Generalizations of weighted trapezoidal inequality for mappings of bounded variation and their applications

MATHEMATICAL AND COMPUTER MODELLING

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Mathematical and Computer Modelling 40 (2004) 77-84 www.elsevier.com/locate/mcm

Generalizations of Weighted Trapezoidal Inequality for Mappings of Bounded Variation and Their Applications KUEI-LIN TSENG Department of Mathematics, Aletheia University Tamsui, Taiwan 25103 kltseng©email, au. edu. tw

G o U - S H E N G YANG Department of Mathematics, Tamt~ng University Tamsui, Taiwan 25137 S. S. D R A G O M I R School of Computer Science & Mathematics, Victoria University Melbourne, Victoria 8001, Australia sever@matilda, wa. edu. au

(Received March 2003; revised and accepted December 2003) A b s t r a c t - - I n this paper, we establish some generalizations of weighted trapezoid inequality for mappings of bounded variation, and give several applications for r-moment, the expectation of a continuous random variable and the Beta mapping. @ 2004 Elsevier Ltd. All rights reserved. K e y w o r d s - - T r a p e z o i d inequality, Mappings of bounded variation, Random variable, r-moments, Expectation, Beta function.

1. I N T R O D U C T I O N The trapezoid inequality states that if f " exists and is bounded on (a, b), then,

ff

b- a

f (x) d x - T [ f

(a) + f (b)] <_

(b - a) ~ 12 ]lf"llo~

(i.i)

where

i]f"[]~ := sup If"l < oo. ze(a,b)

Now, if we assume t h a t In : a = Xo < xl < ... < xn = b is a partition of the interval [a, b] and f is as above, then, we can approximate the integral J: f (x) dx by the trapezoidal quadrature formula AT (f, IN), having an error given by RT (f, In), where 1 n-1

AT (f, In) := ~ ~

[f (xi) + f (x,+l)] li,

i=0 0895-7177/04/$ - see front matter (~) 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2003.12.004

Typeset by ~4~-TEX

K.-L. TSP.NG, e~ al.

78

and the remainder satisfies the estimation n--1

IRT (S, In)l _<

JLS"ll 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 [l-S]. Recently, Cerone, Dragomir, and Pearce [3] proved the following two trapezoid type inequalities. THEOREM 1. Let f : [a, b] -~ R be a mapping of bounded variation. Then,

for all x e [a, b], where Vb(f) denotes the total variation o f f on the interval [a, b]. The constant 1/2 is the best possible. Let In, li (i = 0 , 1 , . . . , n - 1) be as above and let ~i e [xi,x~+l] (i = 0 , 1 , . . . , n - 1) be intermediate points. Define the sum n--1

Tp (f, In, ~) := ~

[(a - x~) f (x~) + (x~+l - [~) f (x~+l)].

i=0

We have the following result concerning the approximation of the integral f : f(x) dx in terms of Tp. THEOREM 2. Let f be defined as in Theorem 1, then, we have

~

b f(x) dx = Tp (f, In, ~) + R p (f, In, ~).

The remainder term R p (f, In, ~), satisfies the estimate ' R P ( f ' I ~ ' ~ ) [ < [ l ~ ( l ) + m a x ~i=0,1,...,~-1 i x ~ + x i + l l v b ( f ) ] < v 2( l ) V b ( f ) -

_

,

(1.3)

where u(I) := max{ll]i = 0, 1 , . . . , n - 1). The constant 1/2 is the best possible. In this paper, we establish weighted generalizations of Theorems 1-2, and give several applications for r-moment, the expectation of a continuous random variable and the Beta mapping. 2. S O M E

INTEGRAL

INEQUALITIES

We may state the following result. THEOREM 3. Let g : [a, b] --~ R be nonnegative and continuous and let h : [a, b] ~

R be

differentiable such that h'(t) = g(t) on [a, b]. Suppose f is defined as in Theorem 1. Then, f(t)g (t) dt - [(x - h(a)) / (a) + (h(b) - x) f (b)] (2.1) <

~

g(t) d r + x -

2

for aii x 6 [h(a), h(b)]. The constant 1/2 is the best possible.

Weighted Trapezoidal InequMity

79

PROOF. Let x E [h(a), h(b)]. Using integration by parts, we have the following identity

~ b(~ - h(t)) d/(t) = I(~ - h(t)) f (t)]b~ + f bf(t)g (t) dt (2.2)

b

= ~ f(t)g (t) dt - [(x - h(a)) f (a) + (h(b) - x) f (b)]. It is well known [9, p. 159] that if #, ~ : [a, b] --* R are such that # is continuous on [a, b] and ~, is of bounded variation on [a,b], then f : #(t)d~(t) exists and [9, p. 177]

ff #(t) d~, (t)

(2.3)

_< sup I#(t)lvb(p). ~e[a,b]

Now, using identity (2.2) and inequality (2.3), we have

ff f(t)g (t) dt -

[(x - h(a)) f (a) + (h(b) - x)

f (b)] <

sup Ix - h(t)] V~ ( f ) . te[~,b]

(2.4)

Since x - h(t) is decreasing on [a, b], h(a) < x < h(b), and h'(t) = g(t) on [a, b], we have sup Ix - h (t)l = max{x - h (a), h (b) - x} te[a,b]

h(b)-h(a) --

2

x +

1 [~

h(a)+h(b)

(2.5)

2

h (a) + h (b)

= 2 Ja g(t) dr+ x

2

"

Thus, by (2.4) and (2.5), we obtain (2.1). Let

g (t) - 1, t e [a, b], h(t)=t, f(t)=

te[a,b], 0,

as t ~ a,

1,

aste(a,b),

O, as t = b, and x = (a + b)/2. Then, we can see that the constant 1/2 is best possible. This completes the proof. | REMARK 1. (1) If we choose g(t) -- 1, h(t) = t on [a, hi, then, the inequality (2.1) reduces to (1.2). (2) If we choose x -- (h(a) + h(b))/2, then, we get

~ b f(t)g (t) dt

f (a) +2 f (b) ~ b g (t) dt <_ ~l f~ b g (t) dr. Vb (f) ,

which is the "weighted trapezoid" inequality. Under the conditions of Theorem 3, we have the following corollaries.

(2.6)

80

K.-L. TSENO, et al.

COROLLARY 1. Let f E C (1) [a, b]. Then, we have the inequality

ff

f(t)g (t) d t - [(x - h(a)) f (a) + (h(b) -

x) f (b)] (2.7)

<_

g(t) dt+

for a/l x e [h(a), h(b)], where I1" II1 is the

-

2 " "

[If'Ill,

L1 - norm, name/y, /. b

[If'Ill := Ja If' (t)l dr. COROLLARY 2. Let f : [a, b] --* R be a Lipschitzian mapping with the constant L > O. Then, we

have the inequality

li

bf(t)g(t) dt - [ ( x - h(a)) f (a) + (h(b) - x) f (b)l

(2.8)

< [ l~i b

g (t) dt + x

h(a)+ 2 h(b)](b-a)L,

for all x • [h(a), h(b)l. COROLLARY 3. Let f : [a, b] --* R be a monotonic mapping. Then, we have the inequality

l

b f(t)g

(t) dt - [(x - h(a)) f (a) + (h(b) - x) f (b)] (2.9)

<_

g (t) dt +

-

=for ali x • [h(a), h(b)]. REMARK 2. The following inequality is well known in literature as the Fej~r inequality (see for example [10]):

f (_~)

/ b g (t) dt <_f b f(t)g (t) dt <_ f (a)-q2 f (b) f f f g (t) dr,

(2.10)

where f : [a, b] -~ R is convex and g : [a, b] --* R is nonnegative integrable and symmetric to (a + b)/2. Using the above results and (2.6), we obtain the following error bound of the second inequality in (2.10),

O -< f (a) + l f~ b g (t) dr. Vba (f), 2 f (b) i b g (t) dt - i b f(t)g (t) dt <_ -~

(2.11)

provided that f is of bounded variation on [a, b]. REMARK 3. If f is convex and Lipschitzian with the constant L on [a, b], g is defined as in Remark 2 and x = (h(a) + h(b))/2, then, we get from (2.8) and (2.10) that

-

2

g (t) d t -

f(t)g (t) dt<

~

g (t) dr.

(2.12)

REMARK 4. If f is convex and monotonic on [a, b], g is defined as in Remark 2 and x = (h(a) + h(b))/2, then, we get from (2.9) and (2.10) that

o -< f (a) +2 f (b) ~b

g (t) dt -

f bf(t)g

(t)

dt <_ if (b) _2 f (a)l f b

g (t) dr.

(2.13)

REMARK 5. If f is continuous, differentiable, convex on [a, b] and f' • Ll(a, b), g is defined as in Remark 2 and x = (h(a) + h(b))/2, then, we get ~ o m (2.7) and (2.10) that

o< - f (a)"k 2 f (b) ~ b g (t) d t - i b f(t)g (t) d t < Ill'Ill 2 ~ b g (t) dt.

(2.14)

Weighted Trapezoidal Inequality

3. A P P L I C A T I O N S

FOR

81

QUADRATURE

RULES

Throughout this section, let g and h be defined as in Theorem 3. Let f : [a,b] ~ R, and let In : a = x0 < xl < ... < xn = b be a partition of [a,b] and ~ e [h(x~), h(x~+l)] (i = 0, 1 , . . . , n - 1) be intermediate points. Put t~ :-- h(xi+]) - h(xi) = f~/+l g(t) dt and define the sum n--i

Tp (f, g, h, In, ~) := E [(~i - h(xi)) f (x~) + (h(x~+l) - ~) f (x~+t)].

i=O We have the following concerning the approximation of the integral f : f(t)g(t) dt in terms of Tp. THEOREM 4. Let f be de~ned as in Theorem 3 and let

f b/(~)g (t) dt = Tp (f, g, h, ~ , ~) + n p (/, g, h, I~, ~).

(3.1)

Then, the remainder term R p ( f , g, h, In, ~) satisfies the estimate IRP(f'g'h'I~'~)l-< [ l u (/) + i=0,L...,n-lmax

- h(xi)+h(xi+l)2

V~(f) _
(3.2)

where u(1) := max{/~li = 0, 1,... , n - 1}. The conste~ut 1/2 is the best possible. PROOF. Apply Theorem 3 on the intervals [x~, x~+~] (i = 0, 1 , . . . , n - 1) to get

f~+~ f(t)g (t) dt -

[(~i - h(xi)) f (xi) + (h(xi+l) - ~) f

i

(x~+~)]

for all i e { 0 , 1 , . . . , n - 1}. Using this and the generalized triangle inequality, we have n--1

IRp (f, g, h, In, ~)1 <- E

f(t)g (t) dt - [(~i - h(x~)) f (x~) + (h(xi+l) - ~)

i=0 J x~

<-- En-l [2 li + ~i --

h(xi) + h(xi+l) ] y~x~+~ x~

i=0

< --

max 1l h(x~) ~=o,1.....~-1 ~ ~ + ~ -

[ I

•

<

~U (1) +

max

(xi+l)

N" V~+ ~ A-~

i=O

x~

(f)

--

i=0,1,...,n--1

- -

?

2

'

and the first inequality in (3.2) is proved. For the second inequality in (3.2), we observe that

I~i-h(x~)+h(x~+1)l
(i=0,1,

•

.. n - l ) ; ,

and then

max

~ _ h(x~) ?(x~+~) I < 1 (1).

i=0,1,...,n-- 1

Thus, the theorem is proved.

|

we choose g(t) ==-1, h(t) = t on [a, b], then, the inequality (3.2) reduces to (1.3). The following corollaries can be useful in practice.

R E M A R K 6. I f

K.-L. TSENG, et

82

al.

COROLLARY 4. Let f : [a, b] --~ R be a Lipschitzian mapping with the constant L > O, I,~ be defined as above and choose ~ = (h(x~) + h(~+~))/2 (i = O, 1 , . . . , n - 1). Then, we ha~e the formula b f(t)g (t) dt = Tp (f, g, h, In, ~) + R p (f, g, h, In, ~)

s(~) + s(~+~)

=

i=O

r

(3.3)

]

~÷~ g(t) dt + R p ( f , g , h , In,~)

and the remainder satisfies the estimate ]Rp (S,g, h, In,~)[ < ~ (1). L . (b - a) -

i3.4 )

2

COROLLARY 5. Let f : [a, b] --~ R be a monotonic mapping and let ~ ( i = O, 1 , . . . , n - 1)

be defined as in Corollary 4. Then, we have the formula (3.3) and the remainder satisfies the estimate (3.5) IRp (f,g,h, In,~)l -< ~(l) 2 "[f(b)-f(a)l The case of equidistant division is embodied in the following corollary and remark. COROLLARY 6. Suppose that g(t) > O, x i = h -1 h ( a ) +

i(h(b) - h(a)) ] n

(i= O,1,...,n)

and

z~ :-- h(x~+l) - h(x~) - h(b) -n h(a) = n1 --jo~ git) dt

(i = 0 , 1 , . . . , n -

1).

Let f be defined as in Theorem 4 and choose ~ = ( h( ~, ) + h( x,+ ~) ) /2 (i = 0, 1,..., n- 1). Then, we have the formula b f(t)g (t) dt --- Tp (f, g, h, In, ~) + Rp (f, g, h, In, ~)

1 /b

=2-~n

(3.6)

n-1

g(t) dt.~_~[f(x~)+f(x~+l)]+Rp(f,g,h,I,~,(), i-~-O

and the remainder satisfies the estimate

1/b git) dt'Vbif)"

]Rp(f,g,h, In,~)l <_~n

(3.7)

REMARK 7. If we want to approximate the integral J~ f (t) g (t) dt by Tp if, g, h, In, ~) with an accuracy less that e > 0, we need at least ne C N points for the partition In, where

n~:=

[1; K

git) dt. V2if)

and [r] "denotes the Gaussian integer of r E R.

1

+1,

Weighted Trapezoidal Inequality

83

4. SOME INEQUALITIES FOR R A N D O M VARIABLES Throughout this section, let 0 < a < b, r E R, and let X be a continuous random variable having the continuous probability density function g : [a, b] --* R and the r-moment b

E~ ( x ) :=

f trg (t) dr,

which is assumed to be finite. THEOREM 5. The inequality

E~(X)

a~ +b~ t < i Ib~-a~l 2 -3

(4.1)

holds. PROOF. If we put f(t) = U, h(t) = f~t 9(x) dx (t E [a, b]) and x = (h(a) + h(b))/2 in Corollary 3, we obtain the inequality

/b

f(t)g(t) dt

f (a) + f (b) / b dt 1/b 2 g(t) <_ -~ g(t) dt. I f ( b ) - f ( a ) ] .

(4.2)

Since b ./. f ( t ) g ( t ) d t = Er ( X ) ,

f(a)+f(b) a~ +b ~ - - 2 2 '

r b

./. g(t) d t = 1, and

If (b) - f (a)l =

Ib r

- arl,

(4.1) follows from (4.2), immediately. This completes the proof. If we choose r = 1 in Theorem 5, then, we have the following remark. REMARK 8. The inequality

(4.3) --

2

where E(X) is the expectation of the random variable X. The following mapping is well known in literature as the Beta mapping:

fl(p,q)

1 :=

t P - l ( 1 - t ) q-1 dt,

p>O,

q>O.

THEOREM 6. Let p, q >_ 1. Then, the inequality 1

(P' q) - ~np i=o

+

1__(~_~)1/p

q-1

<--2np'

(4.4)

holds for any positive integer n. PROOF. Let p, q > 1. If we put a = 0, b = 1, f(t) = (1 - t) q-l, g(t) = t p-1 and h(t) := tP/p (t E [0, 1]) in Corollary 6, we obtain the inequality (4.4). This completes the proof. |

84

K.-L. TSENC, et al.

REFERENCES 1. N.S. Barnett, S.S. Dragomir and C.E.M. Pearce, A quasi-trapezoid inequality for double integrals, ANZIAM J. 44, 355-364, (2003). 2. P. Cerone, Weighted three point identities and their bounds, The SUT J. o/Math. 38 (1), 17-37. 3. P. Cerone, S.S. Dragomir and C.E.M. Pearce, Generalizations of the trapezoid inequality for mappings of bounded variation and applications, Turkish Journal of Mathematics 24 (2), 147-163, (2000). 4. P. Cerone and S.S. Dragomir, Handbook o/Analytic-Computational Methods in Applied Mathematics, (Edited by G. Anastassiou), CRC Press, New York, (2000). 5. P. Cerone, J. Roumeliotis and G. Hanna, On weighted three point quadrature rules, ANZIAM J 42 (E), C340-361, (2000). 6. S.S. Dragomir, On the trapezoid formula for mappings of bounded variation and applications, Kragujevac J. Math. 23, 25-36, (2001). 7. S.S. Dragomir, P. Cerone and A. So/o, Some remarks on the trapezoid rule in numerical integration, Indian J. o/Pure and Appl. Math. 31 (5), 475-494, (2000); RGMIA Res. Rep. Coll. 5 (2), Article 1, (1999). 8. S.S. Dragomir and T.C. Peachey, New estimation of the remainder in the trapezoidal formula with applications, Studia Universitatis Babes-Bolyai Ma~hema$iea XLV (4), 31-42, (2000). 9. T.M. Apostol, Mathematical Analysis, Second Edition, Addision-Wesley Publishing Company, (1975). 10. L. Fej~r, Uberdie Fourierreihen, II (In Hungarian), Math. Na~ur. Ungar. Akad Wiss. 24, 369-390, (1906).