8 102)
CCQ5 (7, x0 ) =
≈ 8.12 · 10−9 ,
3793305600
√
43 − 3 102
≈ 1.28 · 10−9 .
CCQ5 (8, x0 ) =
9957427200
Remark 1. An interesting fact to point out is that out of all the 3-point quadrature formulae, Maclaurin’s formula gives the least estimate of error in Theorem 3.1; among
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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the corrected 3-point quadrature formulae, the corrected Maclaurin’s formula has the
same property.
Further, among the closed 4-point quadrature formulas, the Simpson 3/8 formula
gives the best estimate and the corrected Simpson 3/8 formula is the optimal corrected closed 4-point quadrature formula.
Finally, the node x = 1/5 produces the closed 5-point
√ quadrature formula with
the best error estimate, while the node x = 3/7 − 2/7 produces the corrected
closed 5-point quadrature formula with the same property.
The proofs are similar to those in [5], [6], [7] and [8], respectively.
In view of the previous remark, let us consider the case α = Q5, n = 5, 6 and
x = 1/5. We have:
1
1
1
4
1
Q5
∆n
=
27f (0) + 125f
+ 128f
+ 125f
+ 27f (1)
5
432
5
2
5
and
1
1
CQ5 5,
=
≈ 8.68 · 10−7 ,
5
1152000
1
1
CQ5 6,
=
≈ 1.98 · 10−7 .
5
5040000
√
Further, for α = CQ5, n = 7, 8 and x = x0 := 3/7 − 2/7 we obtain:
∆CQ5
(x0 ) = 0.10143 [f (0) + f (1)] + 0.259261 [f (x0 ) + f (1 − x0 )]
n
1
+ 0.278617 f
+ 3.07832 · 10−3 [f 0 (1) − f 0 (0)]
2
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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and
√
27 − 16 2
≈ 1.15 · 10−9 ,
CCQ5 (7, x0 ) =
3793305600
√
11 − 6 2
CCQ5 (8, x0 ) =
≈ 2.53 · 10−10 .
9957427200
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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References
[1] M. ABRAMOWITZ AND I.A. STEGUN (Eds), Handbook of Mathematical
Functions with Formulae, Graphs and Mathematical Tables, National Bureau
of Standards, Applied Math. Series 55, 4th printing, Washington, 1965.
[2] S.S. DRAGOMIR AND R.P. AGARWAL, Two inequalities for differentiable
mappings and applications to special means of real numbers and to trapezoid
formula, Appl. Mat. Lett., 11(5) (1998), 91–95.
[3] Lj. DEDIĆ, C.E.M. PEARCE AND J. PEČARIĆ, Hadamard and DragomirAgarwal inequalities, higher-order convexity and the Euler formula, J. Korean
Math. Soc., 38 (2001), 1235–1243.
[4] Lj. DEDIĆ, C.E.M. PEARCE AND J. PEČARIĆ, The Euler formulae and convex functions, Math. Inequal. Appl., 3(2) (2000), 211–221.
[5] I. FRANJIĆ, J. PEČARIĆ AND I. PERIĆ, Quadrature formulae of Gauss type
based on Euler identitites, Math. Comput. Modelling, 45(3-4) (2007), 355–370.
[6] I. FRANJIĆ, J. PEČARIĆ AND I. PERIĆ, General 3-point quadrature formulas
of Euler type, (submitted for publication).
[7] I. FRANJIĆ, J. PEČARIĆ AND I. PERIĆ, General closed 4-point quadrature
formulae of Euler type, Math. Inequal. Appl., 12 (2009), 573–586.
[8] I. FRANJIĆ, J. PEČARIĆ AND I. PERIĆ, On families of quadrature formulas
based on Euler identities, (submitted for publication).
[9] V.I. KRYLOV, Approximate Calculation of Integrals, Macmillan, New YorkLondon, 1962.
Dragomir-Agarwal Type Inequalities
I. Franjić and J. Pečarić
vol. 10, iss. 3, art. 65, 2009
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[10] J. PEČARIĆ AND A. VUKELIĆ, Hadamard and Dragomir-Agarwal inequalities, the general Euler two point formulae and convex functions, Rad Hrvat.
akad. znan. umjet., 491. Matematičke znanosti, 15 (2005), 139–152.
[11] J. PEČARIĆ AND A. VUKELIĆ, On generalizations of Dragomir-Agarwal inequality via some Euler-type identitites, Bulletin de la Sociètè des Mathèmaticiens de R. Macèdonie, 26 (LII) (2002), 463–483.
Dragomir-Agarwal Type Inequalities
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vol. 10, iss. 3, art. 65, 2009
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