Two-Point Quadrature Rules for Riemann-Stieltjes Integrals with Lp-error estimates
aa r X i v : . [ m a t h . C A ] M a r TWO-POINT QUADRATURE RULES FOR RIEMANN–STIELTJESINTEGRALS WITH L p –ERROR ESTIMATES M.W. ALOMARI
Abstract.
In this work, we construct a new general two-point quadrature rules for theRiemann–Stieltjes integral R ba f ( t ) du ( t ), where the integrand f is assumed to be satisfiedwith the H¨older condition on [ a, b ] and the integrator u is of bounded variation on [ a, b ]. Thedual formulas under the same assumption are proved. Some sharp error L p –Error estimatesfor the proposed quadrature rules are also obtained. Introduction
The number of proposed quadrature rules that provides approximation for the Riemann–Stieltjes integral ( RS –integral) R ba f ( t ) du ( t ) using derivatives or without using derivativesare very rare in comparison with the large number of methods available to approximate theclassical Riemann integral R ba f ( t ) dt .The problem of introducing quadrature rules for RS -integral R ba f dg was studied via theoryof inequalities by many authors. Two famous real inequalities were used in this approach,which are the well known Ostrowski and Hermite-Hadamard inequalities and their modifica-tions. For this purpose and in order to approximate the RS -integral R ba f ( t ) du ( t ), a gener-alization of closed Newton-Cotes quadrature rules of RS -integrals without using derivativesprovides a simple and robust solution to a significant problem in the evaluation of certainapplied probability models was presented by Tortorella in [32].In 2000, Dragomir [16] introduced the Ostrowski’s approximation formula (which is ofOne-point type formula) as follows: Z ba f ( t ) du ( t ) ∼ = f ( x ) [ u ( b ) − u ( a )] ∀ x ∈ [ a, b ] . Several error estimations for this approximation had been done in the works [15] and [16].From different point of view, the authors of [17] (see also [11, 12]) considered the problemof approximating the Stieltjes integral R ba f ( t ) du ( t ) via the generalized trapezoid formula: Z ba f ( t ) du ( t ) ∼ = [ u ( x ) − u ( a )] f ( a ) + [ u ( b ) − u ( x )] f ( b ) . Many authors have studied this quadrature rule under various assumptions of integrandsand integrators. For full history of these two quadratures see [6] and the references therein.
Date : March 18, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Quadrature formula, Riemann-Stieltjes integral, Ostrowski’s inequality.
Another trapezoid type formula was considered in [20], which reads: Z ba f ( t ) du ( t ) ∼ = f ( a ) + f ( b )2 [ u ( b ) − u ( a )] ∀ x ∈ [ a, b ] . Some related results had been presented by the same author in [18] and [19]. For otherconnected results see [13] and [14].In 2008, Mercer [27] introduced the following trapezoid type formula for the RS -integral Z ba f dg ∼ = [ G − g ( a )] f ( a ) + [ g ( b ) − G ] f ( b ) , (1.1)where G = b − a R ba g ( t ) dt .Recently, Alomari and Dragomir [4], proved several new error bounds for the Mercer–Trapezoid quadrature rule (1.1) for the RS -integral under various assumptions involved theintegrand f and the integrator g .Follows Mercer approach in [27], Alomari and Dragomir [10] introduced the followingthree-point quadrature formula: Z ba f ( t ) dg ( t ) ∼ = [ G ( a, x ) − g ( a )] f ( a ) + [ G ( x, b ) − G ( a, x )] f ( x )+ [ g ( b ) − G ( x, b )] f ( b ) (1.2)for all a < x < b , where G ( α, β ) := β − α R βα g ( t ) dt .Several error estimations of Mercer’s type quadrature rules for RS -integral under variousassumptions about the function involved have been considered in [4] and [7].Motivated by Guessab-Schmeisser inequality (see [22]) which is of Ostrowski’s type, Alo-mari in [5] and [9] presented the following approximation formula for RS -integrals: Z ba f ( t ) du ( t ) ∼ = (cid:20) u (cid:18) a + b (cid:19) − u ( a ) (cid:21) f ( x ) + (cid:20) u ( b ) − u (cid:18) a + b (cid:19)(cid:21) f ( a + b − x ) , (1.3)for all x ∈ (cid:2) a, a + b (cid:3) . For other related results see [6]. For different approaches variantquadrature formulae the reader may refer to [1], [8], [21] and [28].Among others the L ∞ -norm gives the highest possible degree of precision; so that it isrecommended to be ‘almost’ the norm of choice. However, in some cases we cannot accessthe L ∞ -norm, so that L p -norm (1 ≤ p < ∞ ) is considered to be a variant norm in errorestimations.In this work, several L p -error estimates (1 ≤ p < ∞ ) of general two and three pointsquadrature rules for Riemann-Stieltjes integrals are presented. The presented proofs dependon new triangle type inequalities for RS -integrals.Let f be defined on [ a, b ]. If P := { x , x , · · · , x n } is a partition of [ a, b ], write∆ f i = f ( x i ) − f ( x i − ) , for i = 1 , , · · · , n . A function f is said to be of bounded p -variation if there exists a positivenumber M such that (cid:18) n P i =1 | ∆ f i | p (cid:19) p ≤ M , (1 ≤ p < ∞ ) for all partition of [ a, b ], (see [26]). WO-POINT QUADRATURE RULES FOR THE RIEMANN–STIELTJES INTEGRAL 3
Let f be of bounded p -variation on [ a, b ], and let P ( P ) denote the sum (cid:18) n P i =1 | ∆ f i | p (cid:19) p corresponding to the partition P of [ a, b ]. The number b _ a ( f ; p ) = sup nX ( P ) : P ∈ P ([ a, b ]) o , ≤ p < ∞ is called the total p –variation of f on the interval [ a, b ], where P ([ a, b ]) denotes the set of allpartitions of [ a, b ]. For p = 1 it is the usual variation of f ( x ) that was introduced by Jordan(see [24], [25]). For very constructive systematic study of Jordan variation we recommendthe interested reader to refer to [29].In special case, we define the variation of order ∞ of f along [ a, b ] in the classical sense,i.e., if there exists a positive number M such that n X i =1 Osc (cid:16) f ; h x ( n ) i − , x ( n ) i i(cid:17) = n X i =1 (sup − inf) f ( t i ) ≤ M, t i ∈ h x ( n ) i − , x ( n ) i i , for all partition of [ a, b ], then f is said to be of bounded ∞ –variation on [ a, b ]. The number b _ a ( f ; ∞ ) = sup nX ( P ) : P ∈ P [ a, b ] o := Osc ( f ; [ a, b ]) , is called the oscillation of f on [ a, b ]. Equivalently, we may define the oscillation of f as, (see[23]): b _ a ( f ; ∞ ) = lim p →∞ b _ a ( f ; p ) = sup x ∈ [ a,b ] { f ( x ) } − inf x ∈ [ a,b ] { f ( x ) } = Osc ( f ; [ a, b ]) . Let W p denotes the class of all functions of bounded p -variation (1 ≤ p ≤ ∞ ). For anarbitrary p ≥ W p was firstly introduced by Wiener in [30], where he had shownthat W p can only have discontinuities of the first kind. More generally, if f is a real functionof bounded p -variation on an interval [ a, b ], then: • f is bounded, and Osc ( f ; [ a, b ]) ≤ b _ a ( f ; p ) ≤ b _ a ( f ; 1) . This fact follows by Jensen’s inequality applied for h ( p ) = W ba ( f ; p ) which is log-convex and decreasing for all p >
1. Moreover, the inclusions W ∞ ( f ) ⊂ W q ( f ) ⊂ W p ( f ) ⊂ W ( f )are valid for all 1 < p < q < ∞ , (see [31]). • f is continuous except at most on a countable set. • f has one-sided limits everywhere (limits from the left everywhere in ( a, b ], and fromthe right everywhere in [ a, b ); • The derivative f ′ ( x ) exists almost everywhere (i.e. except for a set of measure zero). M.W. ALOMARI • If f ( x ) is differentiable on [ a, b ], then b _ a ( f ; p ) = (cid:18)Z ba | f ′ ( t ) | p dt (cid:19) p = k f ′ k p , ≤ p < ∞ . Lemma 1. [2] Fix 1 ≤ p < ∞ . Let f, g : [ a, b ] → R be such that f is continuous on [ a, b ]and g is of bounded p –variation on [ a, b ]. Then the Riemann–Stieltjes integral R ba f ( t ) dg ( t )exists and the inequality: (cid:12)(cid:12)(cid:12)(cid:12)Z ba w ( t ) dν ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k w k ∞ · Osc ( ν ; [ a, b ]) ≤ k w k ∞ · b _ a ( ν ; p ) , (1.4)holds. The constant ‘1’ in the both inequalities is the best possible. Lemma 2. [2]
Let ≤ p < ∞ . Let w, ν : [ a, b ] → R be such that is w ∈ L p [ a, b ] and ν has aLipschitz property on [ a, b ] . Then the inequality (cid:12)(cid:12)(cid:12)(cid:12)Z ba w ( t ) dν ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ L ( b − a ) − p · k w k p , (1.5) holds and the constant ‘1 ’ in the right hand side is the best possible, where k w k p = (cid:18)Z ba | w ( t ) | p dt (cid:19) /p , (1 ≤ p < ∞ ) . In this paper, we establish two–point of Ostrowski’s integral inequality for the Riemann-Stieltjes integral R ba f ( t ) du ( t ), where f is assumed to be of r - H -H¨older type on [ a, b ] and u is of bounded variation on [ a, b ], are given. The dual formulas under the same assumptionare proved. Some sharp error L p –Error estimates for the proposed quadrature rules are alsoobtained. 2. The Results
Consider the quadrature rule Z ba f ( s ) du ( s ) = Q [ a,b ] ( f, u ; t , x, t ) + R [ a,b ] ( f, u ; t , x, t ) (2.1)where Q [ a,b ] ( f, u ; t , x, t ) is the quadrature formula Q [ a,b ] ( f, u ; t , x, t ) = [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) (2.2)for all a ≤ t ≤ x ≤ t ≤ b .Hence, the remainder term R [ a,b ] ( f, u ; t , x, t ) is given by R [ a,b ] ( f, u ; t , x, t ) := Z ba f ( s ) du ( s ) − [ u ( x ) − u ( a )] f ( t ) − [ u ( b ) − u ( x )] f ( t ) (2.3)The following Two-point Ostrowski’s inequality for Riemann-Stieltjes integral holds. WO-POINT QUADRATURE RULES FOR THE RIEMANN–STIELTJES INTEGRAL 5
Theorem 1.
Let f : [ a, b ] → R be H¨older continuous of order r , (0 < r ≤ , and u : [ a, b ] → R is a mapping of bounded p -variation (1 ≤ p ≤ ∞ ) on [ a, b ] . Then we have the inequality (cid:12)(cid:12) R [ a,b ] ( f, u ; t , x, t ) (cid:12)(cid:12) (2.4) ≤ H max (cid:26)(cid:20) x − a (cid:12)(cid:12)(cid:12)(cid:12) t − a + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (cid:20) b − x (cid:12)(cid:12)(cid:12)(cid:12) t − x + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) r · b _ a ( u ; p ) for all a ≤ t ≤ x ≤ t ≤ b . Furthermore, the first half of each max-term is the best possiblein the sense that it cannot be replaced by a smaller one, for all r ∈ (0 , .Proof. Using the integration by parts formula for Riemann–Stieltjes integral, we have Z xa [ f ( t ) − f ( s )] du ( s ) + Z bx [ f ( t ) − f ( s )] du ( s )= Z xa f ( t ) du ( s ) + Z bx f ( t ) du ( s ) − Z ba f ( s ) du ( s )= [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) − Z ba f ( s ) du ( s )= −R [ a,b ] ( f, u ; t , x, t ) , It is well known that if p : [ c, d ] → R is continuous and ν : [ c, d ] → R is of p -boundedvariation (1 ≤ p < ∞ ), then the Riemann-Stieltjes integral R dc p ( t ) dν ( t ) exists and thefollowing inequality holds: (cid:12)(cid:12)(cid:12)(cid:12)Z dc p ( t ) dν ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup t ∈ [ c,d ] | p ( t ) | d _ c ( ν ) . (2.5)Applying the inequality (2.5) for ν ( t ) = u ( t ), p ( t ) = f ( t ) − f ( s ), for all s ∈ [ a, x ]; and thenfor p ( t ) = f ( t ) − f ( s ), ν ( t ) = u ( t ) for all t ∈ ( x, b ], we get (cid:12)(cid:12)(cid:12)(cid:12) [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) − Z ba f ( s ) du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z xa [ f ( t ) − f ( s )] du ( s ) + Z bx [ f ( t ) − f ( s )] du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z xa [ f ( t ) − f ( s )] du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z bx [ f ( t ) − f ( s )] du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup s ∈ [ a,x ] | f ( t ) − f ( s ) | · x _ a ( u ; p ) + sup s ∈ [ x,b ] | f ( t ) − f ( s ) | · b _ x ( u ; p ) . (2.6)As f is of r - H -H¨older type, we havesup s ∈ [ a,x ] | f ( t ) − f ( s ) | ≤ sup s ∈ [ a,x ] [ H | t − s | r ]= H max { ( x − t ) r , ( t − a ) r } = H [max { ( x − t ) , ( t − a ) } ] r = H (cid:20) x − a (cid:12)(cid:12)(cid:12)(cid:12) t − a + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r , M.W. ALOMARI and sup s ∈ [ x,b ] | f ( t ) − f ( s ) | ≤ sup s ∈ [ x,b ] [ H | t − s | r ]= H max { ( t − x ) r , ( b − t ) r } = H [max { ( t − x ) , ( b − t ) } ] r = H (cid:20) b − x (cid:12)(cid:12)(cid:12)(cid:12) t − x + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r . Therefore, by (2.6), we have (cid:12)(cid:12)(cid:12)(cid:12) [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) − Z ba f ( s ) du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ H (cid:20) x − a (cid:12)(cid:12)(cid:12)(cid:12) t − a + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r · x _ a ( u ; p ) + H (cid:20) b − x (cid:12)(cid:12)(cid:12)(cid:12) t − x + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r · b _ x ( u ; p ) ≤ H max (cid:26)(cid:20) x − a (cid:12)(cid:12)(cid:12)(cid:12) t − a + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r , (cid:20) b − x (cid:12)(cid:12)(cid:12)(cid:12) t − x + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r (cid:27) · b _ a ( u ; p )= H max (cid:26)(cid:20) x − a (cid:12)(cid:12)(cid:12)(cid:12) t − a + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (cid:20) b − x (cid:12)(cid:12)(cid:12)(cid:12) t − x + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) r · b _ a ( u ; p )To prove the sharpness of the constant r for any r ∈ (0 , C >
0, that is, (cid:12)(cid:12)(cid:12)(cid:12) [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) − Z ba f ( s ) du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ H max (cid:26)(cid:20) C ( x − a ) + (cid:12)(cid:12)(cid:12)(cid:12) t − a + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (cid:20) C ( b − x ) + (cid:12)(cid:12)(cid:12)(cid:12) t − x + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) r · b _ a ( u ; p ) . (2.7)Choose f ( t ) = t r , r ∈ (0 , t ∈ [0 ,
1] and u : [0 , → [0 , ∞ ) given by u ( t ) = , t ∈ (0 , − , t = 0As | f ( x ) − f ( y ) | = | x r − y r | ≤ | x − y | r , ∀ x ∈ [0 , , r ∈ (0 , , it follows that f is r - H -H¨older type with the constant H = 1.By using the integration by parts formula for Riemann-Stieltjes integrals, we have: Z f ( t ) du ( t ) = f (1) u (1) − f (0) u (0) − Z u ( t ) df ( t ) = 0 , and W ( u ; p ) = 1. Consequently, by (2.7), we get | t r | ≤ max (cid:26)h Cx + (cid:12)(cid:12)(cid:12) t − x (cid:12)(cid:12)(cid:12)i , (cid:20) C (1 − x ) + (cid:12)(cid:12)(cid:12)(cid:12) t − x + 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) r , ∀ t ∈ [0 , . For t = x and t = x = 1 we get r ≤ C r , which implies that C ≥ . WO-POINT QUADRATURE RULES FOR THE RIEMANN–STIELTJES INTEGRAL 7
It remains to prove the second part, so we consider u ( t ) = , t ∈ [0 , , t = 1therefore as we have obtained previously Z f ( t ) du ( t ) = 0 and _ ( u ; p ) = 1 . Consequently, by (2.4), we get | t r | ≤ max (cid:26)h Cx + (cid:12)(cid:12)(cid:12) t − x (cid:12)(cid:12)(cid:12)i , (cid:20) C (1 − x ) + (cid:12)(cid:12)(cid:12)(cid:12) t − x + 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) r , ∀ t ∈ [0 , . For t = x = 0 and t = we get r ≤ C r , which implies that C ≥ , and the theorem iscompletely proved. (cid:3) The following inequalities are hold:
Corollary 1.
Let f and u as in Theorem 1. In 2.4 choose(1) t = a and t = b , then we get the following trapezoid type inequality (cid:12)(cid:12) R [ a,b ] ( f, u ; a, x, b ) (cid:12)(cid:12) ≤ H (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r · b _ a ( u ; p ) . or equivalently, we may write using parts formula for Riemann-Stieltjes integral (cid:12)(cid:12)(cid:12)(cid:12) [ f ( b ) − f ( a )] u ( x ) − Z ba u ( s ) df ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ H (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r · b _ a ( u ; p ) . The constant is the best possible for all r ∈ (0 , .(2) x = a + b , then we get the following mid-point type inequality (cid:12)(cid:12)(cid:12)(cid:12) R [ a,b ] (cid:18) f, u ; t , a + b , t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ H max (cid:26)(cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) t − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) t − a + 3 b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) r · b _ a ( u ; p ) . The constant is the best possible for all r ∈ (0 , . For instance, setting t = y and t = a + b − y , we get (cid:12)(cid:12)(cid:12)(cid:12) R [ a,b ] (cid:18) f, u ; y, a + b , a + b − y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ H (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) y − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r · b _ a ( u ; p ) . for all y ∈ (cid:2) a, a + b (cid:3) . M.W. ALOMARI (3) t = a + x and t = x + b , then (cid:12)(cid:12)(cid:12)(cid:12) R [ a,b ] (cid:18) f, u ; a + x , x, x + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ H r (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r · b _ a ( u ; p ) Both constants r and are the best possible for all r ∈ (0 , . Corollary 2.
Let f be a H¨older continuous function of order r (0 < r ≤ , on [ a, b ] , and g : [ a, b ] → R is continuous on [ a, b ] . Then we have the inequality (cid:12)(cid:12)(cid:12)(cid:12) f ( t ) Z xa g ( s ) ds + f ( t ) Z bx g ( s ) ds − Z ba f ( s ) g ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ H max (cid:26)(cid:20) x − a (cid:12)(cid:12)(cid:12)(cid:12) t − a + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (cid:20) b − x (cid:12)(cid:12)(cid:12)(cid:12) t − x + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) r · k g k p , for all a ≤ t ≤ x ≤ t ≤ b , where k g k p = (cid:16)R ba | g ( t ) | p dt (cid:17) /p .Proof. Define the mapping u : [ a, b ] → R , u ( t ) = R ta g ( s ) ds . Then u is differentiable on ( a, b )and u ′ ( t ) = g ( t ). Using the properties of the Riemann-Stieltjes integral, we have Z ba f ( t ) du ( t ) = Z ba f ( t ) g ( t ) dt, and b _ a ( u ; p ) = (cid:18)Z ba | u ′ ( t ) | p dt (cid:19) /p = (cid:18)Z ba | g ( t ) | p dt (cid:19) /p , which gives the required result. (cid:3) Theorem 2.
Let ≤ p < ∞ . Let f, u : [ a, b ] → R be such that is f ∈ L p [ a, b ] and u has aLipschitz property on [ a, b ] . If f is r – H –H¨older continuous, then the inequality (cid:12)(cid:12) R [ a,b ] ( f, u ; t , x, t ) (cid:12)(cid:12) ≤ HL ( x − a ) − p ( t − a ) rp +1 + ( x − t ) rp +1 rp + 1 ! p + ( b − x ) − p ( t − x ) rp +1 + ( b − t ) rp +1 rp + 1 ! p (2.8) holds for all p > and r ∈ (0 , . WO-POINT QUADRATURE RULES FOR THE RIEMANN–STIELTJES INTEGRAL 9
Proof.
From Lemma 2 we have (cid:12)(cid:12)(cid:12)(cid:12) [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) − Z ba f ( s ) du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z xa [ f ( t ) − f ( s )] du ( s ) + Z bx [ f ( t ) − f ( s )] du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z xa [ f ( t ) − f ( s )] du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z bx [ f ( t ) − f ( s )] du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ L " ( x − a ) − p (cid:18)Z xa | f ( t ) − f ( s ) | p ds (cid:19) p + ( b − x ) − p (cid:18)Z bx | f ( t ) − f ( s ) | p ds (cid:19) p ≤ HL " ( x − a ) − p (cid:18)Z xa | t − s | rp ds (cid:19) p + ( b − x ) − p (cid:18)Z bx | t − s | rp ds (cid:19) p = HL ( x − a ) − p ( t − a ) rp +1 + ( x − t ) rp +1 rp + 1 ! p + ( b − x ) − p ( t − x ) rp +1 + ( b − t ) rp +1 rp + 1 ! p . which proves the required result. (cid:3) Corollary 3.
Let f and u as in Theorem 2. In (2.8) choose(1) t = a and t = b , then we get the following trapezoid type inequality (cid:12)(cid:12) R [ a,b ] ( f, u ; a, x, b ) (cid:12)(cid:12) ≤ HL ( x − a ) − p ( x − a ) rp +1 rp + 1 ! p + ( b − x ) − p ( b − x ) rp +1 rp + 1 ! p . or equivalently, we may write using parts formula for Riemann-Stieltjes integral (cid:12)(cid:12)(cid:12)(cid:12) [ f ( b ) − f ( a )] u ( x ) − Z ba u ( s ) df ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ HL ( x − a ) − p ( x − a ) rp +1 rp + 1 ! p + ( b − x ) − p ( b − x ) rp +1 rp + 1 ! p . (2) x = a + b , then we get the following mid-point type inequality (cid:12)(cid:12)(cid:12)(cid:12) R [ a,b ] (cid:18) f, u ; t , a + b , t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ HL (cid:18) b − a (cid:19) − p ( t − a ) rp +1 + (cid:0) a + b − t (cid:1) rp +1 rp + 1 ! p + (cid:18) b − a (cid:19) − p (cid:0) t − a + b (cid:1) rp +1 + ( b − t ) rp +1 rp + 1 ! p . For instance, setting t = y and t = a + b − y , we get (cid:12)(cid:12)(cid:12)(cid:12) R [ a,b ] (cid:18) f, u ; y, a + b , a + b − y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ HL (cid:18) b − a (cid:19) − p ( t − a ) rp +1 + (cid:0) a + b − t (cid:1) rp +1 rp + 1 ! p . for all y ∈ (cid:2) a, a + b (cid:3) .(3) t = a + b , x = a + b and t = a +3 b , then (cid:12)(cid:12)(cid:12)(cid:12) R [ a,b ] (cid:18) f, u ; 3 a + b , a + b , a + 3 b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ HL ( b − a ) r r + p ( rp + 1) p . Now, let I be a real interval such that [ a, b ] ⊆ I ◦ the interior of I , a, b ∈ R with a < b .Consider U p ( I ) ( p >
1) be the space of all positive n -th differentiable functions f whose n -thderivatives f ( n ) is positive locally absolutely continuous on I ◦ with R ba (cid:0) f ( n ) ( t ) (cid:1) p dt < ∞ , and f ( n ) ( a ) = f ( n ) ( b ) = 0. L p -error estimates for Riemann–Stieltjes R ba f ( t ) du ( t ) where f belongs to U p ( I ) is consid-ered in the following result. Theorem 3.
Let ≤ p < ∞ . Let f, u : [ a, b ] → R be such that is f ∈ U p ( I ) and u has aLipschitz property on [ a, b ] . If f is r – H –H¨older continuous, then the inequality holds for all p > and r ∈ (0 , . (cid:12)(cid:12) R [ a,b ] ( f, u ; t , x, t ) (cid:12)(cid:12) ≤ L p sin (cid:16) πp (cid:17) π p √ p − n (cid:26) ( x − a ) − p (cid:20) x − a (cid:12)(cid:12)(cid:12)(cid:12) t − x + a (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n + ( b − x ) − p (cid:20) b − x (cid:12)(cid:12)(cid:12)(cid:12) t − x + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n (cid:27) (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) p, [ a,b ] (2.9) WO-POINT QUADRATURE RULES FOR THE RIEMANN–STIELTJES INTEGRAL 11
Proof.
As in the proof of Theorem 2, we have by Lemma 2 (cid:12)(cid:12)(cid:12)(cid:12) [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) − Z ba f ( s ) du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ L " ( x − a ) − p (cid:18)Z xa | f ( t ) − f ( s ) | p ds (cid:19) p + ( b − x ) − p (cid:18)Z bx | f ( t ) − f ( s ) | p ds (cid:19) p ≤ L ( x − a ) − p p sin (cid:16) πp (cid:17) π p √ p − n (cid:20) x − a (cid:12)(cid:12)(cid:12)(cid:12) t − x + a (cid:12)(cid:12)(cid:12)(cid:12) n (cid:21) (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) p, [ a,x ] + ( b − x ) − p p sin (cid:16) πp (cid:17) π p √ p − n (cid:20) b − x (cid:12)(cid:12)(cid:12)(cid:12) t − x + b (cid:12)(cid:12)(cid:12)(cid:12) n (cid:21) (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) p, [ x,b ] ≤ L p sin (cid:16) πp (cid:17) π p √ p − n (cid:26) ( x − a ) − p (cid:20) x − a (cid:12)(cid:12)(cid:12)(cid:12) t − x + a (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n + ( b − x ) − p (cid:20) b − x (cid:12)(cid:12)(cid:12)(cid:12) t − x + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n (cid:27) (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) p, [ a,b ] which proves the required result, where we have used that fact that if h ∈ U p ( I ) then for all ξ ∈ ( a, b ) we have Z ba | h ( t ) − h ( ξ ) | p dt ≤ p p sin p (cid:16) πp (cid:17) π p ( p − n (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) ξ − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) np · Z ba (cid:0) h ( n ) ( x ) (cid:1) p dx. (2.10)In case n = 1, the inequality (2.10) is sharp, see [3]. (cid:3) Remark 1. If f ∈ U p ( I ) and f ( n ) is bounded on I , so that as p → ∞ in (2.9) , then since lim p →∞ p sin ( πp ) p √ p − = π , therefore we have (cid:12)(cid:12) R [ a,b ] ( f, u ; t , x, t ) (cid:12)(cid:12) ≤ L (cid:26) ( x − a ) (cid:20) x − a (cid:12)(cid:12)(cid:12)(cid:12) t − x + a (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n + ( b − x ) (cid:20) b − x (cid:12)(cid:12)(cid:12)(cid:12) t − x + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n (cid:27) (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) ∞ , [ a,b ] (2.11)In what follows we observe several general quadrature rules for the Riemann–Stieltjes in-tegral R ba f ( t ) du ( t ) where f is n -times differentiable whose derivatives belongs ton L p ([ a, b ]).To the best of our knowledge, this is the first time of such result concerning Riemann–Stieltjesintegral without using interpolation. Corollary 4.
Let f and u as in Theorem 3. In (2.9) choose (1) t = a and t = b , then we get the following trapezoid type inequality (cid:12)(cid:12) R [ a,b ] ( f, u ; a, x, b ) (cid:12)(cid:12) ≤ L p sin (cid:16) πp (cid:17) π p √ p − n n ( x − a ) n +1 − p + ( b − x ) n +1 − p o (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) p, [ a,b ] . or equivalently, we may write using parts formula for Riemann-Stieltjes integral (cid:12)(cid:12)(cid:12)(cid:12) [ f ( b ) − f ( a )] u ( x ) − Z ba u ( s ) df ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ L p sin (cid:16) πp (cid:17) π p √ p − n n ( x − a ) n +1 − p + ( b − x ) n +1 − p o (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) p, [ a,b ] . (2) x = a + b , then we get the following mid-point type inequality (cid:12)(cid:12)(cid:12)(cid:12) R [ a,b ] (cid:18) f, u ; t , a + b , t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L (cid:18) b − a (cid:19) − p p sin (cid:16) πp (cid:17) π p √ p − n (cid:26)(cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) t − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n + (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) t − a + 3 b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n (cid:27) (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) p, [ a,b ] . For instance, setting t = y and t = a + b − y , we get (cid:12)(cid:12)(cid:12)(cid:12) R [ a,b ] (cid:18) f, u ; y, a + b , a + b − y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L (cid:18) b − a (cid:19) − p p sin (cid:16) πp (cid:17) π p √ p − n (cid:26)(cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) y − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n + (cid:20) b − a (cid:12)(cid:12)(cid:12)(cid:12) y − a + 3 b (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) n (cid:27) (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) p, [ a,b ] for all y ∈ (cid:2) a, a + b (cid:3) .(3) t = a + b , x = a + b and t = a +3 b , then (cid:12)(cid:12)(cid:12)(cid:12) R [ a,b ] (cid:18) f, u ; 3 a + b , a + b , a + 3 b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L n − p ( b − a ) n +1 − p p sin (cid:16) πp (cid:17) π p √ p − n (cid:13)(cid:13) f ( n ) (cid:13)(cid:13) p, [ a,b ] The dual assumptions
In this section, L p -error estimates of Two-point quadrature rules for the Riemann–Stieltjesintegral R ba f ( t ) du ( t ), where the integrand f is of bounded variation on [ a, b ] and the inte-grator u is assumed to be satisfied the H¨older condition on [ a, b ]. WO-POINT QUADRATURE RULES FOR THE RIEMANN–STIELTJES INTEGRAL 13
Theorem 4.
Let u : [ a, b ] → R be a H¨older continuous of order r , (0 < r ≤ , and f : [ a, b ] → R is a mapping of bounded p -variation (1 ≤ p ≤ ∞ ) on [ a, b ] . Then we have theinequality (cid:12)(cid:12) R [ a,b ] ( f, u ; t , x, t ) (cid:12)(cid:12) ≤ H max (cid:26) ( t − a ) , (cid:20) t − t (cid:12)(cid:12)(cid:12)(cid:12) x − t + t (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , ( b − t ) (cid:27) r · b _ a ( f ; p ) (3.1) for all a ≤ t ≤ x ≤ t ≤ b . Furthermore, the constant is the best possible in the sense thatit cannot be replaced by a smaller one, for all r ∈ (0 , .Proof. Using the integration by parts formula for Riemann–Stieltjes integral, we have Z t a [ u ( s ) − u ( a )] df ( s ) = [ u ( t ) − u ( a )] f ( t ) − Z t a f ( s ) du ( s ) Z t t [ u ( s ) − u ( x )] df ( s ) = [ u ( t ) − u ( x )] f ( t ) − [ u ( t ) − u ( x )] f ( t ) − Z t t f ( s ) du ( s ) Z bt [ u ( s ) − u ( b )] df ( s ) = [ u ( b ) − u ( t )] f ( t ) − Z bt f ( s ) du ( s ) , Adding these identities, we get Z t a [ u ( s ) − u ( a )] df ( s ) + Z t t [ u ( s ) − u ( x )] df ( s ) + Z bt [ u ( s ) − u ( b )] df ( s )= [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) − Z ba f ( s ) du ( s ) (3.2)Applying the triangle inequality on the above identity and then use Lemma 1, for each termseparately, we get (cid:12)(cid:12)(cid:12)(cid:12) [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) − Z ba f ( s ) du ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t a [ u ( s ) − u ( a )] df ( s ) + Z t t [ u ( s ) − u ( x )] df ( s ) + Z bt [ u ( s ) − u ( b )] df ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z t a [ u ( s ) − u ( a )] df ( s ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z t t [ u ( s ) − u ( x )] df ( s ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z bt [ u ( s ) − u ( b )] df ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup s ∈ [ a,t ] | u ( s ) − u ( a ) | · t _ a ( f ; p ) + sup s ∈ [ t ,t ] | u ( s ) − u ( x ) | · t _ t ( f ; p ) (3.3)+ sup s ∈ [ t ,b ] | u ( t ) − u ( b ) | · b _ t ( f ; p ) . As u is of r - H –H¨older type, we havesup s ∈ [ a,t ] | u ( s ) − u ( a ) | ≤ sup s ∈ [ a,t ] [ H | s − a | r ] = H ( t − a ) r , sup s ∈ [ t ,t ] | u ( s ) − u ( x ) | ≤ sup s ∈ [ t ,t ] [ H | s − x | r ]= H max { ( t − x ) r , ( x − t ) r } = H [max { ( t − x ) , ( x − t ) } ] r = H (cid:20) t − t (cid:12)(cid:12)(cid:12)(cid:12) x − t + t (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r , and sup s ∈ [ t ,b ] | u ( s ) − u ( b ) | ≤ sup s ∈ [ t ,b ] [ H | s − b | r ] = H ( b − t ) r , Therefore, by (3.3), we have (cid:12)(cid:12)(cid:12)(cid:12) [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) − Z ba f ( s ) du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ H ( t − a ) r · t _ a ( f ; p ) + H (cid:20) t − t (cid:12)(cid:12)(cid:12)(cid:12) x − t + t (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r · t _ t ( f ; p ) + H ( b − t ) r · b _ t ( f ; p ) ≤ H max (cid:26) ( t − a ) r , (cid:20) t − t (cid:12)(cid:12)(cid:12)(cid:12) x − t + t (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) r , ( b − t ) r (cid:27) · b _ a ( f ; p )= H max (cid:26) ( t − a ) , (cid:20) t − t (cid:12)(cid:12)(cid:12)(cid:12) x − t + t (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , ( b − t ) (cid:27) r · b _ a ( f ; p ) . To prove the sharpness of the constant 1 for any r ∈ (0 , C >
0, that is, (cid:12)(cid:12)(cid:12)(cid:12) [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) − Z ba f ( s ) du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C max (cid:26) ( t − a ) , (cid:18) t − t (cid:12)(cid:12)(cid:12)(cid:12) x − t + t (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , ( b − t ) (cid:27) r · b _ a ( f ; p ) . (3.4)Choose u ( t ) = t r , r ∈ (0 , t ∈ [0 ,
1] and f : [0 , → [0 , ∞ ) given by f ( t ) = , t ∈ (0 , , t = 0As | u ( x ) − u ( y ) | = | x r − y r | ≤ | x − y | r , ∀ x ∈ [0 , , r ∈ (0 , , it follows that u is r - H -H¨older type with the constant H = 1.By using the integration by parts formula for Riemann-Stieltjes integrals, we have: Z f ( t ) du ( t ) = f (1) u (1) − f (0) u (0) − Z u ( t ) df ( t ) = 0 , WO-POINT QUADRATURE RULES FOR THE RIEMANN–STIELTJES INTEGRAL 15 and W ( f ; p ) = 1. Consequently, by (3.4), we get | t r | ≤ C max (cid:26) t , (cid:18) t − t (cid:12)(cid:12)(cid:12)(cid:12) x − t + t (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , (1 − t ) (cid:27) r , ∀ t , t ∈ [0 , , with t ≤ t . Assume first max (cid:26) t , (cid:18) t − t (cid:12)(cid:12)(cid:12)(cid:12) x − t + t (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , (1 − t ) (cid:27) r = t r so that we get 1 ≤ C .Now, assume thatmax (cid:26) t , (cid:18) t − t (cid:12)(cid:12)(cid:12)(cid:12) x − t + t (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , (1 − t ) (cid:27) r = (1 − t ) r . choose t = 1 − t , so that we get 1 ≤ C .Finally, we assume thatmax (cid:26) t , (cid:18) t − t (cid:12)(cid:12)(cid:12)(cid:12) x − t + t (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , (1 − t ) (cid:27) r = (cid:18) t − t (cid:12)(cid:12)(cid:12)(cid:12) x − t + t (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) r . Define f : [0 , → [0 , ∞ ) given by f ( t ) = , t ∈ (0 , , t = 0 , W ( f ; p ) = 2. Therefore, for t = 0 and t = 1, so that we get 1 ≤ C (cid:0) + (cid:12)(cid:12) x − (cid:12)(cid:12)(cid:1) r /p .Choosing x = and r = p or p = r , it follows that 1 ≤ C (cid:0) (cid:1) r r , i.e., C ≥
1. Hence, theinequality (3.1) is sharp, and the theorem is completely proved. (cid:3)
Theorem 5.
Let ≤ p < ∞ . Let f, u : [ a, b ] → R be such that is u ∈ L p [ a, b ] and f has aLipschitz property on [ a, b ] . If u is r - H –H¨older continuous, then the inequality (cid:12)(cid:12) R [ a,b ] ( f, u ; t , x, t ) (cid:12)(cid:12) ≤ LH ( t − a ) r +1 ( rp +1) p + ( t − t ) − p (cid:16) ( t − x ) rp +1 − ( t − x ) rp +1 rp +1 (cid:17) p + ( b − t ) r +1 ( rp +1) p , a ≤ x ≤ t ≤ t ≤ b ( t − a ) r +1 ( rp +1) p + ( t − t ) − p (cid:16) ( x − t ) rp +1 +( t − x ) rp +1 rp +1 (cid:17) p + ( b − t ) r +1 ( rp +1) p , a ≤ t ≤ x ≤ t ≤ b ( t − a ) r +1 ( rp +1) p + ( t − t ) − p (cid:16) ( x − t ) rp +1 − ( x − t ) rp +1 rp +1 (cid:17) p + ( b − t ) r +1 ( rp +1) p , a ≤ t ≤ t ≤ x ≤ b (3.5) holds for all p > and r ∈ (0 , with constant H > . Proof.
As in the proof of Theorem 4, we have by Lemma 2 (cid:12)(cid:12)(cid:12)(cid:12) [ u ( x ) − u ( a )] f ( t ) + [ u ( b ) − u ( x )] f ( t ) − Z ba f ( s ) du ( s ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z t a [ u ( s ) − u ( a )] df ( s ) + Z t t [ u ( s ) − u ( x )] df ( s ) + Z bt [ u ( s ) − u ( b )] df ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z t a [ u ( s ) − u ( a )] df ( s ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z t t [ u ( s ) − u ( x )] df ( s ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z bt [ u ( s ) − u ( b )] df ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ L " ( t − a ) − p (cid:18)Z t a | u ( s ) − u ( a ) | p ds (cid:19) p + ( t − t ) − p (cid:18)Z t t | u ( s ) − u ( x ) | p ds (cid:19) p + ( b − t ) − p (cid:18)Z bt | u ( s ) − u ( b ) | p ds (cid:19) p ≤ LH " ( t − a ) − p (cid:18)Z t a | s − a | rp ds (cid:19) p + ( t − t ) − p (cid:18)Z t t | s − x | rp ds (cid:19) p + ( b − t ) − p (cid:18)Z bt | s − b | rp ds (cid:19) p . Simple computations yield that Z t a | s − a | rp ds = Z t a ( s − a ) rp ds = ( t − a ) rp +1 rp + 1 , Z t t | s − x | rp ds = R t t ( s − x ) rp ds, a ≤ x ≤ t R xt ( x − s ) rp ds + R t x ( s − x ) rp ds, t ≤ x ≤ t R t t ( x − s ) rp ds, t ≤ x ≤ b = ( t − x ) rp +1 − ( t − x ) rp +1 rp +1 a ≤ x ≤ t x − t ) rp +1 +( t − x ) rp +1 rp +1 , t ≤ x ≤ t x − t ) rp +1 − ( x − t ) rp +1 rp +1 , t ≤ x ≤ b , and Z bt | s − b | rp ds = Z bt ( b − s ) rp ds = ( b − t ) rp +1 rp + 1 . Combining these equalities with the last inequality above we get the required result. (cid:3)
WO-POINT QUADRATURE RULES FOR THE RIEMANN–STIELTJES INTEGRAL 17
Corollary 5.
Let ≤ p < ∞ . Let f, u : [ a, b ] → R be such that is u ∈ L p [ a, b ] and f has aLipschitz property on [ a, b ] . If u is r - H –H¨older continuous, then the inequality (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) f ( t ) + ( b − x ) f ( t ) − Z ba s r − f ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ LH ( t − a ) r +1 ( rp +1) p + ( t − t ) − p (cid:16) ( t − x ) rp +1 − ( t − x ) rp +1 rp +1 (cid:17) p + ( b − t ) r +1 ( rp +1) p , a ≤ x ≤ t ≤ t ≤ b ( t − a ) r +1 ( rp +1) p + ( t − t ) − p (cid:16) ( x − t ) rp +1 +( t − x ) rp +1 rp +1 (cid:17) p + ( b − t ) r +1 ( rp +1) p , a ≤ t ≤ x ≤ t ≤ b ( t − a ) r +1 ( rp +1) p + ( t − t ) − p (cid:16) ( x − t ) rp +1 − ( x − t ) rp +1 rp +1 (cid:17) p + ( b − t ) r +1 ( rp +1) p , a ≤ t ≤ t ≤ x ≤ b (3.6) holds for all p > and r ∈ (0 , with constant H > .Proof. Setting u ( t ) = t r , t ∈ [ a, b ], r ∈ (0 , (cid:3) Corollary 6.
Let ≤ p < ∞ . Let f, u : [ a, b ] → R be such that is u ∈ L p [ a, b ] and f has aLipschitz property on [ a, b ] . If u is K -Lipschitz continuous on [ a, b ] , then the inequality (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) f ( t ) + ( b − x ) f ( t ) − Z ba f ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ LK ( t − a ) ( p +1) p + ( t − t ) − p (cid:16) ( t − x ) p +1 − ( t − x ) p +1 p +1 (cid:17) p + ( b − t ) ( p +1) p , a ≤ x ≤ t ≤ t ≤ b ( t − a ) ( p +1) p + ( t − t ) − p (cid:16) ( x − t ) p +1 +( t − x ) p +1 p +1 (cid:17) p + ( b − t ) ( p +1) p , a ≤ t ≤ x ≤ t ≤ b ( t − a ) ( p +1) p + ( t − t ) − p (cid:16) ( x − t ) rp +1 − ( x − t ) p +1 rp +1 (cid:17) p + ( b − t ) ( p +1) p , a ≤ t ≤ t ≤ x ≤ b (3.7) holds for all p > and constant K > .Proof. Setting r = 1c in Corollary 5, we get the required result. (cid:3) Remark 2.
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