ON WEIGHTED SIMPSON TYPE INEQUALITIES AND ... - RGMIA

ON WEIGHTED SIMPSON TYPE INEQUALITIES AND APPLICATIONS KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR

Abstract. In this paper we establish some weighted Simpson type inequalities and give several applications for the r − moments and the expectation of a continuous random variable. An approximation for Euler’s Beta mapping is given as well.

1. Introduction The Simpson’s inequality, states that if f (4) exists and is bounded on (a, b), then Z

b b − a f (a) + f (b) a + b (b − a)5

(4) + 2f (1.1) f (t)dt − ≤

f , a 3 2 2 2880 ∞ where

(4)

f

∞

:= sup f (4) (t) < ∞. t∈(a,b)

Now if we assume that In : a = x0 < x1 < · · · < xn = b is a partition of the Rb interval [a, b] and f is as above, then we can approximate the integral a f (t) dt by the Simpson’s quadrature formula AS (f, In ), having an error given by RS (f, In ), where n−1 X li f (xi ) + f (xi+1 ) xi + xi+1 + 2f , (1.2) AS (f, In ) := 3 2 2 i=0 Rb and the remainder RS (f, In ) = a f (t) dt − AS (f, In ) satisfies the estimation (1.3)

|RS (f, In )| ≤

n−1 1

(4) X 5 li ,

f 2880 ∞ i=0

with li := xi+1 − xi for i = 0, 1, . . . , n − 1. For some recent results which generalize, improve and extend this classic inequality (1.1), see the papers [2] – [7] and [9] – [12]. Recently, Dragomir [6], (see also the survey paper authored by Dragomir, Agarwal and Cerone [7]) has proved the following two Simpson type inequalities for functions of bounded variation: Theorem 1. Let f : [a, b] → R be a mapping of bounded variation. Then Z b b _ b − a f (a) + f (b) a + b 1 (1.4) f (t)dt − + 2f ≤ (b − a) (f ) , a 3 3 2 2 a Date: March 15, 2004. 2000 Mathematics Subject Classification. Primary 26D15, 26D10; Secondary 41A55. Key words and phrases. Simpson Inequality, Weighted Inequalities, Quadrature Rules. 1

2

KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR

Wb where a (f ) denotes the total variation of f on the interval [a, b] . The constant is the best possible.

1 3

Let In , li (i = 0, 1, . . . , n − 1), AS (f, In ) and RS (f, In ) be as above. We have Rb the following result concerning the approximation of the integral a f (t)dt in terms of AS (f, In ) . Theorem 2. Let f be defined as in Theorem 1. Then the remainder Z b (1.5) RS (f, In ) = f (x)dx − AS (f, In ) a

satisfies the estimate b

(1.6)

|RS (f, In )| ≤

_ 1 ν (l) (f ) , 3 a

where ν (l) := max {li |i = 0, 1, . . ., n − 1 } . The constant

1 3

is best posible in (1.6).

In this paper, we establish some generalizations of Theorems 1 – 2, and give several applications for the r − moments and expectation of a continuous random variable. Approximations for Euler’s Beta mapping are also provided. 2. Some Integral Inequalities We may state and prove the following main result: Rx Theorem 3. Let g : [a, b] → R be positive and continuous and let h(x) = a g(t)dt, x ∈ [a, b]. Let f be as in Theorem 3. Then Z Z b b 1 f (a) + f (b) −1 (2.1) f (t)g (t) dt − + 2f h (x) g (t) dt a 3 2 a b 1 h (b) _ ≤ h(b) + x − · (f ) , 3 2 a i h Wb h(b) 5h(b) for all x ∈ 6 , 6 , where a (f ) denotes the total variation of f on the interval [a, b] . The constant 13 is the best possible. i h 5h(b) Proof. Fix x ∈ h(b) , . Define 6 6 ( h (t) − h(b) , t ∈ a, h−1 (x) 6 . s (t) := −1 h (t) − 5h(b) (x) , b 6 , t∈ h By integration by parts, we have the following identity Z b (2.2) s (t) df (t) a " # Z h−1 (x) h (b) h−1 (x) = h (t) − f (t) |a − f (t)g (t) dt 6 a " # Z b 5h (b) b + h (t) − f (t) |h−1 (x) − f (t)g (t) dt 6 h−1 (x)

SIMPSON TYPE INEQUALITIES

3

Z b 1 f (a) + f (b) h (b) + 2f h−1 (x) − f (t)g (t) dt 3 2 a Z b Z b 1 f (a) + f (b) −1 + 2f h (x) g (t) dt − f (t)g (t) dt. = 3 2 a a =

It is well known (see for instance [1, p. 159]) that, if µ, ν : [a, b] → R are such that Rb µ is continuous on [a, b] and ν is of bounded variation on [a, b], then a µ (t) dν (t) exists and [1, p. 177] Z b b _ (ν) . (2.3) µ (t) dν (t) ≤ sup |µ (t)| a t∈[a,b] a Now, using (2.2) and (2.3), we have Z Z b b 1 f (a) + f (b) (2.4) f (t)g (t) dt − + 2f h−1 (x) g (t) dt a 3 2 a ≤ sup |s (t)| t∈[a,b]

b _

(f ) .

a

−1 Since h (t)− h(b) (x) , h (t)− 5h(b) is increasing on h−1 (x) , b 6 is increasing on a, h 6 1 and the fact that max{c, d} = c+d 2 + 2 |c − d| for any real c and d, hence we have h (b) h (b) 5h (b) sup |s (t)| = max ,x − , −x 6 6 6 t∈[a,b] and

(2.5)

h (b) h (b) 5h (b) ,x − , −x 6 6 6 h (b) 5h (b) = max x − , −x 6 6 h (b) 5h (b) 1 x− + −x = 2 6 6 1 h (b) 5h (b) + x− − − x 2 6 6 h (b) h(b) = + x − 3 2 Z b Z 1 1 b g (t) dt + x − g(t)dt . = 3 a 2 a

sup |s (t)| = max t∈[a,b]

Thus, by (2.4) and (2.5), we obtain the desired inequality (2.1). Let us consider the particular functions: g (t) ≡ 1, t ∈ [a, b] , h (t) = t − a, t ∈ [a, b] , ∪ 1 as t ∈ a, a+b 2 f (t) = −1 as t = a+b 2

a+b 2 ,b

4


and x = b−a 2 . Since for these choices we get equality in (2.1), it is easy to see that the constant 13 is the best possible constant in (2.1). This completes the proof. Remark 1. (1) If we choose g (t) ≡ 1, h (t) = t − a on [a, b] and x = b−a 2 , then the inequality (2.1) reduces to (1.4). (2) If we choose x = h(b) 2 , then we get Z Z b b 1 f (a) + f (b) h(b) −1 (2.6) f (t)g (t) dt − + 2f h g (t) dt a 3 2 2 a Z b _ 1 b ≤ g (t) dt · (f ) . 3 a a Under the conditions of Theorem 3, we have the following corollaries. Corollary 1. Let f ∈ C (1) [a, b] . Then we have the inequality Z Z b b 1 f (a) + f (b) −1 + 2f h (x) g (t) dt (2.7) f (t)g (t) dt − a 3 2 a " Z # h(b) 1 b g (t) dt + x − kf 0 k1 , ≤ 3 a 2 h i 5h(b) , , where k·k1 is the L1 −norm, namely for all x ∈ h(b) 6 6 0

kf k1 :=

Z

b

|f 0 (t)| dt.

a

Corollary 2. Let f : [a, b] → R be a Lipschitzian mapping with the constant M > 0. Then we have the inequality Z Z b b 1 f (a) + f (b) −1 (2.8) f (t)g (t) dt − + 2f h (x) g (t) dt a 3 2 a " Z # 1 b h(b) ≤ g (t) dt + x − (b − a) M, 3 a 2 i h 5h(b) for all x ∈ h(b) , . 6 6 Corollary 3. Let f : [a, b] → R be a monotonic mapping. Then we have the inequality Z Z b b 1 f (a) + f (b) −1 (2.9) f (t)g (t) dt − + 2f h (x) g (t) dt a 3 2 a " Z # 1 b h(b) ≤ g (t) dt + x − · |f (b) − f (a)| 3 a 2 i h 5h(b) . for all x ∈ h(b) 6 , 6


5

3. Applications for Quadrature Formulae Throughout this section, let g, h be as in Theorem 3, f : [a, b] → R, R xand let In : a = x0 < x1 < · · · < xn = b be a partition of [a, b] , and hi (x) = xi g(t)dt, h i x ∈ [xi , xi+1 ], ξ i ∈ h(x6i+1 ) , 5h(x6i+1 ) (i = 0, 1, . . ., n − 1) are intermediate points. Rx Put Li := hi (xi+1 ) = xii+1 g (t) dt and define the sum

AS (f, g, In , ξ) :=

n−1 X i=0

Li f (xi ) + f (xi+1 ) + 2f h−1 (ξ i ) 3 2

and Z

b

f (t)g(t)dx − AS (f, g, In , ξ) .

RS (f, g, In , ξ) = a

We have the following approximation of the integral

Rb a

f (t)g (t) dt.

Theorem 4. Let f be defined as in Theorem 3 and let b

Z (3.1)

f (t)g (t) dt = AS (f, g, In , ξ) + RS (f, g, In , ξ) . a

Then, the remainder term RS (f, g, h, In , ξ) satisfies the estimate |RS (f, g, h, In , ξ)| b _ 1 ξ i − hi (xi+1 ) ν (L) + max (f ) ≤ i=0,1,...,n−1 3 2 a

(3.2)

b

≤

_ 2 ν (L) (f ) , 3 a

where ν (L) := max {Li |i = 0, 1, . . ., n − 1 } . The constant of (3.2) is the best possible.

1 3

in the first inequality

Proof. Apply Theorem 3 on the intervals [xi , xi+1 ] (i = 0, 1, . . ., n − 1) to get Z

xi+1

xi

li f (xi ) + f (xi+1 ) −1 f (t)g (t) dt − + 2f hi (ξ i ) 3 2 xi+1 1 hi (xi+1 ) _ ≤ Li + ξ i − (f ) , 3 2 x i

6


for all i = 0, 1, . . ., n − 1. Using this and the generalized triangle inequality, we have |RS (f, g, In , ξ)| n−1 X Z xi+1 Li f (xi ) + f (xi+1 ) −1 f (t)g (t) dt − + 2f hi (ξ i ) ≤ 3 2 x i i=0 xi+1 n−1 X 1 hi (xi+1 ) _ ≤ Li + ξ i − (f ) 3 2 xi i=0 n−1 xi+1 1 hi (xi+1 ) X _ ≤ max Li + ξ i − (f ) i=0,1,...,n−1 3 2 i=0 xi b _ 1 ξ i − hi (xi+1 ) ≤ ν (L) + max (f ) i=0,1,...,n−1 3 2 a and the first inequality in (3.2) is proved. For the second inequality in (3.2), we observe that ξ i − hi (xi+1 ) ≤ 1 Li (i = 0, 1, . . ., n − 1); 3 2 and then

ξ i − h(xi ) + h(xi+1 ) ≤ 1 ν (L) . 3 i=0,1,...,n−1 2 Thus the theorem is proved. max

Remark 2. If we choose g (t) ≡ 1, then h (t) = t − a on [a, b] , ξ i = 0, 1, . . ., n − 1), and the first inequality in (3.2) reduces to (1.6).

xi+1 −xi 2

(i =

The following corollaries are useful in practice. Corollary 4. Let f : [a, b] → R be a Lipschitzian mapping with the constant M > 0, In be defined as above and choose ξ i = hi (x2i+1 ) (i = 0, 1, . . ., n − 1). Then we have the formula Z b (3.3) f (t)g (t) dt = AS (f, g, In , ξ) + RS (f, g, In , ξ) a

=

n−1 X i=0

Li f (xi ) + f (xi+1 ) −1 + 2f hi (ξ i ) + RS (f, g, In , ξ) 3 2

and the remainder satisfies the estimate (3.4)

|RS (f, g, In , ξ)| ≤

ν (L) · M · (b − a) . 3

Corollary 5. Let f : [a, b] → R be a monotonic mapping and let ξ i (i = 0, 1, . . ., n − 1) be defined as in Corollary 4. Then we have the formula (3.3) and the remainder satisfies the estimate ν (L) · |f (b) − f (a)| . 3 The case of equidistant division is embodied in the following corollary and remark:

(3.5)

|RS (f, g, In , ξ)| ≤


7

Rx Corollary 6. Suppose that G(x) = a g(t)dt, x ∈ [a, b], ! Z i b −1 xi = G g(t)dt (i = 0, 1, . . ., n), n a Z

x

g(t)dt, x ∈ [xi , xi+1 ], (i = 0, 1, . . ., n − 1),

hi (x) = xi

and Li := hi (xi+1 ) = G(xi+1 ) − G(xi ) =

1 n

Z

b

g (t) dt (i = 0, 1, . . ., n − 1) . a

Let f be defined as in Theorem 4 and choose ξ i = hi (x2i+1 ) (i = 0, 1, . . . , n − 1). Then we have the formula Z b (3.6) f (t)g (t) dt = AS (f, g, h, In , ξ) + RS (f, g, h, In , ξ) a

Z b n−1 1 X f (xi ) + f (xi+1 ) hi (xi+1 ) −1 = + 2f hi g (t) dt 3n i=0 2 2 a + RS (f, g, h, In , ξ) and the remainder satisfies the estimate b

(3.7)

|RS (f, g, h, In , ξ)| ≤

1 _ (f ) 3n a

Z

b

g (t) dt. a

Rb Remark 3. If we want to approximate the integral a f (t) g (t) dt by AS (f, g, h, In , ξ) with an error less that ε > 0, then we need at least nε ∈ N points for the partition In , where " Z # b b _ 1 nε := g (t) dt · (f ) + 1 3ε a a and [r] denotes the Gaussian integer of r ∈ R. 4. Some Inequalities for Random Variables Throughout this section, let 0 < a < b , r ∈ R , and let X be a continuous random variable having the continuous probability density function g : [a, b] → [0, ∞) and assume the r−moment, defined by Z b Er (X) := tr g (t) dt, a

is finite. Theorem 5. The inequality r 1 r r r Er (X) − 1 ar + 4 h−1 1 (4.1) + b ≤ 3 |b − a | 6 2 Rt holds, where h (t) = a g (x) dx (t ∈ [a, b]).

8


1 Proof. If we put f (t) = tr and x = h(b) 2 = 2 in Corollary 3, then we obtain the inequality Z Z b b 1 f (a) + f (b) 1 −1 (4.2) f (t)g (t) dt − + 2f h g (t) dt a 3 2 2 a Z b 1 g (t) dt. ≤ |f (b) − f (a)| 3 a Since Z b Z b f (t)g (t) dt = Er (X) , g (t) dt = 1, a

a

f (a) + f (b) ar + br = , and |f (b) − f (a)| = |br − ar | , 2 2 (4.1) follows from (4.2). If we choose r = 1 in Theorem 5, then we have the following remark: Remark 4. If E(X) is the expectation of random variable X, then b−a 1 1 −1 (4.3) + b ≤ . E (X) − 6 a + 4h 2 3 5. Inequality for the Beta Mapping The following mapping is well-known in the literature as the Beta mapping: Z 1 q−1 tp−1 (1 − t) dt, p > 0, q > 0. β (p, q) := 0

The following result may be stated: Theorem 6. Let p > 0, q > 1. Then the inequality  "  p1 #q−1 " p1 #q−1 n−1  X 1 1 i i+1  (5.1) β (p, q) − 1− + 1−  np 6 n n i=0  " 1 #q−1  2 2i + 1 p ≤ 1 + 1−  3np 3 2n holds for any positive integer n. q−1

Proof. If we put a = 0, b = 1, f (t) = (1 − t) , g(t) = tp−1 and G (t) = (t ∈ [0, 1]) in Corollary 6, then, p1 Z b 1 i g(t)dt = , xi = (i = 0, 1, . . ., n), p n a nxp − i (x ∈ [xi , xi+1 ], i = 0, 1, . . ., n − 1), np 1 hi (xi+1 ) 2i + 1 p −1 = (i = 0, 1, . . ., n − 1) hi 2 2n

hi (x) =

and

Wb

a (f )

= 1, so that the inequality (5.1) holds.

tp p


9

References [1] T. M. Apostol, Mathematical Analysis, Second Edition, Addision-Wesley Publishing Company, 1975. [2] D. Cruz-Urible and C. J. Neugebauer, Sharp error bounds for the trapezoidal rule and Simpson’s rule, J. Inequal. Pure Appl. Math. 3(2002), no. 4, Article 49, 22 pp. [Online:http://jipam.vu.edu.au/v3n4/031 02.html]. ˇ [3] V. Culjak, J. Peˇ cari´ c and L. E. Persson, A note on Simpson’s type numerical integration, Soochow J. Math. 29(2003), no. 2, 191-200. [4] S. S. Dragomir, On Simpson’s quadrature formula and applications, Mathematica 43(66) (2001), no. 2, 185-194 (2003). [5] S. S. Dragomir, On Simpson’s quadrature formula for Lipschitzian mappings and applications, Soochow J. Math. 25(1999), no. 2, 175-180. [6] S. S. Dragomir, On Simpson’s quadrature formula for mappings of bounded variation and applications, Tamkang J. of Math. 30(1999), no. 1, 53-58. [7] S. S. Dragomir, R. P. Agarwal and P. Cerone, On Simpson’s inequality and applications, J. Inequal. Appl. 5(2000), no. 6, 533-579. [8] L. Fej´ er, Uberdie Fourierreihen, II, Math. Natur. Ungar. Akad Wiss. 24(1906), 369-390. [In Hungarian]. [9] J. Peˇ cari´ c and S. Varoˇsanec, A note on Simpson’s inequality for Lipschitzian functions, Soochow J. Math. 27(2001), no. 1, 53-57. [10] J. Peˇ cari´ c and S. Varoˇsanec, A note on Simpson’s inequality for functions of bounded variation, Tamkang J. of Math. 31(2000), no. 3, 239-242. [11] N. Ujevi´ c, New bounds for Simpson’s inequality, Tamkang J. of Math. 33(2002), no. 2, 129138. [12] G. S. Yang and H. F. Chu, A note on Simpson’s inequality for function of bounded variation, Tamsui Oxford J. Math. Sci. 16(2000), no. 2, 229-240. (Kuei-Lin Tseng) Department of Mathematics, Aletheia University, Tamsui, Taiwan 25103 E-mail address, Kuei-Lin Tseng: [email protected] (Gou-Sheng Yang) Department of Mathematics, Tamkang University, Tamsui, Taiwan 25137 (Sever S. Dragomir) School of Computer Science & Mathematics, Victoria University, Melbourne, Victoria, Australia E-mail address, Sever S. Dragomir: [email protected] URL: http://rgmia.vu.edu.au/SSDragomirWeb.html