GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS. (b a)3 [f(a) + f(b)] f x (a,b)

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1 GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Abstrct. In this pper, we estblish some generliztions of weighted trpezoid inequlity for monotonic mppings, nd give severl pplictions for r moment, the expecttion of continuous rndom vrible nd the Bet mpping. 1. Introduction The trpezoid inequlity, sttes tht if f exists nd is bounded on (, b), then (1.1) f(x)dx b (b )3 [f() f(b) f 1, where f := sup f <. x (,b) Now if we ssume tht I n : = x 0 < x 1 < < x n = b is prtition of the intervl [, b nd f is s bove, then we cn pproximte the integrl f (x) dx by the trpezoidl qudrture formul A T (f, I n ), hving n error given by R T (f, I n ), where (1.) A T (f, I n ) := 1 [f ( ) f (1 ) l i, nd the reminder stisfies the estimtion (1.3) R T (f, I n ) 1 1 f li 3, with l i := 1 for i = 0, 1,..., n 1. For some recent results which generlize, improve nd extend this clssic inequlity (1.1), see the ppers [ [8. Recently, Cerone-Drgomir [3 proved the following two trpezoid type inequlities: 1991 Mthemtics Subject Clssifiction. Primry: 6D15; Secondry: 41A55. Key words nd phrses. Trpezoid inequlity, Monotonic mppings, r moment nd the expecttion of continuous rndom vrible, the Bet mpping. 1

2 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Theorem A. Let f : [, b R be monotonic non-decresing mpping. Then (1.4) f(t)dt [(x ) f () (b x) f (b) (b x) f (b) (x ) f () sgn (x t) f (t) dt (x ) [f (x) f () (b x) [f (b) f (x) (b ) x b [f (b) f (), for ll x [, b. The bove inequlities re shrp. Let I n, l i (i = 0, 1,..., ) be s bove nd let ξ i [, 1 (i = 0, 1,..., ) be intermedite points. Define the sum T P (f, I n, ξ) := [(ξ i ) f ( ) (1 ξ i ) f (1 ). We hve the following result concerning the pproximtion of the integrl f(x)dx in terms of T P. Theorem B. Let f be defined s in Theorem A, then we hve (1.5) f(x)dx = T P (f, I n, ξ) R P (f, I n, ξ). The reminder term R P (f, I n, ξ) stisfies the estimte (1.6) R P (f, I n, ξ) [(1 ξ i ) f (1 ) (ξ i ) f ( ) xi1 sgn (ξ i t) f (t) dt (ξ i ) [f (ξ i ) f ( ) (1 ξ i ) [f (1 ) f (ξ i ) l i ξ i x i 1 [f (1 ) f ( ) ν (l) mx,1,..., ξ i x i 1 [f (b) f () ν (l) [f (b) f () where ν (l) := mx {l i i = 0, 1,..., n 1}. In this pper, we estblish weighted generliztions of Theorems A-B, nd give severl pplictions for r moments nd the expecttion of continuous rndom vrible, the Bet mpping nd the Gmm mpping.. Some Integrl Inequlities Theorem 1. Let g : [, b R be non-negtive nd continuous with g (t) > 0 on (, b) nd let h : [, b R be differentible such tht h (t) = g (t) on [, b.

3 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS 3 (.1) (.) () Suppose f : [, b R is monotonic non-decresing mpping. Then f(t)g (t) dt [(x h()) f () (h(b) x) f (b) (h(b) x) f (b) (x h()) f () sgn ( h 1 (x) t ) f (t) g (t) dt (x h()) [f ( h 1 (x) ) f () (h(b) x) [f (b) f ( h 1 (x) ) [ 1 b g (t) dt h() h(b) x [f (b) f () for ll x [h(), h(b). (b) Suppose f : [, b R is monotonic non-incresing mpping. Then f(t)g (t) dt [(x h()) f () (h(b) x) f (b) (x h()) f () (h(b) x) f (b) sgn ( t h 1 (x) ) f (t) g (t) dt (x h()) [f () f ( h 1 (x) ) (h(b) x) [f ( h 1 (x) ) f (b) [ 1 b g (t) dt h() h(b) x [f () f (b) for ll x [h(), h(b). The bove inequlities re shrp. Proof. (1) () Let x [h(), h(b). identity Using integrtion by prts, we hve the following (.3) (.4) (x h(t)) df (t) = (x h(t)) f (t) = b f(t)g (t) dt f(t)g (t) dt [(x h()) f () (h(b) x) f (b). It is well known [3, p. 813 tht if µ, ν : [, b R re such tht µ is continuous on [, b nd ν is monotonic non-decresing on [, b, then µ (t) dν (t) µ (t) dν (t).

4 4 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR (.5) Now, using identity (.3) nd inequlity (.4), we hve f(t)g (t) dt [(x h()) f () (h(b) x) f (b) = x h(t) df (t) h 1 (x) (x h(t)) df (t) (h(t) x) df (t) h 1 (x) = (x h(t)) f (t) (h(t) x) h 1 (x) b h 1 (x) h 1 (x) f (t) g (t) dt h 1 (x) = (h(b) x) f (b) (x h()) f () f (t) g (t) dt sgn ( h 1 (x) t ) f (t) g (t) dt nd the first inequlities in (.1) re proved. As f is monotonic non-decresing on [, b, we obtin nd then h 1 (x) h 1 (x) f (t) g (t) dt f ( h 1 (x) ) h 1 (x) g (t) dt = (x h()) f ( h 1 (x) ) f (t) g (t) dt f ( h 1 (x) ) h 1 (x) g (t) dt = (h(b) x) f ( h 1 (x) ), sgn ( h 1 (x) t ) f (t) g (t) dt (x h()) f ( h 1 (x) ) (x h(b)) f ( h 1 (x) ). Therefore, (.6) (h(b) x) f (b) (x h()) f () sgn ( h 1 (x) t ) f (t) g (t) dt (h(b) x) f (b) (x h()) f () (x h()) f ( h 1 (x) ) (x h(b)) f ( h 1 (x) ) = (x h()) [f ( h 1 (x) ) f () (h(b) x) [f (b) f ( h 1 (x) ) which proves tht the second inequlity in (.1). As f is monotonic non-decresing on [, b, we hve f () f ( h 1 (x) ) f (b)

5 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS 5 (.7) nd (x h()) [f ( h 1 (x) ) f () (h(b) x) [f (b) f ( h 1 (x) ) mx{x h(), h(b) x} [f ( h 1 (x) ) f () f (b) f ( h 1 (x) ) [ h(b) h() = h() h(b) x [f (b) f () [ 1 b = g (t) dt h() h(b) x [f (b) f (). Thus, by (.5), (.6) nd (.7), we obtin (.1). Let g (t) 1, t [, b h (t) = t, t [, b { 0, t [, b) f(t) = 1, t = b nd x = b. Then f(t)g (t) dt [(x h()) f () (h(b) x) f (b) = (h(b) x) f (b) (x h()) f () sgn ( h 1 (x) t ) f (t) g (t) dt = (x h()) [f ( h 1 (x) ) f () (h(b) x) [f (b) f ( h 1 (x) ) [ 1 b = g (t) dt h() h(b) x [f (b) f () = b which proves tht the inequlities (.1) re shrp. (b) If f is replced by f in (), then (.) is obtined from (.1). This completes the proof. Remrk 1. If we choose g (t) 1, h (t) = t on [, b, then the inequlities (.1) reduce to (1.4). Corollry 1. If we choose x = h()h(b), then we get f () f (b) b (.8) f(t)g (t) dt g (t) dt 1 1 g (t) dt [f (b) f () ( ( ) ) h() h(b) sgn h 1 t f (t) g (t) dt g (t) dt [f (b) f ()

6 6 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR where f nd g re defined s in () of Theorem 1, nd f () f (b) b (.9) f(t)g (t) dt g (t) dt 1 1 g (t) dt [f () f (b) ( ( )) h() h(b) sgn t h 1 f (t) g (t) dt g (t) dt [f () f (b) where f nd g re defined s in (b) of Theorem 1. The inequlities (.8) nd (.9) re the weighted trpezoid inequlities. Note tht the trpezoid inequlity (.8) nd (.9) re, in sense, the best possible inequlities we cn obtin from (.1) nd (.). Moreover, the constnt 1 is the best possible for both inequlities in (.8) nd (.9), respectively. Remrk. The following inequlity is well-known in the literture s the Fejér inequlity (see for exmple [9): ( ) b b f () f (b) b (.10) f g (t) dt f(t)g (t) dt g (t) dt, where f : [, b R is convex nd g : [, b R is positive integrble nd symmetric to b. Using the bove results nd (.8) (.9), we obtin the following error bound of the second inequlity in (.10): (.11) f () f (b) g (t) dt g (t) dt [f (b) f () f(t)g (t) dt ( ) sgn h 1 h() h(b) ( ) t f (t) g (t) dt g (t) dt [f (b) f () provided tht f is monotonic non-decresing on [, b. (.1) f () f (b) g (t) dt g (t) dt [f () f (b) f(t)g (t) dt ( ) sgn t h 1 h() h(b) ( ) f (t) g (t) dt g (t) dt [f () f (b) provided tht f is monotonic non-incresing on [, b.

7 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS 7 3. Applictions for Qudrture Formul Throughout this section, let g nd h be defined s in Theorem 1. Let f : [, b R, nd let I n : = x 0 < x 1 < < x n = b be prtition of [, b nd ξ i [h( ), h(1 ) (i = 0, 1,..., n 1) be intermedite points. Put l i := h(1 ) h( ) = 1 g (t) dt nd define the sum T P (f, g, h, I n, ξ) := [(ξ i h( )) f ( ) (h(1 ) ξ i ) f (1 ). We hve the following result concerning the pproximtion of the integrl f(t)g (t) dt in terms of T P. Theorem. Let ν (l) := mx {l i i = 0, 1,..., n 1}, f be defined s in Theorem 1 nd let (3.1) f(t)g (t) dt = T P (f, g, h, I n, ξ) R P (f, g, h, I n, ξ). Then, the reminder term R P (f, g, h, I n, ξ) stisfies the following estimtes: () Suppose f is monotonic non-decresing on [, b, then (3.) R P (f, g, h, I n, ξ) [(h (1 ) ξ i ) f (1 ) (ξ i h ( )) f ( ) xi1 sgn ( h 1 (ξ i ) t ) f (t) g (t) dt (ξ i h ( )) [ f ( h 1 (ξ i ) ) f ( ) (h (1 ) ξ i ) [ f (1 ) f ( h 1 (ξ i ) ) l i ξ i h (x i) h (1 ) [f (1 ) f ( ) ν (l) mx,1,..., ξ i h (x i) h (1 ) [f (b) f () ν (l) [f (b) f (). (b) Suppose f is monotonic non-incresing on [, b, then (3.3) R P (f, g, h, I n, ξ) [(ξ i h ( )) f ( ) (h (1 ) ξ i ) f (1 ) xi1 sgn ( t h 1 (ξ i ) ) f (t) g (t) dt

8 8 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR (ξ i h ( )) [ f ( ) f ( h 1 (ξ i ) ) (h (1 ) ξ i ) [ f ( h 1 (ξ i ) ) f (1 ) l i ξ i h (x i) h (1 ) [f ( ) f (1 ) ν (l) mx,1,..., ξ i h (x i) h (1 ) [f () f (b) ν (l) [f () f (b). Proof. (1) () Apply Theorem 1 on the intervls [, 1 (i = 0, 1,..., n 1) to get xi1 f(t)g (t) dt [(ξ i h( )) f ( ) (h(1 ) ξ i ) f (1 ) (h(1 ) ξ i ) f (1 ) (ξ i h( )) f ( ) xi1 sgn ( h 1 (ξ i ) t ) f (t) g (t) dt (ξ i h( )) [f ( h 1 (ξ i ) ) f ( ) (h(1 ) ξ i ) [f (1 ) f ( h 1 (ξ i ) ) l i ξ i h(x i) h(1 ) [f (1 ) f ( ) for ll i {0, 1,..., n 1}.

9 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS 9 (3.4) (3.5) Using this nd the generlized tringle inequlity, we hve R P (f, g, h, I n, ξ) xi1 f(t)g (t) dt [(ξ i h( )) f ( ) (h(1 ) ξ i ) f (1 ) [(h (1 ) ξ i ) f (1 ) (ξ i h ( )) f ( ) xi1 sgn ( h 1 (ξ i ) t ) f (t) g (t) dt (ξ i h ( )) [ f ( h 1 (ξ i ) ) f ( ) (h (1 ) ξ i ) [ f (1 ) f ( h 1 (ξ i ) ) l i ξ i h (x i) h (1 ) [f (1 ) f ( ) ν (l) mx,1,..., ξ i h (x i) h (1 ) [f (b) f (). Next, we observe tht ξ i h() h(1 ) 1 l i (i = 0, 1,..., n 1); nd then (3.6) mx,1,..., ξ i h() h(1 ) 1 ν (l). Thus, by (3.4), (3.5) nd (3.6), we obtin (3.). (b) The proof is similr s () nd we omit the detils. This completes the proof. Remrk 3. If we choose g (t) 1, h (t) = t on [, b, then the inequlities (3.) reduce to (1.6). Now, let ξ i = h(xi)h(xi1) (i = 0, 1,..., n 1) nd let T P W (f, g, h, I n ) nd R P (f, g, h, I n ) be defined s T P W (f, g, h, I n ) = T P (f, g, h, I n, ξ) = 1 xi1 [f ( ) f (1 ) g (t) dt nd R P W (f, g, h, I n ) = R P (f, g, h, I n, ξ) = f (t) g (t) dt 1 xi1 [f ( ) f (1 ) g (t) dt. If we consider the weighted trpezoidl formul T P W (f, g, h, I n ), then we hve the following corollry:

10 10 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Corollry. Let f, g, h be defined s in Theorem nd let ξ i = h(xi)h(xi1) (i = 0, 1,..., n 1). Then f (t) g (t) dt = T P W (f, g, h, I n ) R P W (f, g, h, I n ) where the reminder stisfies the following estimtes: (3.7) (3.8) () Suppose f is monotonic non-decresing on [, b, then R P W (f, g, h, I n ) 1 ( xi1 ) g (t) dt [f (1 ) f ( ) 1 xi1 ( xi1 ν (l) g (t) dt [f (b) f (). ( ( ) ) h sgn h 1 (xi ) h (1 ) t f (t) g (t) dt ) [f (1 ) f ( ) (b) Suppose f is monotonic non-incresing on [, b, then R P W (f, g, h, I n ) 1 ( xi1 ) g (t) dt [f ( ) f (1 ) 1 xi1 ( xi1 ν (l) g (t) dt [f () f (b). ( ( )) h sgn t h 1 (xi ) h (1 ) f (t) g (t) dt ) [f ( ) f (1 ) Remrk 4. In Corollry, suppose f is monotonic on [, b, [ = h 1 i (h(b) h()) h () (i = 0, 1,..., n), n nd l i := h(1 ) h( ) = h(b) h() n = 1 n g (t) dt. (i = 0, 1,..., n 1). If we wnt to pproximte the integrl f (t) g (t) dt by T P W (f, g, h, I n ) with n ccurcy less tht ε > 0, we need t lest n ε N points for the prttion I n, where n ε := [ 1 ε g (t) dt f (b) f () nd [r denotes the Gussin integer of r (r R). 1

11 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS Some Inequlities for Rndom Vribles Throughout this section, let 0 < < b, r R, nd let X be continuous rndom vrible hving the continuous probbility density mpping g : [, b R with g (t) > 0 on (, b), h : [, b R with h (t) = g (t) for t (, b) nd the r moment which is ssumed to be finite. E r (X) := Theorem 3. The inequlities E r (X) r b r (4.1) 1 (br r ) t r g (t) dt, 1 (br r ) s r 0 ( ( ) ) 1 sgn h 1 t t r g (t) dt nd (4.) E r (X) r b r 1 (r b r ) 1 (r b r ) s r < 0, ( ( )) 1 sgn t h 1 t r g (t) dt hold. Proof. If we put f (t) = t r (t [, b), h (t) = t g (x) dx (t [, b)nd x = h()h(b) = 1 in Corollry 1, then we obtin (4.1) nd (4.). This completes the proof. The following corollry which is specil cse of Theorem 3. Corollry 3. The inequlities (4.3) E (X) b b ( ( ) ) 1 sgn h 1 t tg (t) dt b hold where E (X) is the expecttion of the rndom vrible X. 5. Inequlities for Bet Mpping nd Gmm Mpping The following two mppings re well-known in the literture s the Bet mpping nd the Gmm mpping, respectively: β (x, y) := Γ(x) = t x 1 (1 t) y 1 dt, x > 0, y > 0. e t t x 1 dt, x > 0. The following inequlity which is n pplicton of Theorem 1 for the Bet mpping holds:

12 1 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Theorem 4. Let p, q > 0. Then we hve the inequlity (5.1) for ll x [ 1 0, p1. β (p 1, q 1) x x sgn [t ((p 1) x) 1 p1 ( 1 x p 1 x 1 (p 1) x 1 (p 1) t p (1 t) q dt ) [ 1 ((p 1) x) 1 q p1 Proof. If we put = 0, b = 1, f(t) = (1 t) q, g(t) = t p nd h (t) = tp1 p1 (t [0, 1) [ 1 in Theorem 1, we obtin the inequlity (5.1) for ll x 0, p1. This completes the proof. The following remrk which is n pplicton of Theorem 4 for the Gmm mpping holds: Remrk 5. Tking into ccount tht β (p 1, q 1) = Γ(p1)Γ(q1) Γ(pq), the inequlity (5.1) is equivlent to i.e., Γ(p 1)Γ(q 1) Γ(p q ) x x ( 1 x p 1 x 1 (p 1) x 1 (p 1) sgn [t ((p 1) x) 1 p1 t p (1 t) q dt ) [ 1 ((p 1) x) 1 q p1 (p 1)Γ(p 1)Γ(q 1) x(p 1)Γ(p q ) [ b x sgn [t ((p 1) x) 1 p1 t p (1 t) q dt (p 1)Γ(p q ) [ ( ) 1 [ x p 1 x 1 ((p 1) x) 1 q p1 (p 1)Γ(p q ) x(p 1) 1 Γ(p q ) nd s (p 1)Γ(p 1) = Γ(p ), we get (5.) Γ(p )Γ(q 1) x(p 1)Γ(p q ) [ b x sgn [t ((p 1) x) 1 p1 t p (1 t) q dt (p 1)Γ(p q ) [ ( ) 1 [ x p 1 x 1 ((p 1) x) 1 q p1 (p 1)Γ(p q ) x(p 1) 1 Γ(p q ).

13 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS 13 References [1 T.M. Apostol, Mthemticl Anlysis, Second Edition, Addision-Wesley Publishing Compny, [ N.S. Brnett, S.S. Drgomir nd C.E.M. Perce, A qusi-trpezoid inequlity for double integrls, submitted. [3 P. Cerone nd S.S. Drgomir, Trpezoidl Type Rules from An Inequlities Point of View, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, Editor: G. Anstssiou, CRC Press, N.Y., (000), [4 S.S. Drgomir, P. Cerone nd A. Sofo, Some remrks on the trpezoid rule in numericl integrtion, Indin J. of Pure nd Appl. Mth., 31(5) (000), [5 S.S. Drgomir nd T.C. Pechey, New estimtion of the reminder in the trpezoidl formul with pplictions, Studi Universittis Bbes-Bolyi Mthemtic, XLV(4) (000), [6 S.S. Drgomir, On the trpezoid qudrture formul nd pplictions, Krgujevc J. Mth., 3 (001), [7 S.S. Drgomir, On the trpezoid qudrture formul for Lipschitzin mppings nd pplictions, Tmkng J. of Mth., 30 () (1999), [8 P. Cerone, S.S. Drgomir nd C.E.M. Perce, A generlized trpezoid inequlity for functions of bounded vrition, Turkish J. Mth., 4() (000), [9 L. Fejér, Uberdie Fourierreihen, II, Mth. Ntur. Ungr. Akd Wiss., 4 (1906), [In Hungrin Deprtment of Mthemtics, Alethei University, Tmsui, Tiwn E-mil ddress: kltseng@emil.u.edu.tw Deprtment of Mthemtics, Tmkng University, Tmsui, Tiwn School of Computer Science nd Mthemtics, Victori University of Technology, PO Box 1448, MCMC 8001, Victori, Austrli. E-mil ddress: sever@mtild.vu.edu.u URL:

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