Volume 10 (2009), Issue 3, Article 65, 11 pp.
DRAGOMIR-AGARWAL TYPE INEQUALITIES FOR SEVERAL FAMILIES OF
QUADRATURES
I. FRANJIĆ AND J. PEČARIĆ
FACULTY OF F OOD T ECHNOLOGY AND B IOTECHNOLOGY
U NIVERSITY OF Z AGREB
P IEROTTIJEVA 6, 10000 Z AGREB , C ROATIA
ifranjic@pbf.hr
FACULTY OF T EXTILE T ECHNOLOGY
U NIVERSITY OF Z AGREB
P RILAZ BARUNA F ILIPOVI ĆA 28 A
10000 Z AGREB , C ROATIA
pecaric@hazu.hr
Received 05 May, 2009; accepted 15 September, 2009
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Dragomir-Agarwal-type inequalities, k-convex functions, General 3-point, 4-point and 5-point quadrature formulae, Corrected quadrature formulae.
2000 Mathematics Subject Classification. 26D15, 65D30, 65D32.
1. I NTRODUCTION
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
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
The research of the authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grants 0581170889-1050 (first author) and 117-1170889-0888 (second author).
125-09
2
I. F RANJI Ć AND J. P E ČARI Ć
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
f (a) + f (b)
b − a |f 0 (a)|q + |f 0 (b)|q q
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 Euler-Simpson, EulerMaclaurin, 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 Euler-twopoint 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".
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].
2. P RELIMINARIES
General 3-point quadrature formulas were obtained in [5] and general corrected 3-point 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 1
Z
1 1 α
α
Fn (x, t)df (n−1) (t),
f (t)dt − Qα (x) + Tn−1 (x) =
(2.1)
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 =
,
2
6(1 − 2x)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 )
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 65, 11 pp.
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D RAGOMIR -AGARWAL T YPE I NEQUALITIES
3
−6B2 (x)f (0) + f (x) + f (1 − x) − 6B2 (x)f (1)
,
12x(1 − x)
30B4 (x)f (0) + f (x) + f (1 − x) + 30B4 (x)f (1)
QCQ4 (x) := QC (0, x, 1 − x, 1) =
,
60x2 (1 − x)2
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)
QQ4 (x) := Q(0, x, 1 − x, 1) =
and
b(n−1)/2c
α
Tn−1
(x)
Fnα (x, t)
(2.2)
=
=
X
k=1
α
Gn (x, t)
1
Gα (x, 0) [f (2k−1) (1) − f (2k−1) (0)],
(2k)! 2k
− Gαn (x, 0),
and finally,
(2.3)
GQ3
n (x, t)
(2.4)
GCQ3
(x, t) =
n
(2.5)
GQ4
n (x, t) =
(2.6)
GCQ4
(x, t) =
n
(2.7) GQ5
n (x, t) =
(2.8)
GCQ5
(x, t)
n
=
Bn∗ (x − t) + 24B2 (x) · Bn∗ 12 − t + Bn∗ (1 − x − t)
,
6(1 − 2x)2
7Bn∗ (x − t) − 480B4 (x) · Bn∗ 12 − t + 7Bn∗ (1 − x − t)
,
30(1 − 2x)2 (1 + 4x − 4x2 )
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)
,
60x2 (1 − x)2
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
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
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 65, 11 pp.
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4
I. F RANJI Ć AND J. P E ČARI Ć
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.
Lemma 2.1. For x ∈ {0} ∪ [1/6, 1/2) and n ≥ 2, GQ3
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 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
15/10],
2n−1 (x, t) > 0 for x ∈ [0, 1/2 −
CQ3
(−1)n+1 G2n−1
(x, t) > 0 for x ∈ [1/6, 1/2).
√
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
3/6],
2n−1 (x, t) > 0 for x ∈ (0, 1/2 −
(−1)n GQ4
2n−1 (x, t) > 0 for x ∈ [1/3, 1/2].
√
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
5/10],
2n−1 (x, t) > 0 for x ∈ (0, 1/2 −
(−1)n GCQ4
2n−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
(−1)n+1 GQ5
2n−1 (x, t) > 0 for x ∈ [1/5, 1/2) .
√
√
CQ5
For x ∈ (0, 1/2 − 21/14] ∪ [3/7 − 2/7, 1/2) and n ≥ 4, G2n−1
(x, t) has no zeros in
variable t on (0, 1/2). The sign of the function is determined by:
i
√
n CQ5
(−1) G2n−1 (x, t) > 0 for x ∈ 0, 1/2 − 21/14 ,
h
√
(−1)n+1 GCQ5
(x,
t)
>
0
for
x
∈
3/7
−
2/7,
1/2
,
2n−1
CQ3
Q4
CQ4
Q5
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), G2n−1 as in
(2.7) and GCQ5
2n−1 as in (2.8).
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
Gαn−j (x, t), j = 1, 2, . . . , n,
j
∂t
(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
α
on (0, 1/2) and (1/2, 1), have constant sign on (0, 1), so the functions |F2n
(t)| attain their
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 65, 11 pp.
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D RAGOMIR -AGARWAL T YPE I NEQUALITIES
5
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
t|F2n
(x, t)|dt = |Gα2n (x, 0)|,
0
0
Z 1
Z 1
1 α
α
|G2n−1 (x, t)|dt = 2
t|Gα2n−1 (x, t)|dt = |F2n
(x, 1/2)| .
n
0
0
(2.11)
(2.12)
α
|F2n
(x, t)|dt
=2
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.
3. M AIN R ESULT
To shorten notation, we denote the left-hand side of (2.1) by
R1
0
f (t)dt − ∆αn (x), i.e.
α
(x)
∆αn (x) := Qα (x) − Tn−1
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
• n ≥ 3 and
(1) α = Q3 and x ∈ {0} ∪ [1/6,√1/2),
(2) α = Q4 and x ∈ (0, 1/2 − 3/6] ∪ [1/3, 1/2],
• 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 − 21/14] ∪ [3/7 − 2/7, 1/2),
then we have
(3.1)
Z
1
f (t)dt −
0
∆αn (x)
≤ Cα (n, x) ·
|f (n) (0)|p + |f (n) (1)|p
2
p1
while if, under same conditions, |f (n) | is concave, then
(3.2)
Z
0
1
f (t)dt −
∆αn (x)
(n) 1 ,
≤ Cα (n, x) · f
2 where
2 α
1 Cα (2k − 1, x) =
F
x,
(2k)! 2k
2 and
Cα (2k, x) =
1
|Gα (x, 0)|
(2k)! 2k
α
with functions F2k
defined as in (2.2) and Gα2k as in (2.3), (2.4), (2.5), (2.6), (2.7) and (2.8).
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 65, 11 pp.
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6
I. F RANJI Ć AND J. P E ČARI Ć
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 Z
≤
|f
0
Z
≤
1
|Fnα (x, t)|dt
(n)
p
((1 − t) · 0 + t · 1)| ·
p1
|Fnα (x, t)|dt
0
1
1− p1 Z 1
Z
(n)
p
α
(n)
p
α
|Fn (x, t)|dt
|f (0)|
(1 − t)|Fn (x, t)|dt + |f (1)|
0
0
1
p1
t|Fnα (x, t)|dt .
0
Inequality (3.1) now follows from (2.11) and (2.12).
To prove (3.2), apply Jensen’s integral inequality to (2.1) to obtain
Z 1
Z 1
α
n! f (t)dt − ∆n (x) ≤
|Fnα (x, t)| · |f (n) ((1 − t) · 0 + t · 1)|dt
0
0
!
R1
Z 1
α
((1
−
t)
·
0
+
t
·
1)|F
(x,
t)|dt
n
0
≤
|Fnα (x, t)|dt · f (n)
.
R1
α
|F
(x,
t)|dt
0
n
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
1
3
−
3
1
3
−
3
3
−
= f
+ f
∆Q3
n
6
2
6
2
6
and
CQ3
CQ3
√ !
√
3− 3
9−4 3
3,
=
≈ 1.2 · 10−3 ,
6
1728
√ !
3− 3
1
4,
=
≈ 2.3 · 10−4 .
6
4320
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
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 65, 11 pp.
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D RAGOMIR -AGARWAL T YPE I NEQUALITIES
7
and
1
≈ 8.68 · 10−6 ,
115200
1
CCQ3 (6, 0) =
≈ 1.65 · 10−6 .
604800
CCQ3 (5, 0) =
For x = 1/6, the following estimates for the corrected Maclaurin’s formula are produced:
1
1
1
5
1 0
1
CQ3
=
27f
+ 26f
+ 27f
+
[f (1) − f 0 (0)]
∆n
6
80
6
2
6
240
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
1
1
3
1 0
CQ3
∆n
=
8f
−f
+ 8f
+
[f (1) − f 0 (0)]
4
15
4
2
4
120
and
1
1
CCQ3 5,
=
≈ 8.68 · 10−6 ,
4
115200
1
31
=
≈ 1.6 · 10−6 .
CCQ3 6,
4
19353600
√
For x = 1/2 − 15/10 (⇔ GCQ3
(x, 0) = 0), the following estimates for the Gauss 3-point
2
formula are produced:
"
√ !
√ !
√ !#
5
−
15
1
5
−
15
1
5
+
15
∆CQ3
=
5f
+ 8f
+ 5f
n
10
18
10
2
10
and
CCQ3
CCQ3
√ !
√
5 − 15
25 − 6 15
=
≈ 3.06 · 10−6 ,
5,
10
576000
√ !
5 − 15
1
6,
=
≈ 4.96 · 10−7 .
10
2016000
√ .
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
p
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 65, 11 pp.
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8
I. F RANJI Ć AND J. P E ČARI Ć
and
CCQ3
p
√ !
225 − 30 30
5,
30
p
p
√
√
√
46 225 − 30 30 − 120 30 − 4 30 + 150 30 − 825
≈ 7.86 · 10−6 ,
=
1728000
p
√ !
√
45 − 7 30
15 − 225 − 30 30
CCQ3 6,
=
≈ 1.47 · 10−6 .
30
4536000
15 −
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.
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
1
2
1 0
CQ4
∆n
=
13f (0) + 27f
+ 27f
+ 13f (1) −
[f (1) − f 0 (0)]
3
80
3
3
120
and
1
1
CCQ4 5,
=
≈ 1.45 · 10−6 ,
3
691200
1
1
=
≈ 3.67 · 10−7 .
CCQ4 6,
3
2721600
√
For x = 1/2 − 5/10 ⇔ GCQ4
(x,
0)
=
0
, the following estimates for the Lobatto 4-point
2
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
=
≈ 3.88 · 10−6 ,
5,
10
576000
√ !
5− 5
1
6,
=
≈ 6.61 · 10−7 .
10
1512000
3.5. CASE α = Q5 and n = 5, 6. For x = 1/4, the following estimates for Boole’s formula
from [11] are recaptured.
√
3.6. CASE α = CQ5 and n = 7, 8. For x = 1/2 − 21/14 ⇔ GCQ5
(x,
0)
=
0
, the
2
following estimates for the Lobatto 5-point formula are produced:
"
#
√ !
√ !
1
1
7
−
21
1
7
+
21
=
9f (0) + 49f
+ 64f
+ 49f
+ 9f (1)
∆CQ5
n
4
180
14
2
14
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 65, 11 pp.
http://jipam.vu.edu.au/
D RAGOMIR -AGARWAL T YPE I NEQUALITIES
9
and
CCQ5
CCQ5
√ !
√
12 21 − 49
7 − 21
=
7,
≈ 4.74 · 10−9 ,
14
1264435200
√ !
7 − 21
1
=
≈ 7.03 · 10−10 .
8,
14
1422489600
For x = 1/4, the following estimates for the corrected Boole’s formula are produced:
∆CQ5
n
1
1
1
3
1
=
217f (0) + 512f
+ 432f
+ 512f
+ 217f (1)
4
1890
4
2
4
1 0
[f (1) − f 0 (0)]
−
252
and
1
17
=
≈ 3.49 · 10−9 ,
CCQ5 7,
4
4877107200
1
1
=
≈ 6.15 · 10−10 .
CCQ5 8,
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:
∆CQ5
n
"
√ !
7− 7
1
=
37f (0) + 98f
14
270
√ !
7− 7
+ 98f
14
#
√ !
7+ 7
+ 37f (1)
14
−
1 0
[f (1) − f 0 (0)]
180
and
CCQ5
CCQ5
√ !
√
7− 7
343 − 16 7
7,
=
≈ 8.81 · 10−9 ,
14
34139750400
√ !
7− 7
1
8,
=
≈ 1.41 · 10−9 .
14
711244800
Finally, for
1
x = x0 := −
2
p
√
45 − 2 102
14
⇔ 98x4 − 196x3 + 102x2 − 4x − 1
= 98
!
!
!
√
√
1
102
1
102
x2 − x +
−
x2 − x +
+
=0 ,
49
98
49
98
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 65, 11 pp.
http://jipam.vu.edu.au/
10
I. F RANJI Ć AND J. P E ČARI Ć
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)
≈ 8.12 · 10−9 ,
CCQ5 (7, x0 ) =
3793305600
√
43 − 3 102
CCQ5 (8, x0 ) =
≈ 1.28 · 10−9 .
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 the corrected 3point 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
=
27f (0) + 125f
+ 128f
+ 125f
+ 27f (1)
∆n
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
and
√
27 − 16 2
CCQ5 (7, x0 ) =
≈ 1.15 · 10−9 ,
3793305600
√
11 − 6 2
CCQ5 (8, x0 ) =
≈ 2.53 · 10−10 .
9957427200
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 65, 11 pp.
http://jipam.vu.edu.au/
D RAGOMIR -AGARWAL T YPE I NEQUALITIES
11
R EFERENCES
[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 Dragomir-Agarwal 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
(submitted for publication).
AND
I. PERIĆ, General 3-point quadrature formulas of Euler type,
[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 York-London, 1962.
[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
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463–483.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 65, 11 pp.
http://jipam.vu.edu.au/