TAMKANG JOURNAL OF MATHEMATICS Volume 31, Number 3, Autumn 2000
A NOTE ON SIMPSON’S INEQUALITY FOR FUNCTIONS OF BOUNDED VARIATION ˇ ´ AND S. VAROSANEC ˇ J. PECARI C
Abstract. Inequalities of Simpson’s type for functions whose n-th derivative, n ∈ {0, 1, 2, 3} is of bounded variation are given.
1. Introduction One of fundamental results in numerical integration is Simpson’s inequality which states if f (4) exists and is bounded on (a, b) then Z b b−a a+b 1 f (x)dx − [f (a) + 4f + f (b)] ≤ (b − a)5 ||f (4) k∞ . (1) a 2880 6 2
The disadvantage that this estimation can not be applied if the fourth derivate of f either does not exist on (a, b) or is not bounded there, was removed in result of Dragomir, [1]. Namely, in [1] the following result was proven. Theorem A. Let f : [a, b] → R be a mapping of bounded variation on [a, b]. Then Z 1 b b−a a+b f (x)dx − [f (a) + 4f + f (b)] ≤ (b − a)Vab (f ), (2) 3 a 6 2
where Vab (f ) denotes the total variation of f on the interval [a, b].
In the proof of the previous result the following identity is used: −
Z
a
b
s(x)df (x) =
Z
a
b
b−a a+b f (x)dx − f (a) + f (b) + 4f 6 2
where s(x) =
x− x−
5a+b 6 a+5b 6
x ∈ [a, a+b 2 ) x ∈ [ a+b 2 , b].
Received September 8, 1999. 2000 Mathematics Subject Classification. Primary 26D15, Secondary 41A55. Key words and phrases. Simpson’s inequality, function of bounded variation. 239
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240
The right-hand side of (3) is denoted by R(f ). Similar identity can be found in the book [2, p.174] but for absolute continuous function f so that on the left-hand side of Rb (3) we have − a s(x)f ′ (x)dx. In the same book some other identities in connection with Simpson’s inequality are given. Here we give versions of those identities which are suitable for our purpose: 1 R(f ) = 2
a+b 2
Z
a
1 R(f ) = − 6
Z
1 24
Z
R(f ) =
2a + b ′ 1 )df (x) + (x − a)(x − 3 2
a+b 2
b
(x − b)(x − a+b 2
(x − a)2 (x −
a + b ′′ 1 )df (x) − 2 6
(x − a)3 (x −
a + 2b ′′′ 1 )df (x) + 3 24
a a+b 2
Z
a
Z
a + 2b ′ )df (x) 3
b a+b 2
Z
(x − b)2 (x −
a + b ′′ )df (x) 2
b a+b 2
(x − b)3 (x −
(4) (5)
2a + b ′′′ )df (x) (6) 3
where f is a function such that f ′ , f ′′ , f ′′′ is of bounded variation, respectively. These identities are proven using integration by parts. In [2], result related to (2) is given. Namely, using identity (6) it is proven that if f ′′′ is an absolutely continuous with total variation V3 , then |R(f )| ≤
1 (b − a)4 V3 . 1152
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Here, we state results related to inequalities (2) and (7) which give an error estimate of R(f ) expressed by total variation of either function f or its derivatives. 2. Main Results Theorem 1. Let n ∈ {0, 1, 2, 3}. Let f be a real function on [a, b] such that f (n) is function of bounded varation. Then Z b b−a a+b f (x)dx − f (x) + 4f + f (b) ≤ Cn (b − a)n+1 Vab (f (n) ), a 6 2
where
1 1 1 1 , C1 = , C2 = , C3 = 3 24 324 1152 b (n) (n) and Va (f ) is the total variation of function f . C0 =
Proof. For n = 0 the proof is given in [1]. For n = 3 the proof is similar to that one from [2]. If n = 1, using identity (4) under notation ( 1 (x − a)(x − 2a+b 3 ), s1 (x) = 12 2b+a 2 (x − b)(x − 3 ),
x ∈ [a, a+b 2 ) x ∈ [ a+b 2 , b].
SIMPSON’S INEQUALITY FOR FUNCTIONS OF BOUNDED VARIATION
we have
Z b ′ |R(f )| = s1 (x)df (x) ≤ max |s1 (x)|Vab (f ′ ). a x∈[a,b]
241
2
(b−a) a+b Maximum of function | 12 (x − a)(x − 2a+b 3 )| on interval [a, 2 ] is 24 , and the same 1 2a+b a+b value is maximum of function | 2 (x−b)(x− 3 )| on interval [ 2 , b]. So, maxx∈[a,b] |s1 (x)|
=
(b−a)2 24
and (b − a)2 b ′ Va (f ). 24
|R(f )| ≤
If n = 2 using identity (5) we have Z b |R(f )| = s2 (x)df ′′ ≤ max |s2 (x)|Vab (f ′′ ), a x∈[a,b]
where
(
s2 (x) =
− 16 (x − a)2 (x −
− 16 (x − b)2 (x −
a+b 2 ), a+b 2 ),
x ∈ [a, a+b 2 )
x ∈ [ a+b 2 , b].
Global maximum investigation gives that maxx∈[a, a+b ] | − 61 (x − a)2 (x − 2
maxx∈[ a+b ,b] | − 61 (x − b)2 (x − 2 So, the following holds
a+b 2 )|
=
|R(f )| ≤
a+b 2 )|
=
(b−a)3 324 .
1 (b − a)3 Vab (f ′′ ). 324
As a simple consequence of the previous theorem we have the following corollary. Corollary 1. If f is a function on [a, b] such that f ′′′ is of bounded variation then |R(f )| ≤
min
{Cn (b − a)n+1 Vab (f (n) )}.
n∈{0,1,2,3}
Corollary 2. Let n ∈ {0, 1, 2, 3}. If f is a function such that f (n) is an absolute continuous function, then |R(f )| ≤ Cn (b − a)n+1 kf (n+1) k1 , where kgk1 =
Rb a
|g(x)|dx. Cn , n = 0, 1, 2, 3, are constants defined in Theorem 1.
Using inequalities of Simpson’s type from Theorem 1 we can obtain the following estimation of remainder term R(f, σn ) in Simpson’s quadrature formula Z
a
b
f (x)dx = A(f, σn ) + R(f, σn ),
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242
where σn is a partition of the interval [a, b], i.e. σn = {a = x0 , x1 , x2 , . . . , xm = b; a < x1 < · · · < xm−1 < b} and A(f, σn ) is equal to m−1 m−1 1 X 2 X xi + xi+1 [f (xi ) + f (xi+1 )]hi + f hi . 6 i=0 3 i=0 2 We have the following estimations for R(f, σn ). Corollary 3. Let n ∈ {0, 1, 2, 3}. Let f be a function on [a, b] such that f (n) is a function of bounded variation and σn be a partition of [a, b]. Then the remainder term R(f, σn ) in Simpson’s quadrature formula (8) satisfies: |R(f, σn )| ≤ Cn Vab (f (n) ) · max{hn+1 : i = 0, . . . , n − 1} i where hi = xi+1 − xi , i = 0, 1, . . . , m − 1, and Cn are defined as in Theorem 1. Remark 1. Similary we can improve results related to special means given in [1]. References [1] S. S. Dragomir, On Simpson’s quadrature formula for mappings of bounded variation and applications, Tamkang J. Math., 30(1999), 53-58. [2] A. Ghizzetti and A. Ossicini, Quadrature formulae, International series of numerical mathematics, Vol. 13, Birkh¨ auser Verlag Basel-Stuttgart, 1970.
Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia. E-mail:
[email protected] Department of Mathematics, University of Zagreb, Bijeniˇcka 30, 10000 Zagreb, Croatia. E-mail:
[email protected] 