A new Ostrowski Grüss inequality involving 3n knots

A new Ostrowski Grüss inequality involving 3n knots tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn...

Applied Mathematics and Computation 235 (2014) 272–282 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc A new Ostrowski–Grüss inequality involving 3n 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 LMPT, UMR CNRS 7350, Université de Tours, Parc de Grandmont, 37200 Tours, France a r t i c l e i n f o Keywords: Integral inequality Taylor expansion Ostrowski Ostrowski–Grüss Simpson Iyengar Bernoulli polynomial a b s t r a c t This is the fifth and last in our series of notes concerning some classical inequalities such as the Ostrowski, Simpson, Iyengar, and Ostrowski–Grüss inequalities in R In the last note, we propose an improvement of the Ostrowski–Grüss inequality which involves 3n knots where n = is an arbitrary numbers More precisely, suppose that fxk gnkẳ1 & ẵ0; 1; fyk gnkẳ1 & ẵ0; 1, and fak gnkẳ1 & ½0; nŠ are arbitrary sequences with Pn Pn k¼1 ak ¼ n and k¼1 ak xk ¼ n=2 The main result of the present paper is to estimate n 1X ak f a ỵ b aịyk ị n kẳ1 ba Z b f tịdt a n f ðbÞ À f ðaÞ X ak ðyk À xk ị n kẳ1 00 in terms of either f or f Unlike the standard Ostrowski–Grüss inequality and its known R  b variants which basically estimate f ðxÞ À a f ðtÞdt =ðb À aÞ in terms of a correction term as a linear polynomial of x and some derivatives of f, our estimate allows us to freely replace f ðxÞ and the correction term by using 3n knots fxk gnk¼1 ; fyk gnk¼1 and fak gnk¼1 As far as we know, this is the first result involving the Ostrowski–Grüss inequality with three sequences of parameters Ó 2014 Elsevier Inc All rights reserved Introduction It is no doubt that one of the most fundamental concepts in mathematics is inequality However, as mentioned in a recent notes by Qi [23], the development of mathematical inequality theory before 1930 are scattered, dispersive, and unsystematic Loosely speaking, the theory of mathematical inequalities has just formally started since the presence of a book by Hardy et al [7] Since then, the theory of mathematical inequalities has been pushed forward rapidly as a lot of books for inequalities were published worldwide Although the set of mathematical inequalities nowadays is huge, inequalities involving integrals and derivatives for real Rb functions always have their own interest Within this kind of inequalities, the one involving estimates of a f ðtÞdt by bounds of the derivative of its integrand turns out to be fundamental as it has a long history and has received considerable attention from many mathematicians Not long before 1934, at the very beginning of the history of mathematical inequalities, in 1921, Pólya derived an inequalRb ity which can be used to estimate the integral a f ðtÞdt by bounds of the first order derivative f His inequality basically says that the following holds ⇑ Corresponding author at: Department of Mathematics, College of Science, Viêt Nam National University, Hà Nôi, Viet Nam E-mail addresses: nhat_huy85@yahoo.com (V.N Huy), quoc-anh.ngo@lmpt.univ-tours.fr, bookworm_vn@yahoo.com (Q.-A Ngô) http://dx.doi.org/10.1016/j.amc.2014.02.090 0096-3003/Ó 2014 Elsevier Inc All rights reserved V.N Huy, Q.-A Ngô / Applied Mathematics and Computation 235 (2014) 272–282    bÀa  Z b   f ðtÞdt kf k1 ; 5   b À a a 273 ð1:1Þ for any differentiable function f having f aị ẳ f bị ẳ and kf k1 ẳ supx2ẵa;b jf j Later on, in 1938, Iyengar [15] generalized (1.1) by showing that    Z b f aị ỵ f ðbÞ b À a ðf ðbÞ À f ðaÞÞ  f ðtÞdt À kf k1 À 5   b À a a 4ðb À aÞkf k1 ð1:2Þ for any differentiable function f Here the only difference is that the condition f aị ẳ f bị ¼ is no longer assumed in (1.2) Apparently, (1.2) provides a simple error estimate for the so-called trapezoidal rule Also in this year, Ostrowski [21, page 226] proved another type of the Pólya–Iyengar inequality (1.2) which tells us how to R  b approximate the difference f ðxÞ À a f tịdt =b aị for x ẵa; bŠ More precisely, he proved that   Z b     f ðxÞ À f ðtÞdt 5    bÀa a À Á2 ! x aỵb ỵ b aịkf k1 ðb À aÞ ð1:3Þ for all x ½a; bŠ As Rwe have just mentioned, unlike (1.1), the inequality (1.3) provides a bound for the approximation of the b integral average a f ðtÞdt =ðb À aÞ by the value f xị at the point x ẵa; bŠ Similar to the inequality (1.2), the Simpson inequality, which gives an error bound for the well-known Simpson rule, has been considered widely which is given as follows       Z b aỵb  Cc  ỵ f bị  f tịdt f aị ỵ 4f b aị;   b À a a 12 ð1:4Þ where C and c are real numbers such that c < f xị < C for all x ẵa; b In recent years, a number of authors have written about generalizations of (1.1)–(1.4) For example, this topic is considered in [2,3,5,14,16,17,20,19,22,26,29] In this way, some new types of inequalities are formed, such as inequalities of Ostrowski–Grüss type, inequalities of Ostrowski–Chebyshev type, etc The present paper is organized as the following First, still in Section 1, let us use some space of the paper to mention several typical generalizations of (1.1)–(1.4) Later on, we shall review our recent works considering as generalizations of (1.1)–(1.4) which aims to propose a completely new idea in order to generalize these inequalities In the final part of this section, we state our main result of the present paper whose proof is in Section 1.1 Generalization of the Ostrowski inequality (1.3) In the literature, there are several ways to generalize the Ostrowski inequality (1.3) The first and most standard way is to replace the term kf k1 on the right hand side of (1.3) by kf kq for any q = where, throughout the paper, we denote kgkq ẳ Z b !1=q jgtịjq dt ; a for any function g Within this direction, Theorem 1.2 in a monograph by Dragomir and Rassias [4] is the best as they were able to derive the best constant, see also [12, Theorem 2] To be completed, let us recall the inequality that they proved   ! Z b  b aị1=p x apỵ1 b xpỵ1 1=p    f tịdt ỵ kf kq f xị  p þ 1Þ1=p  bÀa a bÀa bÀa with 1=p þ 1=q ¼ The second way to generalize the Ostrowski inequality (1.3) is to consider the so-called Ostrowski–Grüss type inequality R  b ÞÀf ðaÞ The only difference is that the term x aỵb ị f bba will be added to control f ðxÞ À a f ðtÞdt =ðb À aÞ Within this type of gen2 eralization, let us recall a result due to Dragomir and Wang in [5, Theorem 2.1] More precisely, they proved the following     Z b  a ỵ b f ðbÞ À f ðaÞ  f ð x Þ À f ðtÞdt À x À  ðb À aÞðC À cÞ   bÀa a bÀa  for all x ½a; bŠ where f is integrable on ½a; bŠ and c f xị C, for all x ẵa; b and for some constants c; C R ð1:5Þ 274 V.N Huy, Q.-A Ngô / Applied Mathematics and Computation 235 (2014) 272–282 Recently in [26], by using f 00 instead of f and replacing C À c by kf 00 k2 , Ujevic´ proved that the following inequality holds     Z b 3=2  a ỵ b f ðbÞ À f ðaÞ ðb À aÞ  pffiffiffi kf 00 k2 : f ð x Þ À f ðtÞdt À x À  5  bÀa a bÀa  2p ð1:6Þ for all x ½a; bŠ provided f 00 L2 ða; bÞ 1.2 Generalization of the Iyengar inequality (1.2) Concerning the Iyengar inequality (1.2), by adding the term ðf ðbÞ À f ðaÞÞðb À aÞ=8 to the left hand side of (1.2), in [6, Corollary 1], the following Iyengar type inequality was obtained    3 !  Z b  f aị ỵ f bị b a M j Dj   ; f tịdt ỵ f bị f aịị ðb À aÞ À  b À a a  24 bÀa M ð1:7Þ for any f C ẵa; b with jf 00 xịj M and D ẳ f aị 2f a þ bÞ=2Þ þ f ðbÞ Other generalizations for (1.2) can also be found in the literature, for example, in [1] 1.3 Generalization of the Simpson inequality (1.4) Regarding to the Simpson inequality (1.4), there are three types of generalization First, using higher order derivatives of f as in [18, Corollary 3], the following Simpson–Grüss type inequalities for n ¼ 1; 2; have been proved Z      b ba aỵb   nỵ1 ỵ f ðbÞ  C n ðCn À cn Þðb À aị ; f t ịdt f aị ỵ 4f    a ð1:8Þ for any function f : ẵa; b ! R such that f n1ị is an absolutely continuous function and cn f ðnÞ ðtÞ Cn for some real constants cn and Cn and where C ¼ 5=72; C ¼ 1=62, and C ¼ 1=1152 Second, we can estimate the left hand side of (1.4) by using the Chebyshev functional associated to f To be exact, the following inequality holds      3=2 qffiffiffiffiffiffiffiffiffiffiffi Z b bÀa aỵb  b aị  ỵ f bị  f t ịdt f aị ỵ 4f rf Þ;    a 6 ð1:9Þ where the operator r is given by rf ị ẳ kf k22 À kf k21 =ðb À aÞ Third, we can generalize (1.4) by using different points rather than a; a ỵ bị=2, and b In fact, the following inequality was proved in [25, Theorem 3] Z pffiffiffi       b p p ba aỵb  aỵb   00  f ðtÞdt À f À À ðb À aị ỵ f ỵ b aị  kf k1 ðb À aÞ ;    a 2 ð1:10Þ for any twice differentiable function f such that f 00 is bounded and integrable Another generalization that follows this idea was obtained in [24, Theorem 7] by considering kf 00 k2 instead of kf 00 k1 This leads us to the following result Z ! !! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffiffi   b 00 ba aỵb aỵb 49   5=2 f ðtÞdt À f À ðb aị ỵ f ỵ b a ị kf k2 ðb À aÞ : 5    a 2 80 2 ð1:11Þ In the following subsection, we summarize our previous works concerning to some generalizations of all inequalities mentioned above Our aim is to highlight the main idea that has been used through these works and that probably is the source of our inspiration to write this paper 1.4 Our previous works Several years ago, we initiated a new research direction which aims to propose a completely new way to treat inequalities of the type (1.1)–(1.4) Before briefly reviewing our results, let us recall some notations that we introduced in [8] for the first time For each k ¼ 1; n, we choose a knot xk for which xk < We then put Q ðf ; n; x1 ; ; xn ị ẳ and If ị ẳ Z a b f ðtÞdt: n b À aX f a ỵ b aịxk ị n kẳ1 V.N Huy, Q.-A Ngô / Applied Mathematics and Computation 235 (2014) 272–282 275 The basic idea of our research direction is to approximate Iðf Þ by Q ðf ; n; x1 ; ; xn Þ under suitable choices of the knots xk Our mission started in 2009 with a generalization of the inequality (1.10), see [8, Theorem 4] In fact, by assuming further that our knots xk satisfy the following system of algebraic equations x1 ỵ x2 þ Á Á Á þ xn ¼ n2 ; > > > > > > < n xj1 ỵ xj2 þ Á Á Á þ xjn ¼ jþ1 ; > > > Á Á Á > > > : mÀ1 x1 ỵ xm1 ỵ ỵ xm1 ẳ mn ; n we were able to prove that jIð f Þ À Q ðf ; n; x1 ; ; xn Þj m!  mq ỵ 1=q  ỵ m 1ịq ỵ 1=q ! kf mị kp b aị mỵ1=q ; 1:12ị for any mth differentiable function f such that f ðmÞ Lp ða; bÞ and where q is chosen in such a way that 1=p ỵ 1=q ¼ Surprisingly, except for the constant appearing on the right hand side of (1.12) which is not optimal, however, as far as we know, all generalizations of either (1.4) and (1.10) or (1.11) always take the form of (1.12) by selecting suitable xk , see [8] for some examples Moreover, our inequality (1.12) provides a new way to generate new inequalities of the form (1.10) and (1.11) Following this research direction, in 2010, we found a new generalization for (1.8) which basically gives us the following estimate mỵ1 jIðf Þ À Q ðf ; n; x1 ; ; xn ịj 2m ỵ b aị S sị m ỵ 1ị! 1:13ị for any mth differentiable function f : ẵa; b ! R where S :¼ supa x b f mị xị and s :ẳ inf a x b f ðmÞ ðxÞ, see [9, Theorem 2] Here the sequence fxk gk is assumed to satisfy a new system of equations given by x1 ỵ x2 ỵ ỵ xn ẳ n2 ; > > > > > > > > < xj ỵ xj ỵ ỵ xjn ẳ n ; jỵ1 > > > > > > xm1 ỵ xm1 ỵ ỵ xm1 ẳ mn ; > n > : m m m n x1 ỵ x2 ỵ ỵ xn ẳ mỵ1 : As can be seen, the estimate (1.13) allows us to freely use derivatives of any order of f In addition, the set of points fa; ða þ bÞ=2; bg which appears in the original estimate (1.8) is now replaced by our knots fxk gk Later on, also in the year 2010, by keeping the sequence fxk gk which satisfies (1.14) above, we obtained the following generalization for (1.9)   mỵ1=2 q 1 b À aÞ jIð f Þ À Q ðf ; n; x1 ; ; xn Þj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðf ðmÞ Þ m! 2m þ 2m À ð1:14Þ for any m-times differentiable function f : ½a; bŠ ! R such that f ðmÞ L2 ða; bÞ, see [10, Theorem 3] Again, as in (1.13), the estimate (1.14) allows us to use derivatives or any order of f and the set of point fa; ða þ bÞ=2; bg is now the set fxk gk Finally, in 2010, we announced a generalization for (1.7) Our generalization has two folds First we replace the term ðf aị ỵ f bịị=2 by the term Q as in the previous works Second, we replaced f 00 by f 000 to get a new estimate Precisely, we proved in [11, Theorems and 4] the following 3 Ap;q ðb À aÞ Iðf Þ À Q ðf ; n; x1 ; ; xn Þ þ ðb À aÞ pðf ðbÞ À f ðaÞÞ Bp;q ðb À aÞ ð1:15Þ       Iðf Þ À Q ðf ; n; x1 ; ; xn ị ỵ b À aÞ2 q À ðf ðbÞ À f ðaÞÞ K r;q ðb À aÞ4À1=r kf 000 k r   ð1:16Þ and where the constant K r;q depends only on q and r while the constants Ap;q and Bp;q depend on p; q; inf a x b f ðxÞ, and supa x b f ðxÞ Besides, the sequence fxk gnkẳ1 & ẵ0; 1ị is now chosen in such a way that & x1 ỵ x2 ỵ ỵ xn ẳ 2n ; x21 ỵ x22 ỵ ỵ x2n ẳ nq; for some q ½0; 1=2Š 276 V.N Huy, Q.-A Ngơ / Applied Mathematics and Computation 235 (2014) 272–282 While the optimal constants for (1.12), and (1.14)–(1.16) remain unknown, the optimal constant for (1.13) has been recently found For a detail of the progress of finding the optimal constants, we refer the reader to [27,31,28], especially the work [30, Theorem 2.3] It is worth noticing that in [30], a beautiful connection between the optimal constant for (1.13) and the well-known Bernoulli polynomials has been established From our point of view, this could be led to optimal constants for the others inequalities such as (1.12), and (1.14)–(1.16) We hope that we shall soon see some responses on this issue 1.5 Our main result In the last paper of the series, our purpose is to make some improvements of Ostrowski type inequalities such as (1.5) and (1.6) In order to see the idea underlying our generalization, let us take a look at the inequalities (1.5)–(1.11) The main difference between the inequalities (1.5) and (1.6) and the others is the presence of f ðxÞ A prior to this work, what we have Rb already done is to keep the integral a f ðtÞdt fixed but freely prescribed the value of f at certain points using our knots In this work, we make a further step by replacing f ðxÞ in (1.5) and (1.6) by something which is new and depends on more than one parameter A simple choice that one could think about is to replace f ðxÞ by a set of new knots Our present work has three folds First, we generalize (1.5) Before doing so, let us further introduce some notation Let = be satised a1 ỵ a2 ỵ ỵ an ẳ n: 1:17ị For each i ẳ 1; n, we assume yi Instead of using f mentioned above, we then use the following quantity Q ðf ; y1 ; ; yn ị ẳ n b aX ak f a ỵ yk b aịị: n kẳ1 1:18ị We note that this new Q given in (1.18) is different from the previous one by the weights ak Besides, Q ðf ; y1 ; ; yn Þ=ðb À aÞ goes back to f ðxÞ if one sets n ¼ 1; a1 ¼ 1, and y1 ¼ ðx À aÞ=ðb À aÞ We are now in a position to state our main result for this generalization Theorem 1.1 Let I & R be an open interval such that ½a; bŠ & I and let f : I ! R be an differentiable function We also let C ẳ supx2ẵa;b f xị and c ẳ inf x2ẵa;b f ðxÞ Then the following estimate holds     n f ðbÞ À f ðaÞ X  ðQ ðf ; y1 ; y2 ; ; yn Þ À Iðf ÞÞ À ak ðyk À xk Þ ðb À aÞðC À cÞ   b a n kẳ1 1:19ị for arbitrary sequences fxk gnkẳ1 & ẵ0; and fyk gnkẳ1 & ẵ0; with n=1 and n a1 x1 ỵ a2 x2 ỵ ỵ an xn ẳ : Clearly, the estimate in (1.19) still makes use of f on the interval ½a; bŠ However, the term f ðxÞ which appears in (1.5) had been changed to Q ðf ; y1 ; y2 ; ; yn Þ=ðb À aÞ In order to see the difference, let us now consider a very special case of (1.19) By choosing n ¼ and a1 ¼ we see that we have no choice for x1 but x1 ¼ 1=2 If we choose y1 ¼ ðx À aÞ=ðb aị where x ẵa; b then, by changing variables, (1.19) tells us that     Z b   xÀa   f ðxÞ À  f ðt Þdt À ðf ðbÞ À f ðaÞÞ À  bÀa a b À a  ðb À aÞðC À cÞ  |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl }   f ðbÞÀf ðaÞ   xaỵbị ba which is nothing but an OstrowskiGrỹss type inequality of the form (1.5) Second, we generalize (1.6) Unlike the previous approach, for simplicity, we shall use kf 00 kp instead of kf 00 k2 We prove the following result Theorem 1.2 Let I & R be an open interval such that ½a; bŠ & I and let f : I ! R be an twice-times differentiable function such that f 00 Lp ða; bÞ; p Then we have     n f ðbÞ À f ðaÞ X  ðQ ðf ; y1 ; y2 ; ; yn Þ À Iðf ÞÞ À ak ðyk À xk Þ ðb À aÞ2À1=p kf 00 kp   b À a n k¼1 for arbitrary sequences fxk gnkẳ1 & ẵ0; and fyk gnkẳ1 & ½0; 1Š with n = and n a1 x1 ỵ a2 x2 ỵ ỵ an xn ẳ : 1:20ị V.N Huy, Q.-A Ngụ / Applied Mathematics and Computation 235 (2014) 272–282 277 As an immediate application of Theorem 1.2, we also obtain    Z b  1   2À1=p f ðtÞdt À ðf ðbÞ À f ðaÞÞ x À kf 00 kp  b aị f a ỵ ðb À aÞxÞ À  bÀa a  for any x ½a; bŠ and any p In the last part of the present paper, we slightly improve (1.12) and (1.13) with weights ak Concerning (1.13), we prove the following result theorem Theorem 1.3 Let I & R be an open interval such that ½a; bŠ & I and let m = be arbitrary We also let f : I ! R be a mth differentiable function and denote S ẳ supx2ẵa;b f mị xị and s ẳ inf x2ẵa;b f mị xị Then we have mỵ1 jIf ị Q f ; x1 ; ; xn Þj 2m ỵ b aị S sị; m ỵ 1ị! 1:21ị for arbitrary sequences fxk gnkẳ1 & ½0; 1Š with n = and a1 x1 þ a2 x2 þ Á Á Á þ an xn ¼ 2n ; > > > > > > > > Á > > > : n a1 xm1 ỵ a2 xm2 ỵ ỵ an xmn ẳ mỵ1 ; 1:22ị Regarding to (1.12), we prove the following result Theorem 1.4 Let I & R be an open interval such that ½a; bŠ & I and let f : I ! R be a mth differentiable function with m = such that f ðmÞ Lp ða; bÞ; p Then the following estimate holds jIðf Þ À Q ðf ; x1 ; ; xn Þj m!  mq ỵ 1=q  ỵ m 1ịq þ 1=q ! mþ1=q ðb À aÞ kf ðmÞ kp ; 1:23ị for arbitrary sequences fxk gnkẳ1 & ẵ0; satisfying a1 x1 ỵ a2 x2 ỵ ỵ an xn ẳ 2n ; > > > > > > < n a1 xj1 ỵ a2 xj2 ỵ ỵ an xjn ẳ jỵ1 ; > > >ÁÁÁ > > > : a1 xmÀ1 þ a2 xmÀ1 þ Á Á Á þ an xmÀ1 ẳ mn ; n 1:24ị and 1=p ỵ 1=q ¼ Before closing this section, we would like to mention that due to the restriction of the technique that we use, inequalities (1.19), (1.20), (1.23), and (1.21) are not sharp However, the presence of the paper [30] strongly proves that there could be some possibility to get optimal constants for all these inequalities Besides, it turns out that the right hand sides of (1.19), (1.20), (1.23), and (1.21) not depend on n but the regularity of the function f This is because we want to unify all the number of (interpolation) points appearing in all known inequalities mentioned at the beginning of the present paper by n, see (1.8)–(1.11) Finally, it is worth noting that rather than the classical inequalities mentioned above, other classical inequalities such as the Fejér and Hermite–Hadamard inequalities have also been studied, for example, see [13] Proofs We spend this section to prove Theorems (1.1)–(1.4) First, we prove Theorem 1.1 Proof of Theorem 1.1 By using the Taylor formula with the integral remainder, it is not hard to check that f ða ỵ b aịyk ị ẳ f aị ỵ Z baịyk f a ỵ tịdt ẳ f aị ỵ Z baị yk f a ỵ yk t ịdt ẳ f aị ỵ 0 a Therefore, by taking the sum for k from to n, we get n X n X kẳ1 kẳ1 ak f a ỵ b aịyk ị ẳ nf aị ỵ ak Z which can be rewritten using our notation as a b Z ! yk f a1 yk ị ỵ yk t Þdt ; b yk f ðað1 À yk ị ỵ yk tịdt: 278 V.N Huy, Q.-A Ngụ / Applied Mathematics and Computation 235 (2014) 272–282 ! Z b n 1X Qðf ; y1 ; y2 ; ; yn ị ẳ f aị ỵ ak yk f a1 yk ị ỵ yk tịdt : ba n kẳ1 a Similarly, we obtain ! Z b n 1X Qðf ; x1 ; x2 ; ; xn ị ẳ f aị þ ak xk f ðað1 À xk Þ þ xk t ịdt : ba n kẳ1 a 2:1ị Hence, by subtracting, we arrive at    Q ðf ; y ; y ; ; y Þ Q ðf ; x ; x ; ; x Þ f ðbÞ À f ðaÞ X n  n n À À ak ðyk À xk Þ    bÀa bÀa n k¼1     Z b Z b 1 X n n X f ðbÞ À f ðaÞ f bị f aị   ẳ dt À dt  ak yk f ðað1 À yk ị ỵ yk t ị ak xk f a1 xk ị ỵ xk t ị  n k¼1 bÀa n k¼1 bÀa a a Z b Z b X X n n 1 ay C cịdt ỵ ax C cịdt b aịC cị; kẳ1 k k kẳ1 k k n |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} n |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} a a 5n ð2:2Þ ¼n=2 where we have used the fact that f and f bị f aịị=b aị belong to ẵc; CŠ From the estimate (2.2), it is necessary to control Rb Q ðf ; x1 ; x2 ; ; xn Þ This can be done if we use a f ðtÞdt This is the content of the next part of the proof Indeed, thanks to Z b f tịdt ẳ Z a b b tịf tịdt ỵ b aịf aị a and (2.1), some easy calculation first shows that  Z Z ! Z ! Z b   b  b b    f ðtÞdt À Q ðf ; x1 ; x2 ; ; xn Þ ¼  ðb À tÞf ðtÞdt À ðb À tÞdt f ðtÞdt    a  a bÀa a a ! Z ! Z b b b tịdt f tịdt ỵ ba a a |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} M Z b n bÀa X À ak xk f xk ịa ỵ xk t ịdt n a k¼1 ! Z !!! Z b b ak xk dt f xk ịa ỵ xk t Þdt À bÀa a a   !  b b X n bÀa  À ak xk dt f xk ịa ỵ xk t ịdt : kẳ1  n ba a a |{z} Z ! Z N Clearly, M ẳ b aịf ðbÞ À f ðaÞÞ=2 which implies 1 2 ðb À aÞ c M ðb À aÞ C: 2 For the term N, it is clear that N¼ n 1X n k¼1 Z b ak xk dt a ! Z ! b a f ðð1 xk ịa ỵ xk t ịdt ẳ n 1X b aịak xk n kẳ1 which yields ! ! Z b Z b Xn Xn ðb À aÞak xk cdt N ðb À aÞak xk Cdt : k¼1 k¼1 n n a a |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} cðbÀaÞ2 =2 CðbÀaÞ2 =2 Therefore, the difference M À N is now easy to handle as follows jM À Nj ðb À aÞ ðC À cÞ: Z a ! b f ðð1 À xk Þa þ xk t Þdt 279 V.N Huy, Q.-A Ngơ / Applied Mathematics and Computation 235 (2014) 272–282 For the remaining terms in the expansion of Indeed, we can estimate further as follows Rb a f ðtÞdt À Q ðf ; x1 ; x2 ; ; xn Þ above, one may consult the Grüss inequality Z ! Z ! Z b   b b   0 ðb À tÞdt f ðtÞdt  ðb À aÞ ðC À cÞ:  ðb À tÞf ðtÞdt À   a bÀa a a We note that n X ak Z b a k¼1 ! Z ! Z b n b X xk f xk ịa ỵ xk tịdt ak xk dt f xk ịa ỵ xk tịdt ẳ 0: b a kẳ1 a a Hence, all in one, we arrive at  Z   b  f ðtÞdt À Q ðf ; x1 ; x2 ; ; xn Þ   b À a a Z ! Z ! Z b   b  b  0 ðb À tÞdt f ðtÞdt   ðb À tÞf ðtÞdt À  b À a a bÀa a a jM À Nj bÀa ! b aị C cị ỵ ðb À aÞ ðC À cÞ bÀa ỵ ẳ b aịC cị: 2:3ị Having (2.2) and (2.3) yields   Z b  Q ðf ; y ; y ; ; y Þ n f ðbÞ À f ðaÞ X   n À f ðtÞdt À a ðy À x Þ  k k k    bÀa bÀa a n k¼1  Q ðf ; y1 ; y2 ; ; yn Þ Q ðf ; x1 ; x2 ; ; xn Þ  À bÀa bÀa   Z   n f ðbÞ À f ðaÞ X  b   À ak ðyk À xk ị ỵ f tịdt Q f ; x1 ; x2 ; ; xn Þ     n b À a a k¼1 ðb À aÞðC À cÞ: The proof is now complete h We now prove Theorem 1.2 whose proof is basically based on Theorem 1.1 The idea is to control C À c from the above in terms of f 00 Proof of Theorem 1.2 To prove the theorem, we observe from the Hölder inequality that, for all u; v ½a; bŠ satisfying u v , there holds Z v  Z v 1=p   1=q jf uị f v ịj ẳ  f 00 ðtÞdt  jf 00 ðtÞjp dt ðv À uÞ1=q kf 00 kp ðb À aÞ ; u u where 1=p ỵ 1=q ẳ Thanks to C ẳ supx2ẵa;b f xị; c ẳ inf x2ẵa;b f xị, we immediately have C c kf 00 kp ðb À aÞ1=q : Making use of this and Theorem 1.1, we obtain     n f ðbÞ À f ðaÞ X  ðQ ðf ; y1 ; y2 ; ; yn Þ À Iðf ÞÞ À ak ðyk xk ị b aị1ỵ1=q kf 00 kp ;   b À a n k¼1 which now completes the proof because ỵ 1=q ẳ À 1=p h To prove Theorem 1.3, we follow the same idea and method used in [9] and refer the reader to [9] for details Proof of Theorem 1.3 By applying the Taylor formula with the integral remainder to the function If ị ẳ m X kỵ1 b aị f kị aị ỵ k ỵ 1ị! kẳ0 Z bÀa Rx a f ðtÞdt, we arrive at m b a tị mị f a ỵ tịdt: m! ð2:4Þ 280 V.N Huy, Q.-A Ngơ / Applied Mathematics and Computation 235 (2014) 272–282 For each i n, applying the Taylor formula with the integral remainder again to the function f ðxÞ, we now get k mÀ1 k X x ðb À aÞ i f ða ỵ xi b aịị ẳ k! kẳ0 k aị kị f aị ỵ k! kẳ0 Z xi baị m1 X xk b i ẳ f kị aị þ Z bÀa ðxi ðb À aÞ À tÞ m 1ị! m1 f mị a ỵ tịdt m1 xm i ðb À a À tÞ ðm À 1Þ! f mị a ỵ xi tịdt: Then by summing up and thanks to the first m À equations in (1.22), we deduce that n X k mÀ1 X ðb aị iẳ1 kẳ0 f a ỵ xi b aịị ẳ n k ỵ 1ị! f kị aị þ n Z X i¼1 bÀa xmi ðb À a À tÞmÀ1 ðm À 1Þ! f ðmÞ ða þ xi tÞdt: In other words, we have proved that Qðf ; x1 ; ; xn Þ ẳ kỵ1 m1 X n b aị b aX f kị aị ỵ n iẳ1 k ỵ 1ị! kẳ0 Z ba xmi b a tịm1 m 1ị! f mị a ỵ xi tịdt: ð2:5Þ Combining (2.4) and (2.5) gives !   m n xmi ðb À ÁÞmÀ1 ðmÞ ðb À ÁÞ ðmÞ b À aX À Iðf Þ À Q ðf ; x1 ; ; xn ị ẳ I f I f xi ịa ỵ xi ị : n iẳ1 m! m 1ị! Observe that  m m À Á ðb À aÞ ðb À ÁÞ :I f ðmÞ : I ðf ðmÀ1Þ ðbÞ f m1ị aịị ẳ ba m ỵ 1ị! m! Therefore, we can write   ðb À aÞm ðmÀ1Þ bÀa P jIðf Þ À Qðf ; x1 ; ; xn ịj ẳ  f bị f m1ị aịị ỵ M N ; n n m ỵ 1ị! with MẳI Nẳ    m m À Á ðb À ÁÞ ðmÞ ðb À ÁÞ À I f ðmÞ ; I f bÀa m! m! n X I i¼1 xmi ðb À ịm1 m 1ị! ! PẳI !! f mị xi ịa ỵ xi ị m1 xmi b À ÁÞ À I bÀa ðm À 1Þ! ! I f mị xi ịa ỵ xi ÁÞ ; xmi ðb À ÁÞmÀ1 À ðm À 1Þ! Á I f ðmÞ ðð1 À xi Þa þ xi ÁÞ : Making use of the Grüss inequality, see [9, Lemma 5], gives that mỵ1 jMj ðb À aÞ m! ðS À sÞ and that m jNj m n ðb À aÞ xm 1X n b aị i S sị ẳ S sị: iẳ1 m 1ị! m þ 1Þðm À 1Þ! For remaining terms, it is clear that mỵ1 m mỵ1 b aị b aị ðb À aÞ s5 ðf ðmÀ1Þ ðbÞ À f ðmÀ1Þ aịị S; m ỵ 1ị! m ỵ 1ị! m ỵ 1ị! while a direct calculation shows Pẳ n X Á xmi ðb À aÞm À ðmÞ I f xi ịa ỵ xi ị : m! iẳ1 Consequently, thanks to holds Pn k¼1 ak xmk ¼ n=ðm þ 1Þ and here is the only place we make use of the last equation in (1.22), there V.N Huy, Q.-A Ngô / Applied Mathematics and Computation 235 (2014) 272–282 mỵ1 281 mỵ1 nb aị nb aị s5P5 S: m ỵ 1ị! m ỵ 1ị! In other words, we have proved that   ðb À aÞm À m1ị P b aịmỵ1 m1ị 5  f bị f aị S sị: m ỵ 1ị! n m ỵ 1ị! Thus, Theorem 1.3 follows by using the triangle inequality h We now prove Theorem 1.4 To this purpose, we follow the same idea and method used in [8] and we refer the reader to [8] for details Proof of Theorem 1.4 From the proof of Theorem 1.3 and using the triangle inequality, we obtain     mÀ1  n  ðb À ÁÞm ðmÞ  b À aX  ai xm ðmÞ i ðb À ÁÞ   jIðf Þ À Q ðf ; x1 ; ; xn Þj  f  ỵ f xi ịa ỵ xi ÁÞ :    n m! ðm À 1ị! iẳ1 2:6ị Thanks to [8, Eq (10)], the first term sitting on the right hand side of (2.6) can be estimated as follows !   mqỵ1 1=q ðb À ÁÞm ðmÞ  ðb À aÞ   kf ðmÞ kp :  m! f  m! mq ỵ 1 2:7ị For the second term, we also note from [8] that ( xi kf mị xi ịa ỵ xi ịkp xi kf mị k1 ; if p ẳ 1; kf mị kp ; if p < 1: Thanks to xi ẵ0; 1, we can write xi kf mị xi ịa ỵ xi ịkp kf mị kp in any case Making use of the Hölder inequality, one can estimate the second term on the right hand side of (2.6) as follows   mÀ1  n  ðb À ÁÞ b À aX  ai xm f mị xi ịa ỵ xi ị  i  n iẳ1  m 1ị! n xmi b À aX mÀ1 kf ðmÞ ðð1 xi ịa ỵ xi tịkp kb ị kq n iẳ1 m 1ị! n xm1 b aX mÀ1 i kf ðmÞ kp kðb À ÁÞ kq n iẳ1 m 1ị! !1=q kf mị kp b aịmqỵ1 ẳ : m! m 1ịq ỵ 2:8ị Combining relations (2.6)(2.8), we conclude that mqỵ1 ðb À aÞ jIðf Þ À Q ðf ; x1 ; ; xn Þj m! mq þ !1=q kf ðmÞ kf ðmÞ kp ðb À aịmqỵ1 kp ỵ m! m 1ịq ỵ !1=q and Theorem 1.4 follows h It is interesting to note that a weaker version for the inequality (1.23) can be derived from Theorem 1.3 so long as m P Indeed, similar to the proof of Theorem 1.2, we can estimate f ðmÀ1Þ in terms of kf ðmÞ kp to obtain 1=q S À s kf ðmÞ kp ðb À aÞ : From this, Theorem 1.3 with m replaced by m À 1, and thanks to m P 3, we obtain jIðf Þ À Q ðf ; x1 ; ; xn ịj 2m ỵ mỵ1=q b aị kf mị kp : 4m! Clearly, the preceding estimate is weaker than that of Theorem 1.4 since m! 1=q  1=q ! 1 2m ỵ < ỵ < mq ỵ m 1ịq ỵ m! 4m!  for any m = and any q = Note that here we require m = rather than m = as in Theorem 1.4 since we have to make use of Theorem 1.3 with m replaced by m À Having this fact, the condition m À = automatically leads to m = as claimed 282 V.N Huy, Q.-A Ngô / Applied Mathematics and Computation 235 (2014) 272–282 Before closing our paper, we would like to comment on the similarity between the right hand sides of (1.12) and (1.20) Indeed, more or less, the right hand side of (1.20) is a particular case of that of (1.12) when one uses m ¼ except the power of b À a Note that, in the case m ¼ 2, the right hand side of the estimate (1.12) reads as  2q ỵ 1=q  ỵ qỵ1 1=q ! 31=p b aị kf 00 kp : The difference between the power of b À a in (1.12) and (1.20) relies on the fact that our expansion (2.1) is only up to f One way to equalize these orders is to use some expansion for higher orders together with the Hölder inequality For interested reader, we refer to the proof of (1.12) in [8, Theorem 4] Therefore, we expect that there exists a positive constant C which only depends on p and m such that     n f bị f aị X   mỵ11=p Qf ; y1 ; y2 ; ; yn Þ À Iðf ÞÞ À ðyk À xk Þ Cðb À aÞ kf ðmÞ kp ;   b À a n kẳ1 2:9ị where the sequence fxk gk satises the same system of equations for (1.12) We shall not prove (2.9) here and leave it for interested reader Acknowledgments The authors would like to thank the two anonymous referees for careful reading of our manuscript and the valuable comment and suggestions, especially for pointing out a mistake in the first paragraph after the proof of Theorem 1.4 in the previous version of the paper V.N Huy was partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 101.01-2011.32 Q.A Ngô acknowledges the support of the Région Centre through the convention no 00078780 during the period 2012-2014 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] R.P Agarwal, V Cˇuljak, J Pecˇaric´, Some integral inequalities involving bounded higher order derivatives, Math Comput Model 28 (1998) 51–57 G.A Anastassiou, Ostrowski type inequalities, Proc Am Math Soc 123 (1995) 3775–3781 X.L Cheng, Improvement of some Ostrowski–Grüss type inequalities, Comput Math Appl 42 (2001) 109–114 S.S Dragomir, T.M Rassias, Ostrowski type inequalities and applications in numerical integration, RGMIA Monographs, Victoria University, 2000 S.S Dragomir, S Wang, An inequality of Ostrowski–Grüss’s 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) 15–20 N Elezovic´, J Pecˇaric´, Steffensen’s inequality and estimates of error in trapezoidal rule, Appl Math Lett 11 (1998) 63–69 G.H Hardy, J.E Littlewood, G Pólya, Inequalities, Cambridge University Press, 1934 V.N Huy, Q.A Ngô, New inequalities of Ostrowski-like type involving n knots and the Lp -norm of the mth derivative, Appl Math Lett 22 (2009) 1345– 1350 V.N Huy, Q.A Ngô, A new way to think about Ostrowski-like type inequalities, Comput Math Appl 59 (2010) 3045–3052 V.N Huy, Q.A Ngô, New inequalities of Simpson-like type involving n knots and the mth derivative, Math Comput Model 52 (2010) 522–528 V.N Huy, Q.A Ngô, On an Iyengar-type inequality involving quadratures in n knots, Appl Math Comput 217 (2010) 289–294 V.N Huy, Q.A Ngô, New bounds for the Ostrowski-like type inequalities, Bull Korean Math Soc 48 (2011) 95–104 V.N Huy, Q.A Ngô, Some new results on the Fejér and Hermite–Hadamard inequalities, Rocky Mountain J Math 43 (2013) 1625–1636 S Hussain, A Qayyum, A generalized Ostrowski-Grüss type inequality for bounded differentiable mappings and its applications, J Inequal Appl 2013 (1) (2013) pp K.S.K Iyengar, Note on an inequality, Math Student (1938) 75–76 W Liu, A Tuna, Weighted Ostrowski, trapezoid and Grüss type inequalities on time scales, J Math Inequal (2012) 381–399 W Liu, Y Zhu, J Park, Some companions of perturbed Ostrowski-type inequalities based on the quadratic kernel function with three sections and applications, J Inequal Appl 2013 (226) (2013) 14 pp M Matic´, Improvement of some inequalities of Euler–Grüss type, Comput Math Appl 46 (2003) 1325–1336 M Matic´, J.E Pecˇaric´, N Ujevic´, Improvement and further generalization of some inequalities of Ostrowski–Grüss type, Comput Math Appl 39 (2000) 161–175 G.V Milovanovic´, J.E Pecˇaric´, On generalization of the inequality of A Ostrowski and some related applications, Univ Beogmd Publ Elektrotehn Fak Ser Mat Fiz 544–576 (1976) 155–158 A.M Ostrowski, Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment Math Helv 10 (1938) 226–227 J Park, Generalizations of Ostrowski–Grüss type integral inequalities for twice differentiable mappings, Int J Appl Math Stat 39 (2013) 10–23 F Qi, Polya type integral inequalities: origin, variants, proofs, refinements, generalizations, equivalences, and applications, RGMIA Res Rep Collect 16 (2013) Article 20 (32 pp) N Ujevic´, Error inequalities for a quadrature formula of open type, Rev Colombiana Mat 37 (2003) 93–105 N Ujevic´, Error inequalities for a quadrature formula and applications, Comput Math Appl 48 (2004) 1531–1540 N Ujevic´, New bounds for the first inequality of Ostrowski–Grüss type and applications, Comput Math Appl 46 (2003) 421–427 S.W Vong, A note on some Ostrowski-like type inequalities, Comput Math Appl 62 (2011) 532–535 Z Wang, S.W Vong, On some Ostrowski-like type inequalities involving n knots, Appl Math Lett 26 (2013) 296–300 Z Wang, S.W Vong, On some generalizations of an Ostrowski–Grüss type integral inequality, Appl Math Comput 229 (2014) 239–244 Y Wu, Y Wang, On the optimal constants of Ostrowski-like inequalities involving n knots, Appl Math Comput 219 (2013) 7789–7794 Y.X Xiao, Remarks on Ostrowski-like inequalities, Appl Math Comput 219 (2012) 1158–1162  ... inequalities for a quadrature formula and applications, Comput Math Appl 48 (2004) 1531–1540 N Ujevic´, New bounds for the first inequality of Ostrowski Grüss type and applications, Comput Math Appl... worth noting that rather than the classical inequalities mentioned above, other classical inequalities such as the Fejér and Hermite–Hadamard inequalities have also been studied, for example, see... inequalities, Rocky Mountain J Math 43 (2013) 1625–1636 S Hussain, A Qayyum, A generalized Ostrowski- Grüss type inequality for bounded differentiable mappings and its applications, J Inequal Appl 2013 (1)