Applied Mthemtics E-Notes, 5(005), 53-60 c ISSN 1607-510 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION Nend Ujević Received 15 December 00 Abstrct A generl integrl inequlity for convex functions is derived. New error bounds for the midpoint, trpezoid, verged midpoint-trpezoid nd Simpson s qudrture rules re obtined. Applictions in numericl integrtion re lso given. 1 Introduction In recent yers number of uthors hve considered n error nlysis for some known nd some new qudrture formuls. They used n pproch from n inequlities point of view. For exmple, the midpoint qudrture rule is considered in [1], [3], [15], the trpezoid rule is considered in [], [5], [15], the verged midpoint-trpezoid qudrture rule is considered in [8], [15], [16] nd Simpson s rule is considered in [], [5], [13]. In most cses estimtions of errors for these qudrture rules re obtined by mens of derivtives of integrnds. In this pper we first derive generl integrl inequlity for convex functions. Then we pply this inequlity to obtin new error bounds for the bove mentioned qudrture rules. Finlly, we give pplictions in numericl integrtion. The min property of the obtined error bounds is tht they re expressed in terms function vlues of integrnd f, which hs to be convex function. Hence, we cn pply these qudrture rules (with the obtined error bounds) to integrnds which re not differentible functions. In composite qudrture formuls we use the sme dt for finding n pproximte vlue of integrl nd for finding n estimtion of error. An illustrtive exmple is given tht shows how ccurte the obtined estimtions cn be. A Generl Inequlity We begin with some elementry fcts. Let f :[, b] R be given function. We sy tht f is n even function with respect to the point t 0 =( + b)/ iff( + b t) =f(t) Mthemtics Subject Clssifictions: 6D10, 1A55, 65D30. Deprtment of Mthemtics, University of Split, Teslin 1/III, 1000 Split, Croti 53
5 An Integrl Inequlity for Convex Functions for t [, b]. We sy tht f is n odd function with respect to the point t 0 =( + b)/ if f( + b t) = f(t) fort [, b]. Here we use the term even (odd) function for given function f :[, b] R if f is even (odd) with respect to the point t 0 =( + b)/. Ech function f :[, b] R cn be represented s sum of one even nd one odd function, f(x) =f 1 (x)+f (x), where f 1 (x) = f(x)+f(+b x) is n even function nd f (x) = f(x) f(+b x) is n odd function. It is not difficult to verify the following fcts. If f is n odd function then f is n even function. If f,g re even or odd functions then fg is n even function. If f is n even function nd g is n odd function then fg is n odd function. We now show tht if f is n integrble nd odd function then f(x)dx =0. We hve f(x)dx = f( + b x)dx = f(u)du = Thus, f(x)dx = 0 nd we proved the bove ssertion. If f is n integrble nd even function then we hve (+b)/ f(x)dx = f(x)dx = f(x)dx. (+b)/ We hve (+b)/ f(x)dx = f(x)dx + f(x)dx (+b)/ nd (+b)/ f(x)dx = (+b)/ f( + b x)dx = f(u)du = (+b)/ f(x)dx. (+b)/ f(x)dx. Thus the bove ssertion holds. We now show tht the function K(, b, x) = x α x [, ( + b)/) 0 x =( + b)/ x β x (( + b)/,b] (1) is n odd function if α + β = + b. Forx [, ( + b)/) we hve K(, b, + b x) = + b x β = + b x ( + b α) = x + α = K(, b, x). For x (( + b)/,b]wehve K(, b, + b x) = + b x α = + b x ( + b β) = x + β = K(, b, x).
N. Ujević 55 Finlly, for x =( + b)/, K(, b, + b ( + b)/) = 0 = K(, b, ( + b)/). Hence, K(, b, x) is n odd function. THEOREM 1. Let f C (, b) ndf (x) 0, x [, b], i.e. f is convex function. Let K(, b, x) bedefined by (1). Then b (α )f()+(β α)f(( + b)/) + (b β)f(b) f(x)dx K [f()+f(b) f(( + b)/)], () where K =mx x [,b] K(, b, x). PROOF. Integrting by prts, we obtin (+b)/ K(, b, x)f (x)dx = (x α)f (x)dx + + b = (α )f()+(β α)f If we introduce the nottions then we hve +(b β)f(b) (+b)/ (x β)f (x)dx f(x)dx. (3) f1(x) = f (x)+f ( + b x), f (x) = f (x) f ( + b x) f (x) =f 1(x)+f (x) nd K(, b, x)f 1(x) is n odd function while f (x) nd K(, b, x)f (x) re even functions. Thus, we hve nd K(, b, x)f (x)dx = K(, b, x)f (x)dx = = K(, b, x)[f 1(x)+f (x)] dx K(, b, x)f (x)dx K(, b, x)f(x)dx K K f(x) dx (+b)/ f (x) dx = K f (x)+f ( + b x) dx. () (+b)/
56 An Integrl Inequlity for Convex Functions Note now tht f (x) is n incresing function, since f (x) 0, x [, b]. Thus, (+b)/ f (x)+f ( + b x) dx = From (3)-(5) we esily get (). (+b)/ (f (x)+f ( + b x)) dx = f(b)+f() f( + b ). (5) 3 Applictions to Qudrture Formuls We hve the following results. PROPOSITION 1. (Midpoint inequlity) Let f C (, b)ndf (x) 0, x [, b], i.e. f is convex function. Then we hve f( + b b )(b ) f(x)dx b f()+f(b) f( + b ). (6) PROOF. We choose α =, β = b in (1). Then K = b. From this fct nd () we see tht (6) holds. PROPOSITION. (Trpezoid inequlity) Under the ssumptions of Proposition 1 we hve f()+f(b) b (b ) f(x)dx b f()+f(b) f( + b ). (7) PROOF. We choose α = β = +b in (1). Then K = b. From the lst fct nd () we see tht (7) holds. PROPOSITION 3. (Averged midpoint-trpezoid inequlity) Under the ssumptions of Proposition 1 we hve f()+f( + b b b )+f(b) f(x)dx b f()+f(b) f( + b ). (8) PROOF. We choose α = 3+b, β = +3b in (1). Then K = b. From this fct nd () we see tht (8) holds. PROPOSITION. (Simpson s inequlity) Under the ssumptions of Proposition 1 we hve f()+f( + b b b )+f(b) f(x)dx 6 b f()+f(b) f( + b 3 ). (9)
N. Ujević 57 PROOF. We choose α = 5+b 6, β = +5b 6 in (1). Then K = b 3.Fromthelst fct nd () we see tht (9) holds. REMARK 1. The inequlities obtined in Propositions 1- cn lso be derived using the well-known Hermite-Hdmrd inequlities. Furthermore, if f is convex function then we hve 0 f(x)dx f( + b b )(b ) f()+f(b) f( + b ) nd b f( + b ) f() f(b) f(x)dx f()+f(b) (b ) 0. Applictions in Numericl Integrtion Let π = {x 0 = <x 1 < <x n = b} be given subdivision of the intervl [, b], h i = x i+1 x i, i =0, 1,..., n 1. We define n 1 xi + x i+1 σ n (f) = h i f(x i )+f(x i+1 ) f. (10) THEOREM. Let f C (, b) ndf (x) 0, x [, b]. Let π be given subdivision of the intervl [, b]. Then where nd f(x)dx = A M (π,f)+r M (π,f), n 1 xi + x i+1 A M (π,f)= h i f R M (π,f) 1 σ n(f). PROOF. Apply Proposition 1 to the intervls [x i,x i+1 ]ndsum. THEOREM 3. Under the ssumptions of Theorem we hve where f(x)dx = A T (π,f)+r T (π,f), A T (π,f)= 1 n 1 h i [f(x i )+f(x i+1 )]
58 An Integrl Inequlity for Convex Functions nd R T (π,f) 1 σ n(f). PROOF. Apply Proposition to the intervls [x i,x i+1 ]ndsum. THEOREM. Under the ssumptions of Theorem we hve where nd f(x)dx = A MT (π,f)+r MT (π,f), A MT (π,f)= 1 n 1 h i f(x i )+f( x i + x i+1 )+f(x i+1 ) R MT (π,f) 1 σ n(f). PROOF. Apply Proposition 3 to the intervls [x i,x i+1 ]ndsum. THEOREM 5. Under the ssumptions of Theorem we hve where nd f(x)dx = A S (π,f)+r S (π,f), A S (π,f)= 1 n 1 h i f(x i )+f( x i + x i+1 )+f(x i+1 ) 6 R S (π,f) 1 3 σ n(f). PROOF. Apply Proposition 3 to the intervls [x i,x i+1 ]ndsum. EXAMPLE. Let us now consider the integrl 1 0 ( x)dx. Notethtf(x) = x is convex function on the intervl [0, 1]. Note lso tht we cnnot pply the clssicl estimtions of error (expressed in terms of the first, second,... derivtives), since f, f,... re unbounded on the intervl [0, 1]. The exct vlue is 1 0 ( x)dx = 0.66666666666666. If we use formuls given in theorems -5 with h i = h = 1 n, n = 1000, then we get A M (π,f)= 0.6666685719568 R M (π,f) 0.8369E 05 A T (π,f)= 0.6666601339368 R T (π,f) 0.8369E 05 A MT (π,f)= 0.6666635868 R MT (π,f) 0.18E 05 A S (π,f)= 0.66666575899501 R S (π,f) 0.566E 05
N. Ujević 59 nd the exct errors re R M (π,f) =0.1906E 05, R T (π,f) =0.653E 05, R MT (π,f) =0.3138E 05, R S (π,f) =0.90767E 06. We see tht the estimtions re very ccurte for this exmple. REMARK. Note tht in ll error inequlities we use the sme vlues f(x i )to clculte the pproximtion of the integrl f(t)dt nd to obtin the error bound nd recll tht function evlutions re generlly considered the computtionlly most expensive prt of qudrture lgorithms. On the other hnd, the usul wy to estimte the errors is to find f (k), k {1,,...}. Hence, the presented wy of estimtion of the errors is very simple nd effective. Wehveonlyonerestriction:theintegrndhs to be convex function. References [1] G. A. Anstssiou, Ostrowski type inequlities, Proc. Amer. Mth. Soc., 13(1)(1995), 3775 3781. [] P. Cerone, Three points rules in numericl integrtion, Nonliner Anl., 7()(001), 31 35. [3] P. Cerone nd S. S. Drgomir, Midpoint-type Rules from n Inequlities Point of View, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, Editor: G. Anstssiou, CRC Press, New York, (000), 135 00. [] P. Cerone nd S. S. Drgomir, Trpezoidl-type Rules from n Inequlities Point of View, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, Editor: G. Anstssiou, CRC Press, New York, (000), 65 13. [5] D. Cruz-Uribe nd C. J. Neugebuer, Shrp error bounds for the trpezoidl rule nd Simpson s rule, J. Inequl. Pure Appl. Mth., 3(), Article 9, (00), 1. [6] S. S. Drgomir, An inequlity improving the first Hermite-Hdmrd inequlity for convex functions defined on liner spces nd pplictions for semi-inner products, J. Inequl. Pure Appl. Mth. 3()(00), Article 31, 8 pp. (electronic). [7] S. S. Drgomir, An inequlity improving the second Hermite-Hdmrd inequlity for convex functions defined on liner spces nd pplictions for semi-inner products, J. Inequl. Pure Appl. Mth. 3(3)(00), Article 35, 8 pp. (electronic).
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