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Applied Mathematics and Computation 217 (2010) 289–294 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc On an Iyengar-type inequality involving quadratures in n knots Vu Nhat Huy a, Quô´c-Anh Ngô a,b,* a b Department of Mathematics, College of Science, Viêt Nam National University, Hà Nôi, Viet Nam Department of Mathematics, National University of Singapore, Block S17 (SOC1), 10 Lower Kent Ridge Road, Singapore 119076, Singapore a r t i c l e i n f o a b s t r a c t In this short note, we give an Iyengar-type inequality involving quadratures in n knots, where n is an arbitrary natural number Ó 2010 Elsevier Inc All rights reserved Keywords: Inequality Error Integral Taylor Iyengar Introduction In 1938, Iyengar [6] proved the following interesting integral inequality which has received considerable attention from many researchers Theorem (See [6]) Let f be differentiable on [a, b] and jf0 (x)j M Then Z b f aị ỵ f bị Mðb À aÞ ðf ðbÞ À f ðaÞÞ2 f ðxÞdx À À : 5 b À a a 4Mðb À aÞ ð1Þ Through the years, Iyengar’s inequality (1) has been generalized in various ways Set Iẳ ba Z b f xịdx a ba b aị2 f aị ỵ f bịị ỵ ðf ðbÞ À f ðaÞÞ; in [1,4], the following Iyengar-type inequality was obtained Theorem (See [1,4]) Let f C2[a, b] and j f00 (x)j M Then jIj M j Dj ðb À aị3 ; 24 24M 2ị where D ẳ f aị 2f aỵb ỵ f ðbÞ: * Corresponding author at: Department of Mathematics, National University of Singapore, Block S17 (SOC1), 10 Lower Kent Ridge Road, Singapore 119076, Singapore E-mail addresses: nhat_huy85@yahoo.com (V Nhat Huy), bookworm_vn@yahoo.com (Q.-A Ngô) 0096-3003/$ - see front matter Ó 2010 Elsevier Inc All rights reserved doi:10.1016/j.amc.2010.05.060 290 V Nhat Huy, Q.-A Ngô / Applied Mathematics and Computation 217 (2010) 289–294 Since then, Theorem was generalized and improved by lots of mathematicians, let us mention the works of Cheng in [3] and Franjic´ et al [5] in the literature In those papers, the authors tried to estimate the left hand side of (2) by various ways In contrast to [3,5], we will generalize the left hand side of (2) into a general form and then obtain some new estimates Before stating our main result, let us introduce the following notation For each i ¼ 1; n, we assume xi 1, put Qðf ; x1 ; ; xn ị ẳ n 1X f a ỵ b aịxk ị: n kẳ1 We are in a position to state our main result Theorem Let I & R be an open interval such that [a, b] & I and let f : I ! R be a twice differentiable function such that, for all x [a, b], c f00 (x) C for some positive c and C Assume fxk gnk¼1 & ẵ0; 1ị is such that x1 ỵ x2 ỵ ỵ xn ẳ n ; 3ị and x21 ỵ x22 ỵ ỵ x2n ẳ nq; where q  0; 12 à ð4Þ is a given number Then the following estimate holds Ap;q ðb À aÞ2 bÀa Z b f ðxÞdx À Qðf ; x1 ; x2 ; ; xn ị ỵ b aịpf bị f ðaÞÞ Bp;q ðb À aÞ2 ; ð5Þ a where p is an arbitrary number and Ap;q À Á q C ỵ p ỵ 16 c; > > À Á > < À C þ p À 2q þ 13 c; Á ¼ À > p q ỵ 16 C ỵ 2q c; > > : p 2q C ỵ 16 c; q q q q À 16 = 0; p 2q ỵ 16 = 0; q c ỵ p ỵ 16 C; > > À > < À c þ C p À q þ 1Á; Á 2q ẳ > p q ỵ c þ C; > >À Á : p À 2q c ỵ 16 C; q q q q À 16 = 0; p À 2q þ 16 = 0; À 16 < 0; p À 2q ỵ 16 = 16 < 0; p 2q ỵ 16 = 0; 16 = 0; p 2q ỵ 16 < 0; 16 < 0; p 2q ỵ 16 < 0; and Bp;q 16 = 0; p 2q ỵ 16 < 0; 16 < 0; p 2q ỵ 16 < 0: Remark If we take n = 2, p ¼ 18 and x1 = 0, x2 = then by (4) one has q ¼ 12 Thus (5) tells us that Z b f aị ỵ f bị b a ỵ f bị f aịị Cỵ c b aị2 f xịdx À C À c ðb À aÞ2 ; 24 bÀa a 24 which is nothing but an Iyengar-type inequality of kind (2) Theorem Let I & R be an open interval such that [a, b] & I and let f : I ! R be a thrice differentiable function such that f000 Lr[a, b] for some < r < The given set fxk gnkẳ1 & ẵ0; 1ị is as in Theorem Then the following estimate holds Z b 3rÀ1 q 0 ðf ðbÞ À f ðaÞÞ K r;q ðb À aÞ r kf 000 kr ; À f ðxÞdx À Q ðf ; x1 ; x2 ; ; xn ị ỵ b aị b À a a ð6Þ where K r;q ¼ rÀ1 rÀ1 rÀ1 rÀ1 r q r1 r q r1 r ỵ þ : 4r À 3r À 2r À 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 : ½a; b ! R and let r be a positive integer If f is such that f(rÀ1) is absolutely continuous on [a, b], x0 (a, b) then for all x (a, b) we have V Nhat Huy, Q.-A Ngô / Applied Mathematics and Computation 217 (2010) 289–294 291 f xị ẳ T r1 f ; x0 ; xị þ RrÀ1 ðf ; x0 ; xÞ; where TrÀ1(f, x0, Á) is Taylor’s polynomial of degree r À 1, that is, T r1 f ; x0 ; xị ẳ r1 ðkÞ X f ðx0 Þðx À x0 Þk ; k! k¼0 and the remainder can be given by RrÀ1 ðf ; x0 ; xị ẳ Z x x0 x tÞrÀ1 f ðrÞ ðtÞ dt: ðr À 1Þ! ð7Þ By a simple calculation, the remainder in (7) can be rewritten as Rr1 f ; x0 ; xị ẳ Z xx0 x x0 tịr1 f rị x0 ỵ tÞ dt; ðr À 1Þ! which helps us to deduce a similar representation of f as follows f x ỵ uị ẳ Z u r1 X uk kị u tịr1 rị f xị ỵ f x ỵ tịdt: k! r 1ị! kẳ0 8ị Proof of Theorem Put Fxị ẳ Z x f tịdt: a Applying Lemma to F(x) with x = a and u = b À a and the Fundamental Theorem of Calculus, we get Z b f xịdx ẳ Fbị Faị ẳ b aịf aị ỵ a b aị2 f aị ỵ Similarly, one has f a ỵ b aịxk ị ẳ f aị ỵ ỵ Z a f aị b aịxk ỵ 1! Z ðbÀaÞxk Z a b ðb À xÞ2 00 f ðxÞdx: b aịxk uịf 00 a ỵ uịdu uẳxk xaị ẳ f aị ỵ f aịb aÞxk b x2k ðb À xÞf 00 ðð1 À xk ịa ỵ xk xịdx: Thus, Pn Q f ; x1 ; x2 ; ; xn Þ ẳ f aị ỵ b aị kẳ1 xk n } |{z f aị ỵ n Z b 1X x2 b xịf 00 xk ịa ỵ xk xịdx n kẳ1 a k ẳ12 ẳ f aị þ n Z b bÀa 1X f ðaÞ þ x2 b xịf 00 xk ịa ỵ xk xịdx; n kẳ1 a k and also f bị f aị ẳ Z b f 00 ðxÞdx: a Then bÀa Z b f ðxÞdx À Q ðf ; x1 ; x2 ; ; xn ị ỵ b aịpf bị À f ðaÞÞ Z b q q ðf ðbÞ À f aịị ỵ b aị p ỵ f bị f aịị ẳ f xịdx À Q ðf ; x1 ; x2 ; ; xn ị ỵ b aị ba a 6 Z b Z b n Z b ðb À xÞ2 00 1X q ¼ À x2k ðb À xÞf 00 ðð1 À xk ịa ỵ xk xịdx ỵ b aị f 00 xịdx f xịdx ba a n kẳ1 a a q ỵ b aị p ỵ f bị f ðaÞÞ Z b Z b n Z b ðb À xÞ2 00 1X q À x2k b xịẵf 00 xk ịa ỵ xk xị cdx ỵ b aị ẵf 00 xị cdx ẵf xị cdx ẳ ba a n k¼1 a a q f bị f aịị: ỵ b aị p ỵ a 292 V Nhat Huy, Q.-A Ngô / Applied Mathematics and Computation 217 (2010) 289–294 We can estimate further as follows bÀa 05 Z b a ðb À xÞ2 00 ½f ðxÞ À cdx bÀa Z b a b xị2 C cị ẵC cdx ¼ bÀa Z b a ðb À xÞ2 dx ẳ b aị2 C cị; while n Z b n Z b 1X 1X x2k b xịẵf 00 xk ịa ỵ xk xị cdx = x2 b xịẵC cdx n k¼1 a n k¼1 a k "Z # ! n n b 1X 1X q ðb À aị2 C cịx2k ẳ b aị2 C cị: x2k b xịC cịdx ẳ ¼À n k¼1 n k¼1 2 a 0= À Moreover, if 2q À 16 = then ðb À aÞ q À Z b a Z b q q b aị2 C cị; ẵf 00 xị cdx b aị ẵC cdx ẳ À À a otherwise, Z b q q ðb À aÞ2 C cị b aị ẵf 00 xị À cdx 0: À À 6 a Finally, if p 2q ỵ 16 = then q q q ðb À aÞ p À þ ðf ðbÞ À f ðaÞÞ Cðb aị2 p ỵ ; 6 q q q b aị p ỵ ðf ðbÞ À f ðaÞÞ cðb À aị2 p ỵ ; 6 cb aị2 p ỵ and Cb aị2 p ỵ provided p 2q ỵ 16 Thus Ap;q ðb À aÞ2 bÀa Z b f ðxÞdx À Qðf ; x1 ; x2 ; ; xn ị ỵ b aÞpðf ðbÞ À f ðaÞÞ Bp;q ðb À aÞ2 ; a where Ap;q À Á q C cị ỵ c p 2q þ 16 ; > > À Á > < À q C cị ỵ q C cị ỵ cp q ỵ 1; 2 À Á ¼ > À 2q ðC À cị ỵ C p 2q ỵ 16 ; > > À Á À Á : q À ðC cị ỵ 2q 16 C cị ỵ C p 2q ỵ 16 ; q q q q À 16 = 0; p 2q ỵ 16 = 0; 16 < 0; 6 p 2q ỵ 16 = 0; = 0; p 2q ỵ 16 < 0; < 0; p 2q ỵ 16 < 0; and Bp;q À Á À Á 81 ðC À cị ỵ 2q 16 C cị ỵ C p 2q ỵ 16 ; > > >1 < C cị ỵ C p 2q ỵ 16 ; q ẳ 61 q > > C cị ỵ C cị ỵ c p þ ; > :1 ðC À cÞ þ c p 2q ỵ 16 ; The proof is now complete q q q q À 16 = 0; 6 < 0; p 2q ỵ 16 = 0; p 2q ỵ 16 = 0; P 0; p 2q ỵ 16 < 0; < 0; p 2q ỵ 16 < 0: h Proof of Theorem We now apply Lemma to F(x) with x = a and u = b À a, we obtain Z b f xịdx ẳ Fbị Faị ẳ b aịf aị ỵ a b aị2 b aị3 00 f aị ỵ f aị ỵ Z a b ðb À xÞ3 000 f ðxÞdx: Similarly, one has Z ðbÀaÞxk f ðaÞ f 00 aị b aịxk uị2 000 uẳx xaị b aịxk ỵ ẵb aịxk ỵ f a ỵ uịdu kẳ f aị 1! 2! 2! Z b xk ðb À xÞ2 000 f 00 aị 2 b aị xk ỵ f xk ịa ỵ xk xịdx: ỵ f aịb aịxk ỵ 2 a f a ỵ b aịxk ị ẳ f aị ỵ V Nhat Huy, Q.-A Ngô / Applied Mathematics and Computation 217 (2010) 289–294 293 Thus, Pn Q ðf ; x1 ; x2 ; ; xn ị ẳ f aị ỵ b aị kẳ1 xk n } |{z f aị ỵ ẳ12 Pn n Z b x xk ðb À xÞ2 000 f 00 ðaÞ 1X b aị2 kẳ1 k ỵ f xk ịa ỵ xk xịdx n kẳ1 a n |fflfflfflffl{zfflfflfflffl} q n Z b xk ðb À xÞ2 000 ba q 1X f aị ỵ f 00 aịb aị2 ỵ ẳ f aị ỵ f xk ịa ỵ xk xịdx; 2 n kẳ1 a and f ðbÞ À f ðaÞ ẳ b aịf 00 aị ỵ Z b b À xÞf 000 ðxÞdx: a Then Z b q 0 À ðf ðbÞ À f ðaÞÞ f ðxÞdx À Q ðf ; x1 ; x2 ; ; xn ị ỵ b aị b a a Z b Z b q q 00 000 ðb À aÞ f aị ỵ b aị ẳ f xịdx Qðf ; x1 ; x2 ; ; xn ị ỵ b xịf xịdx b À a a 6 a Z b Z b ðb À xÞ3 n Z b xk ðb À xị2 000 1X q ẳ b À aÞ ðb À xÞf 000 ðxÞdx: f 000 ðxÞdx f xk ịa ỵ xk xịdx ỵ b À a a n k¼1 a 6 a We can estimate further as follows !rÀ1 rÀ1 Z b r Z b ðb À xÞ3 r 3rÀ1 1 rÀ1 r jðb À xÞ3 jrÀ1 dx kf 000 kr ẳ b aị r kf 000 kr ; f 000 ðxÞdx 6ðb À aÞ a b À a a 4r À while # " Z 1X 1 X b ðb À xÞ2 n Z b n xk ðb À xÞ2 000 3 000 xk f xk ịa ỵ xk xịdx f xk ịa ỵ xk xịdx n k¼1 a n k¼1 2 a " # P rÀ1 rÀ1 n n 3rÀ1 3rÀ1 x3 X rÀ1 r rÀ1 r x3k ðb À aÞ r kf 000 kr ¼ ðb À aÞ r kf 000 kr k¼1 k 2n k¼1 3r À 3r À 2n P rÀ1 rÀ1 n 3rÀ1 3rÀ1 x rÀ1 r q rÀ1 r ðb À aÞ r kf 000 kr kẳ1 k ẳ b aị r kf 000 kr ; 3r À 3r À 2n and Z rÀ1 Z b q 1 b 3rÀ1 q q rÀ1 r 000 000 À ðb À aÞ À ðb À aÞ ðb À xÞf ðxÞdx À ðb À xÞf ðxÞdx ðb À aÞ r kf 000 kr : 6 2r À a a Thus Z b 3rÀ1 q 0 f ðxÞdx À Q ðf ; x1 ; x2 ; ; xn Þ þ ðb À aÞ À ðf ðbÞ À f ðaÞÞ K r;q ðb À aÞ r kf 000 kr ; b À a a where K r;q ¼ rÀ1 rÀ1 rÀ1 rÀ1 r q rÀ1 r q rÀ1 r ỵ ỵ : 4r 3r À 2r À The proof is now complete à Acknowledgments The authors would like to appreciate the anonymous referee’s sharp comments which make the statements with correct proofs References [1] R.P Agarwal, V Cˇuljak, J Pecˇaric´, Some integral inequalities involving bounded higher order derivatives, Math Comput Model 28 (1998) 51–57 [2] G.A Anastassiou, S.S Dragomir, On some estimates of the remainder in Taylor’s formula, J Math Anal Appl 263 (2001) 246–263 294 [3] [4] [5] [6] V Nhat Huy, Q.-A Ngô / Applied Mathematics and Computation 217 (2010) 289–294 X.L Cheng, The Iyengar type inequality, Appl Math Lett 14 (2001) 975–978 N Elezovic´, J Pecˇaric´, Steffensen’s inequality and estimates of error in trapezoidal rule, Appl Math Lett 11 (1998) 63–69 I Franjic´, J Pecˇaric´, I Peric´, Note on an Iyengar type inequality, Appl Math Lett 19 (2006) 657–660 K.S.K Iyengar, Note on an inequality, Math Student (1938) 75–76 ... V Nhat Huy, Q.-A Ngô / Applied Mathematics and Computation 217 (2010) 289–294 Since then, Theorem was generalized and improved by lots of mathematicians, let us mention the works of Cheng in. .. (2) into a general form and then obtain some new estimates Before stating our main result, let us introduce the following notation For each i ¼ 1; n, we assume xi 1, put Qðf ; x1 ; ; xn ị ẳ n. .. 24 which is nothing but an Iyengar-type inequality of kind (2) Theorem Let I & R be an open interval such that [a, b] & I and let f : I ! R be a thrice differentiable function such that f000 Lr[a,
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