DSpace at VNU: NEW BOUNDS FOR THE OSTROWSKI-LIKE TYPE INEQUALITIES tài liệu, giáo án, bài giảng , luận văn, luận án, đồ...
Bull Korean Math Soc 48 (2011), No 1, pp 95–104 DOI 10.4134/BKMS.2011.48.1.095 NEW BOUNDS FOR THE OSTROWSKI-LIKE TYPE INEQUALITIES ´c-Anh Ngo ˆ ˆ Vu Nhat Huy and Quo Abstract We improve some inequalities of Ostrowski-like type and further generalize them Introduction In 1938, Ostrowski [8] proved the following interesting integral inequality which has received considerable attention from many researchers Theorem (See [8]) Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b) whose derivative function f ′ : (a, b) → R is bounded on (a, b), i.e., ∥f ′ ∥∞ = supt∈(a,b) |f ′ (t)| < ∞ Then ( ( )2 ) ∫ b x − a+b 1 (1) f (t)dt ≦ + (b − a)∥f ′ ∥∞ f (x) − b−a a (b − a)2 for all x ∈ [a, b] This inequality gives an upper bound for the approximation of the integral ∫b average b−a f (t)dt by the value f (x) at point x ∈ [a, b] The first generalizaa tion of Ostrowski inequality was given by G V Milovanovi´c and J E Peˇcari´c in [7] However, note that estimate (1) can be applied only if f ′ is bounded In the first part of this paper, we will improve (1) by assuming f ′ ∈ Lp (a, b) for some ≦ p < ∞ More precisely, we obtain the following theorem Theorem Assume that ≦ p Let I ⊂ R be an open interval such that [a, b] ⊂ I and let f : I → R be a differentiable function such that f ′ ∈ Lp (a, b) Then we have ∫ b (2) f (x) − f (t)dt ≦ A(x, q)∥f ′ ∥p b−a a Received May 13, 2009 2010 Mathematics Subject Classification 26D10, 41A55, 65D30 Key words and phrases inequality, error, integral, Taylor, Ostrowski c ⃝2011 The Korean Mathematical Society 95 ˆ´C-ANH NGO ˆ VU NHAT HUY AND QUO 96 for all x ∈ [a, b] where A(x, q) = b−a and p + q ( ( )q+1 ) q1 b−a a+b + x− q+1 2 q = Remark limq→+∞ A(x, q) = for each x ∈ [a, b] Example Let us consider the integral ∫ 1√ sin (t2 )dt Then we have f (t) = √ sin (t2 ) and ( ) 2t cos t2 f ′ (t) = √ 3 sin2 (t2 ) such that f ′ (t) → ∞ as t → On the other hand, we have ( ) ∫ ∫ t2 cos t2 16 dt √ ≦ |f ′ (t)| dt ≦ max , 2) sin (t 0≦t≦1 sin (t ) 0 i.e., ∥f ′ ∥L2 ≦ 43 It follows that ( ) √ ∫ 1√ √ 1 3 2 √ + x− sin (x ) − sin (t )dt ≦ 24 for all x ∈ [0, 1] In recent years, a number of authors have written about generalizations of Ostrowski inequality For example, this topic is considered in [1, 3, 4, 6, 11, 5] In this way, some new types of inequalities are formed, such as inequalities of Ostrowski-Griiss type, inequalities of Ostrowski-Chebyshev type, etc The first inequality of Ostrowski-Gră uss type was given by Dragomir and Wang in [4] It was generalized and improved by Mati´c, Peˇcari´c, and Ujevi´c in [6] Cheng gave a sharp version of the mentioned inequality in [3] Recently in [11], Ujevi´c proved the following result which gives much better results than estimations based on [3] Theorem (See [11, Theorem 4]) Let f : I → R, where I ⊂ R is an interval, ◦ be a twice continuously differentiable mapping in the interior I of I with f ′′ ∈ ◦ L2 (a, b) and let a, b ∈ I , a < b Then we have ( ) ∫ b a + b f (b) − f (a) (b − a) ′′ √ ∥f ∥2 (3) f (x) − f (t)dt − x − ≦ b−a a b−a 2π for all x ∈ [a, b] NEW BOUNDS FOR THE OSTROWSKI-LIKE TYPE INEQUALITIES 97 If we assume f is such that f ′′ is of class Lp for some ≦ p < ∞, then we obtain: Theorem Let f : I → R, where I ⊂ R is an interval, be a twice continuously ◦ differentiable mapping in the interior I of I with f ′′ ∈ Lp (a, b), ≦ p < ∞, we have ( ) ∫ b a + b f (b) − f (a) (4) f (x) − f (t)dt − x − ≦ B(q)∥f ′′ ∥p b−a a b−a for all x ∈ [a, b] where ( ) q1 ( ) q1 q+1 2q+1 (b − a) (b − a) B(q) = + q+1 2(b − a) 2q + and p + q = Remark limq→+∞ B(q) = 2(b − a) Proofs Before proving our main theorem, we need an essential lemma below It is well-known in the literature as Taylor’s formula or Taylor’s theorem with the integral remainder Lemma (See [2]) Let f : [a, b] → R and let r be a positive integer If f is such that f (r−1) is absolutely continuous on [a, b], x0 ∈ (a, b), then for all x ∈ (a, b) we have f (x) = Tr−1 (f, x0 , x) + Rr−1 (f, x0 , x) , where Tr−1 (f, x0 , ·) is a Taylor’s polynomial of degree r − 1, that is, Tr−1 (f, x0 , x) = r−1 (k) k ∑ f (x0 ) (x − x0 ) k=0 and the remainder can be given by (5) ∫ x Rr−1 (f, x0 , x) = x0 k! r−1 (x − t) f (r) (t) dt (r − 1)! By a simple calculation, the remainder in (5) can be rewritten as ∫ x−x0 r−1 (r) (x − x0 − t) f (x0 + t) Rr−1 (f, x0 , x) = dt (r − 1)! which helps us to deduce a similar representation of f as following ∫ u r−1 k r−1 ∑ (u − t) u (k) f (x) + f (r) (x + t) dt (6) f (x + u) = k! (r − 1)! k=0 ˆ´C-ANH NGO ˆ VU NHAT HUY AND QUO 98 Proof of Theorem Denote ∫ x f (t)dt F (x) = a By Fundamental Theorem of Calculus I (f ) = F (b) − F (a) Applying Lemma gives ( ) ( ) ∫ b a+b b−a ′ a+b F (b) = F F + + (b − t) F ′′ (t)dt a+b 2 2 which implies that ( F (b) − F We see that ( F (a) = F which yields ( F (a) − F a+b ) a+b ) a+b Therefore, ) ( F (b) − F (a) = (b − a)f ( b−a = f ( a−b ′ + F a−b = f a+b ) ( ∫ a+b a+b a+b ∫ a+b ) ∫ a + a+b ) ∫ a+b + (b − t) f ′ (t)dt (a − t) F ′′ (t)dt (t − a) f ′ (t)dt a ′ ∫ (b − t) f (t)dt − a+b By changing t = a + b − x, we get ∫ b ∫ (b − t) f ′ (t)dt = b + b + a+b ) a+b (t − a) f ′ (t)dt a a+b (t − a) f ′ (a + b − t) dt a which helps us to deduce that ( ) ∫ a+b ∫ b a+b f (t)dt = (b − a)f + (t − a) (f ′ (a + b − t) − f ′ (t)) dt a a On the other hand, ( f (x) − f Then f (x) − ∫ x = a+b b−a ∫ ) ∫ x = f ′ (t)dt a+b b f (t)dt a f (t)dt − b−a ′ a+b ∫ a a+b (t − a) (f ′ (a + b − t) − f ′ (t)) dt NEW BOUNDS FOR THE OSTROWSKI-LIKE TYPE INEQUALITIES Next we consider the case < p < ∞ We first have the following estimates ∫ a+b (t − a) (f ′ (a + b − t) − f ′ (t)) dt a ∫ a+b ≦ ∫ ′ (t − a) f (a + b − t) dt + a a+b ≦ |f ′ (a + b − t)| dt p ) p1 (∫ a ) q1 a+b q |t − a| dt a (∫ a+b + ′ p ) p1 (∫ |f (t)| dt a = (t − a) f ′ (t)dt a (∫ ( a+b a+b ) q1 q |t − a| dt a ( )q+1 ) q1 b−a ∥f ′ ∥p q+1 Clearly, ∫ x a+b f ′ (t)dt ≦ ∫ p x ∫ b ≦ |f ′ (t)| dt p a+b p ′ p 1q dt ∫ |f (t)| dt a+b = ∥f ∥p x − q x a+b q x q dt a+b a ′ ∫ q Hence, ∫ b f (t)dt b−a a ( ( )q+1 ) q1 b − a a+b + x− ≦ b−a q+1 2 f (x) − q ∥f ′ ∥ p If p = 1, then ∫ a+b (t − a) (f ′ (a + b − t) − f ′ (t)) dt a ∫ a+b b−a (|f ′ (a + b − t)| + |f ′ (t)|) dt a b−a ′ = ∥f ∥1 ≦ 99 ˆ´C-ANH NGO ˆ VU NHAT HUY AND QUO 100 and ∫ x a+b f ′ (t)dt ≦ ∥f ′ ∥1 which helps us to claim that f (x) − Corollary If we put x = and ≦ p < ∞, we have ( f a+b ) − b−a Note that b−a ( ∫ b a b−a a+b , ∫ b f (t)dt ≦ a ′ ∥f ∥1 □ then under the assumptions of Theorem 1 f (t)dt ≦ b−a ( ( )q+1 ) q1 b−a ∥f ′ ∥p q+1 ( )q+1 ) q1 ( ) q1 ( )1 1 b−a q b−a = q+1 2 q+1 Proof of Theorem Clearly, by Lemma one has ∫ b 1 f (t)dt = (F (b) − F (a)) b−a a b−a ( ) ∫ b 2 (b − a) ′′ (b − t) ′′′ ′ (b − a)F (a)+ = F (a)+ F (t)dt b−a 2 a ∫ b b−a ′ (b − x) ′′ = f (a) + f (a) + f (t)dt b−a a Similarly, ∫ ′ b f (x) = f (a) + (x − a) f (a) + (b − t) f ′′ (t)dt a and f (b) − f (a) = b−a b−a ( (b − a)f ′ (a) + = f (a) + b−a ′ ∫ ∫ b ) (b − t) f ′′ (t)dt a b (b − t) f ′′ (t)dt a Therefore, ( ) ∫ b a + b f (b) − f (a) f (x) − f (t)dt − x − b−a a b−a ∫ b ∫ b ∫ b x − a+b (b − t) ′′ ′′ f (t)dt − (b − t) f ′′ (t)dt (b − x) f (x) dt − = b−a a b−a a a NEW BOUNDS FOR THE OSTROWSKI-LIKE TYPE INEQUALITIES If < p < ∞, then by the Hăolder inequality, one has ) q1 ( ( b a (b − t) f ′′ (t)dt ≦ ∥f ′′ ∥p and b−a ∫ b a b q (b − t) dt = a q+1 (b − a) q+1 ) q1 ∥f ′′ ∥p , (∫ ) q1 b (b − t) ′′ 2q ′′ f (t)dt ≦ ∥f ∥p (b − t) dt 2(b − a) a ( ) q1 2q+1 (b − a) ∥f ′′ ∥p , = 2(b − a) 2q + and ∫ x − a+b b−a b (b − t) f ′′ (t)dt ≦ a ∫ ≦ b (b − t) f ′′ (t)dt a ( q+1 (b − a) q+1 ) q1 ∥f ′′ ∥p Thus, ( ) a + b f (b) − f (a) f (t)dt − x − b−a a ( ) q1 ( ) q1 q+1 2q+1 (b − a) (b − a) ∥f ′′ ∥ + ≦ p q+1 2(b − a) 2q + f (x) − b−a ∫ b If = p, then again by the Hăolder inequality, one has b b ′′ (b − t) f (t)dt ≦ (b − a) |f ′′ (t)| dt = (b − a)∥f ′′ ∥1 , a and b−a ∫ b a and x − a+b b−a a 2 (b − t) ′′ (b − a) f (t)dt ≦ b−a ∫ a b (b − t) f (t)dt ≦ ′′ ∫ b a ∫ b |f ′′ (t)| dt = a (b − t) f ′′ (t)dt ≦ b − a ′′ ∥f ∥1 , (b − a)∥f ′′ ∥1 Hence, f (x) − b−a ∫ b a ( ) a + b f (b) − f (a) f (t)dt − x − ≦ 2(b − a)∥f ′′ ∥1 b−a 101 ˆ´C-ANH NGO ˆ VU NHAT HUY AND QUO 102 Therefore, ( ) a + b f (b) − f (a) f (t)dt − x − b−a a ( )q ( ) q1 q+1 2q+1 (b − a) (b − a) ∥f ′′ ∥ + ≦ p q+1 2(b − a) 2q + f (x) − b−a ∫ b □ Corollary If we put x = a+b , then under the assumptions of Theorem and ≦ p < ∞, we have ( ) ∫ b a+b f f (t)dt − b−a a ( ) q1 ( ) q1 q+1 2q+1 (b − a) (b − a) ∥f ′′ ∥ ≦ + p q+1 2(b − a) 2q + Applications in numerical integral Let Γ = {x0 = a < x1 < · · · < xn = b} be a given subdivision of the interval [a, b] such that h = xi+1 − xi = b−a n Then we obtain the following theorem by using Corollary Theorem Under the assumptions of Theorem and ≦ p < ∞, we have ( ) ( ) q1 ( )1 ∫ b n xi−1 + xi 1 1∑ b−a q ′ f − f (t)dt ≦ ∥f ∥p n i=1 b−a a 2n q + Proof We have ) ( ) q1 ( )1 ( ∫ xi n b−a q ′ xi−1 + xi − f (t)dt ≦ ∥f ∥p,[xi−1 ,xi ] f b − a xi−1 q+1 2n where ∥f ′ ∥p,[xi−1 ,xi ] = Then, (∫ xi ) p1 |f ′ (t)| dt p xi−1 ( ) ∫ b n xi−1 + xi 1∑ f − f (t)dt n i=1 b−a a ( ) q1 ( )1 n b−a q ∑ ′ ≦ ∥f ∥p,[xi−1 ,xi ] 2n q + 2n i=1 ∫ Put xi αi = xi−1 |f ′ (t)| dt p NEW BOUNDS FOR THE OSTROWSKI-LIKE TYPE INEQUALITIES Then n ∑ i=1 ′ ∥f ∥p,[xi−1 ,xi ] = n ∑ ( p αi ≦ n 1− p i=1 n ∑ 103 ) p1 αi = n1− p ∥f ′ ∥p i=1 Therefore, ( ) ∫ b n 1∑ xi−1 + xi f f (t)dt − n i=1 b−a a ( ) q1 ( )1 b − a q 1− p1 ′ ≦ ∥f ∥p n 2n q + 2n ( ) q1 ( )1 1 b−a q ′ = ∥f ∥p 2n q + □ If we use Corollary 2, we then obtain the following theorem whose proof will be omitted Theorem Under the assumptions of Theorem and ≦ p < ∞, we have ( ) ∫ b n xi−1 + xi 1∑ f − f (t)dt n i=1 b−a a ( ) q1 ( ) q1 q+1 2q+1 (b − a) (b − a) ∥f ′′ ∥ + ≦ 2 p n q+1 2(b − a) 2q + References [1] G A Anastassiou, Ostrowski type inequalities, Proc Amer Math Soc 123 (1995), no 12, 3775–3781 [2] G A Anastassiou and S S Dragomir, On some estimates of the remainder in Taylor’s formula, J Math Anal Appl 263 (2001), no 1, 246–263 [3] X L Cheng, Improvement of some Ostrowski-Gră uss type inequalities, Comput Math Appl 42 (2001), no 1-2, 109–114 [4] S S Dragomir and S Wang, An inequality of Ostrowski-Gruss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Comput Math Appl 33 (1997), no 11, 15–20 [5] V N Huy and Q A Ngˆ o, New inequalities of Ostrowski-like type involving n knots and the Lp -norm of the m-th derivative, Appl Math Lett 22 (2009), no 9, 1345–1350 [6] M Mati´ c, J E Peˇ cari´ c, and N Ujevi´ c, Improvement and further generalization of inequalities of Ostrowski-Gră uss type, Comput Math Appl 39 (2000), no 3-4, 161–175 [7] G V Milovanovi´ c and J E Peˇ cari´ c, On generalization of the inequality of A Ostrowski and some related applications, Univ Beograd Publ Elektrotehn Fak Ser Mat Fiz No 544-576 (1976), 155158 ă [8] A M Ostrowski, Uber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment Math Helv 10 (1938), 226–227 [9] N Ujevi´ c, Error inequalities for a quadrature formula of open type, Rev Colombiana Mat 37 (2003), no 2, 93–105 ˆ´C-ANH NGO ˆ VU NHAT HUY AND QUO 104 , Error inequalities for a quadrature formula and applications, Comput Math Appl 48 (2004), no 10-11, 1531–1540 [11] , New bounds for the first inequality of Ostrowski-Gră uss type and applications, Comput Math Appl 46 (2003), no 2-3, 421–427 [10] Vu Nhat Huy Department of Mathematics College of Science ˆt Nam National University Vie ` No ˆ i, Vie ˆt Nam Ha E-mail address: nhat huy85@yahoo.com ´c-Anh Ngo ˆ ˆ Quo College of Science ˆt Nam National University Vie ` No ˆ i, Vie ˆt Nam Ha and Department of Mathematics National University of Singapore Block S17 (SOC1), 10 Lower Kent Ridge Road 119076, Singapore E-mail address: bookworm vn@yahoo.com ... Error inequalities for a quadrature formula and applications, Comput Math Appl 48 (2004), no 10-11, 1531–1540 [11] , New bounds for the first inequality of Ostrowski-Gră uss type and applications,... − a) (f ′ (a + b − t) − f ′ (t)) dt NEW BOUNDS FOR THE OSTROWSKI-LIKE TYPE INEQUALITIES Next we consider the case < p < ∞ We first have the following estimates ∫ a+b (t − a) (f ′ (a + b − t)... way, some new types of inequalities are formed, such as inequalities of Ostrowski-Griiss type, inequalities of Ostrowski-Chebyshev type, etc The first inequality of Ostrowski-Gră uss type was