Copyright © 2002 L. J. Dedić et al. This work is licensed under a Creative Commons License.Some generalizations of Ostrowski inequality are given, by using some Euler identities involving harmonic sequences of polynomials.L. J. Dedić, M. Matić, J. Pečarić, and A. Vukeli

Dedic, Ljuban

Matic, M.

Pecaric, Josip

Vukelic, A.

Journal of Inequalities and Applications

L. J. Dedić

M. Matić

J. Pečarić

A. Vukelić

Crossref

On generalizations of Ostrowski inequality via Euler harmonic identities

Adelaide Research & Scholarship

J. ofInequal. & Appl., 2002, Vol. 7(6), pp. 787-805 Taylor & FrancisTaylor & Francis GroupOn Generalizations of OstrowskiInequality via Euler HarmonicIdentitiesL. J. DEDIa’*, M. MATI(b’t, J. PE(ARI(c’$ and A. VUKELI(d’’IaDepartment of Mathematics, University of Split, Teslina 12, 21000 Split,Croatia; bFESB, Mathematics Department, University of Split, R. Bo#kovi6aBB, 21000 Split, Croatia; CApplied Mathematics Department, University ofAdelaide, Australia; dFaculty of Food Technology and Biotechnology,Mathematics Department, University of Zagreb, Pierottijeva 6, 10000 Zagreb,Croatia(Received 7 April 2001; Revised 10 July 2001)Some generalizations of Ogtrowski inequality are given, by using some Euler identitiesinvolving harmonic sequences of polynomials.Keywords: Ostrowski inequality; Harmonic polynomialsClassification: 1991 Mathematics Subject Classification: 26D15, 26D20, 26D991 INTRODUCTIONLet Bk(t), k > 0, be the Bernoulli polynomials, Bk Bk(0), k > 0, theBernoulli numbers. The Bernoulli polynomials Bk(t), k > 0 are un-iquely determined by the following identitiesBk(t) kBk_ (t), k > 1; Bo(t) (1.1)* E-mail: ljuban@pmfst.hrCorresponding author. E-mail: mmatic@fesb.hrE-mail: jpecarie@spam.adelaide.edu.auE-mail: avukelie@pbf.hrISSN 1025-5834 print; ISSN 1029-242X. (C) 2002 Taylor & Francis LtdDOI: 10.1080/1025583021000022504788 L.J. DEDIt et al.andBk(t + l)-Bk(t) kt-1, k > O. (1.2)For example, the three first Bemoulli polynomials are given byBo(t) 1, B (t) t- (1/2), Bz(t) 2 / (1/6). For some furtherdetails on the Bernoulli polynomials and the Bernoulli numbers seefor example [1] or [2].Let B*(t), k >_ O, be the periodic fimctions of period 1, related to theBernoulli polynomials as(t) (t), 0 _< < , (t + ) *(t), t R.From the properties of the Bernoulli polynomials it follows thatB 1, B’ is a discontinuous function with a jump of- at each inte-ger, and BT,, k > 2, is a continuous function.Letf" [a, b] R be such thatf"-) is a function of bounded varia-tion on [a, b] for some n > 1. In the recent paper [4] the following twoidentities have been proved:f(x) b a f(t)dt + Tn(x) + R,(x) (1.3)andf(x) b a f(t)dt + Tn- (x) + R2n(x), (1.4)where To(x) 0 andTin(x) Z (b a)k-kk=lxafor < rn < n, while(b a)n-I.x) In! j[a,blxOSTROWSKI INEQUALITY 789andRn(X) (b-n!a)n-1 j x x-aia,b][B*n(b 5 ta) Bn(b a)] df(n-l’(t)"Here, as in the rest of the paper, we write [a,b]g(t)dqg(t) to denote theRiemann-Stieltjes integral with respect to a function qg" [a, b] R ofbbounded variation, and fa g(t) dt for the Riemann integral. The formulae(1.3) and (1.4) hold for every x 6 [a, hi. They are extensions of the wellknown formula for the expansion of an arbitrary functionf with a con-tinuous nth derivative f(n) in Bernoulli polynomials [3, p. 17]:Iif(t) dt + Tn-l(x) + Rn(x),f(x) b awhere(b a)n-1 Ji x xn’ [B*n(b- ta)- Bn(b )]r(n,(t) dr"Let the polynomials Pk(t), k > 0 satisfy the following conditionP(t) Pk-I (t), k >_ 1; Po(t)For a sequence (Pk(t), k > 0) of polynomials satisfying the condition(1.5), we say that it is a harmonic sequence of polynomials. From (1.5),by an easy induction it follows that every harmonic sequence of poly-nomials must be of the formkCi tk-i,P#,(t) Z (k i)----.i=O k>O,where (ck, k > 0) is a sequence of real numbers such that co 1. Infact, ck Pk(0), k > 0. Especially, we have Po(t) 1, Pl(t) + ci,Pz(t) (1/2)t2 -+- cit + cz.The aim of this paper is to generalize the formulae (1.3) and (1.4), byreplacing the Bernoulli polynomials by an arbitrary harmonic sequence790 L.J. DEDI( et al.of polynomials, and using them to prove some generalizations ofOstrowski inequality.2 EULER HARMONIC IDENTITIESAssume that (Pk(t), k > 0) is a harmonic sequence of polynomials i.e.the sequence of polynomials satisfying the condition (1.5). DefineP,(t), k > 0 to be a periodic functions of period 1, related to Pk(t),k>0asP(t)=Pk(t), O < < P(t-t-1) *-Pk(t), tER.Thus, P(t) 1, while for k > 1, P(t) is continuous on R\Z and has ajump ofk Pk(0) Pk(1) (2.1)at every integer t, whenever czk-76 0. Note that --1, sincePl (t) + Cl, for some Cl R. Also, note that from (1.5) it follows,!Pk (t) Pk_l(t), k > 1, E R\Z. (2.2)Let a, b 6 R, a < b, and.]" [a, b] --+ R be such thatf(n-l) is a func-tion of bounded variation on [a, b] for some n >_ 1. For every x 6 [a, b]and < rn < n we introduce the following notationsmk=lwith convention To(x)= 0, andmZ,n(X) Z(b a)’- 0kf(k-l)(x),k=2(2.4)with convention -t’l(X) O.OSTROWSKI INEQUALITY 791LEMM, Let a, b 6 R, a < b, x 6 [a,b], k > O. Define q)k(x;.)’[a, b] R asqgk(x; t) P b- a<t<b.Ifk > 1, then for every continuous function F" [a, b] --+ R, we haveia,b]F(t) dqgk(x; t) F(t)qk_(x; t) dt akF(x),for a < x < b, andF(t) dok(b; t)a,b]F(t)qk_l(b; t) dt kF(a).Proof Let k > and assume that a < x < b. The function qgk(x; .) isdifferentiable on [a,b]\{x} and its derivative is equal to (-1/(b- a))qgk_(x; .), by (2.2). Further, it has a jump of ok(x;x + O)-qgk(x; x- 0)-k at x, which gives the first formula in this case. Forx a the function qk(a; .) is differentiable on (a, b) and its derivative isequal to (-1/(b a))qgk_ (a; .). Further, it has jump of qgk(a; a + 0)ok(a; a)-k at the point a, while qgk(a; b) qok(a; b 0) 0, whichgives the first formula for x a. The second formula is a consequenceof the first one and of the fact that qg(b; .) qg(a; .).THEOREM Let (Pk, k > 0) be a harmonic sequence ofpolynomialsandf" [a, b] R such thatftn-1) is a continuousfunction ofboundedvariation on [a, b] for some n > 1. Then for every x [a, b]f(x) b---- a f(t) dt + ’n(X) + Tn(X) -]-" knl(X), (2.5)f(x) b-a f(t)dt + ’n- (x) + n(X) .qt_ kn2(X), (2.6)792 L.J. DEDI( et al.where n(X) and Zn(X) are defined by (2.3) and (2.4), respectively, andia,b]X x--aProof For < k < n consider the integralsI *(x-t)df(k-)(t)"I(x) ( a)-P a[a,b]By partial integration we get(2.7)First, assume that a < x < b. For every k > we have(;_:)P’ =P-a ) (x-a)=pk(Xb-a)=P*k b-a -aTherefore, using the first fonnula from Lemma 1, we get from (2.7)ax d(k-l)(x) + (b a)k-z l)(t)P2_ ,Z,,a dt. (2.8)Since exi -1, for k (2.8) reduces to(x)I (x) P b L-a If(b) -f(a)] -f(x) + b a f(t) dt. (2.9)OSTROWSKI INEQUALITY 793For k > 2 we have(b a)k-2 Pk-I . (t) Ik_l(x)[a,b]and (2.8) can be rewritten asx-a+ (b a)-f(-l(x) + I_ (x). (2.10)From (2.9) and (2.10) it is easy to obtainnk=l-t- E(b a)k-I of(k-1)(x) f(x) + b a f(t) dr,k=2which is equivalent to (2.5), since In(x) --n(X). Thus, (2.5) holds fora < x < b. If x= b, then we have P,((b-b)/(b-a))- P(O)=P,(O), P,((b a)/(b a)) P,(1) P,(0) Pk(0). Similarly as wedid for a < x < b, using the above equalities and the second formulafrom Lemma 1, we get from (2.7).D,(b) (b a)k-iP,(O)[f(k-1)(b) -f(k-1)(a)]+ (b a)#’- kf(k-l)(a) + Ik_ (b),bfor k > 2, and I1 (b) Pl (O)[f(b) -f(a)] -f(a) + (1 I(b a)) .a f(t) dt.Applying the above identities, we getIn(b) Z(b a)-Pk(O)[f(-l)(b) -f(-l)(a)]k=l+ E(b a)k-1 kf(k-l)(a) --f(a) + b- a f(t) dt.k=2794 L.J. DEDI( et al.We have zi -1 and, by (2.1), Pk(0) Pk(1) + k. Therefore, the lastidentity can be rewritten asnI.(b) Z(b a)k-’Pk(1)[f(k-l)(b) f(k-1)(a)]k=l+ Z(b- a)-e(-(b)-f(b) + b a f(ok=2which is equivalent to (2.5) for x b, since In(b) -n(b).Note thatTherefore, +&(X) --’.(X) ,,_ (X) //(X), SO that the formula (2.6)follows from the formula (2.5).Example 1 Let Pk(t)- (1/k!)Bk(t), k >_0, where Bk(t) are theBernoulli polynomials. From (1.1) it follows that (Pk(t), k > 0) is aharmonic sequence of polynomials. Also, we have 1,23.1.19, 23.1.20]B2j(O) B2j(1) B2j, B2j+l (0) B2j+l(1) O, j >_ 1,which implies that cz 0 for k > 2, while 01 -1 as in the generalcase. Moreover, in this case, for any f" [a, b]--R such that f(n-l) isa continuous function of bounded variation on [a, b], we getL,(x) Tn,(X), "Cm(X) O, m _< n and kin(x) Rn(x),i Rn(x)~2 R2n(X),where T,,,(x), Rn(x) and R2n(x) are defined as in the Introduction. Conse-quently, the formulae (2.5) and (2.6) become (1.3) and (1.4), respec-tively.Example 2 For fixed ? R defineP(t)= (t--?)k, k>0.OSTROWSKI INEQUALITY 795Then (Pk(t), k > 0) is a harmonic sequence of polynomials, ande(0) T., (-r), P,(1)--(1Therefore, in this casea, P,(O) P,(1)-.[(-7)’ (1 7)’], k>l.Further, we havem=(b ,0k!k=l?) ’ [fq’-)(b) fq’- )(a)]andm"Cm(X) E (b a)’-k!k=2 [(_?)k (1 T)kV(k-l)(x),for every x 6 [a, b].3 GENERALIZATIONS OF THE OSTROWSKI INEQUALITYIn this section we shall use the same notations as above.THEOREM 2 Suppose that (Pk(t), k >_ O) is a harmonic sequence ofpolynomials. Letf’[a, b]--R be such thatf(n-) is an L-Lipschitzianfunction on [a, b] for some n >_ 1. Thenf(x) b a f(t) dt Tn- (x) "On(X)<_(b-a) Ien(t)-Pn(b_a (3.1)f(x) b a f(t)dt- ,(x) "Cn(X) <__ (b a)n IPn(t)[ dt. L,(3.2)for every x [a, b].796 L.J. DEDI( et al.Proof If tp:[a,b]--- R is L-Lipschitzian on [a,b], that is[q(x)- p(y)[ < L. Ix- y[, Yx, y [a, hi, then for any integrablefunction g:[a, b] --+ R we haveg(t) dip(t) _< [g(t)[ dt.L. (3.3)Using this estimate we getIkZ <x)l (bx- x a<_ (b -a)n-I IP:(b ta) -Pn(-b--L a)l dt. L.)(t)Since the function P*(.) has period 1, we have[P(y + t)- z] dt ]Pn*(’)- zl dt IP,(t) z dr,for every y, z 6 R. Thereforex x- a Ii x- a[P:(b 5 ta) Pn("’b a,)l dt (b a) IPn(t) Pn(.b a)l dr,which impliesand (3.1) follows from (2.6).The inequality (3.2) follows from (2.5) by the similar argument.Remark 1 Both of the above inequalities for n (the second onewith cl ---((x- a)/(b- a)) are reduced to the Ostrowski inequalityfor a functionf which is L-Lipschitzian on [a, b] (see [10]).Remark 2 (i) The inequality (3.1) for n--2 and cl--(-1/2) wasproved in [4]. Also it is an improvement and extension of the similarresult from [13] (for details see Remark 3 in [4]).OSTROWSKI INEQUALITY 797For x a or x b we have the trapezoid inequality which with itsgeneralization and applications was considered in [5] (see also [16]).For x ((a + b)/2) we have the midpoint inequality and its general-ization and applications were considered in [6].(ii) cl may be chosen depending on fixed x, e.g. 1---((x- a)/(b a)). In this case we get result which is an extension ofthe same result from [11] where the above inequality was proved withL I "11 for a class of functions with bounded second derivatives.THEOREM 3we haveIff’ is L-Lipschitizian on [a, b], then for every x [a, b]f(t) dt-1 (a+b,(x)+-(b-a) x 2< x + 4)"(3.4)Proof If we put n 2 in the first identity from Theorem we havef(x) = b a f(t) dt + P.xb [f(b) -f(a)]--a+ (b- a)P2 (xb a)[f’(b)-f’(a)] + (b- a)zff’(x)whereandPl(t) + Cl, P2(t) (t2/2) + cit + C22 -((112) + c).First we chose C and C2 such that PI((X-a)/(b-a))= 0 andP2((x-a)/ (b a)) O, i.e. Cl -((x- a)/(b a)) and c2=(1/2)((x a)/(b a))2. This gives Pl(t) ((x a)/(b a)),798 L.J. DEDI( et al.P2(t) (1/2)(t ((x a)/(b a)))2(b- a) so that we haveand 2 (x- ((a + b)12))/’(x)f(x) b a f(t) dt + x- 2, X[a,b]Further we consider the identity (3.5) for x b with P(t) and Pz(t)replaced by/51 (t) = + c-1 and P-2(t) (t2/2) + c- + c-z, respectively,i.e.f(b) b a f(t) dt + fil (1)[f(b) -f(a)]+ (b a)fiz(1)[f’(b) -f’(a)]-(b-a,(+Ul’,x,-,b-a,I[a,t,]15(-;-) df’(t).We chose cr and 72 such that (1/2)+ (l- 0 and P2(1)--0, i.e.(l -(1/2) and c2 0. This gives /3 (t) (1/2), fiz(t)(1/2)t(t 1) andf(b) b a f(t) dt + -} [f(b) -f(a)]or equivalentlyf(a) +f(b)2 li’f(t) dt (b a) j [a,b]OSTROWSKI INEQUALITY 799If we multiply (3.6) and (3.7) with (1/2) and then add them up, we getthe identityb- a f(t) dt-- f(x) + +- x-b -a’l[a,b][e(;- ta) q_e(bb -)] (3.8)If we denote the left hand side of (3.8) by R(x), then using the estimate(3.3) we getL [11 a+b[ (b a)31=b-a x 2 + 4-----dtMultiplying the above inequality by b a > 0 we get (3.4). (The equal-ity case in the above expression can be done by elementary but ratherlong calculation and we omit the details).Remark 3 Whenf is a twice differentiable function with bounded andintegrable second derivative, the inequality (3.4) holds with L If"ll.So this inequality is a correction and in the same time an extension ofthe main result from [14]. Namely it is easy to see thata)2 K(x, t), (3.9)where(a + b),) for [a, x],(t a) 2K(x, t)(t b) ( (a + b),) for 6 (x,2 (3.10)800 L.J. DEDI( et al.and for twice differentiable functionf with integrable second derivativethe identity (3.8) multiplied by b- a reduces toif(t) dt- -=-K(x,t)f"(t)dt.a+2 b)f’(x)(3.11)In [14] the incorrect version of the identity (3.11), with -(b-a)(x- (a + b)/2) f’(x) in place of (1/2)(b a) (x (a + b)/2)f’(x),was obtained as a basic result.THEOREM 4 Let f" [a, b]-- R be such that f(n-l) is a continuousfunction of bounded variation on [a, b] for some n >_ 1. Thenf(x)-’--a f(t) dt ’n- (x) Zn(X)< (b a)n-I max IPn(t)- en(xbt[o,l]f(x) b a f(t) dt Tn(x) Zn(X)< (b- a)"-! max [P(t)lVba(f(n-))t[o,]for every x [a, b], where Va (f("- 1)) is the total variation off(n- on[a, b].Proof If F" [a,b]--+ R is bounded and the Stieltjes integralf[a,b] F(t) df(n-)(t) exists, thenF(t) df(n- )(t) _< max IF(t)l" Va(f("-)).te[a,bl(3.12)OSTROWSKI INEQUALITY 801Let us apply this estimation to the second formula of Theorem 1. Wehavef(t) dt ’n- (x)andX X a[kn2(X)[ -(b a)n-1a[e (b ta) en (b ---a) l df(n-1)(t)< (b a)n- max n Pn Vba(f(n-))t[a,b]xma(b a)n-I max ]Pn(t)- Pn (b a)t[0,1]which proves our first assertion. Similarly we prove the second one.Remark 4 The first inequality for n was proved in [9] (see also [7]and [17]). The second one with n and c -((x a)/(b a)) wasproved in [9]. For n 2 and c -((x a)/(b a)) in first inequalitywe have result which is an extension of the result from [12] withVba(f’) I[f"lll for a class of functions f such that f" E LI (a, b).THEOREM 5[a, b] thenIff’ is a continuous function of bounded variation onIS 1[ f(a)f(b)] (a+b)f,(x)f(t) dt-f(x) + (b a) +-(b a) x 2a+bl (b-a) a+bx-. 2 + 2for x [a,14-4qra +(b-a)216...+ 3-/forxe 4a+ b,4 4X4 {’-J 4 a+ 43- + v/b)4 a+ 4802 L.J. DEDI( et al.Proof Using the estimate (3.12) we get from (3.8) and (3.9)2 t[a,b]l’)sup IK(x, t)l,2(b a)t[a,b] (3.13)where R(x) is the left hand side of (3.8) and K(x, t) is defined by (3.10).Further, by a simple calculation we getsup ]g(x, t)[ max[.(bl- a)2t[a,b] 16x-2 +-2 xforx a,.----a+(b a)216forx6 4 a+a+b 2 b-a+ 23 b] ["J[3 ’e/4 4 aq-+ V/b,b]"4Substituting this in (3.13) and then multiplying by b-a, we getproposed inequality.Remark 5 When f is a twice differentiable function with integrablesecond derivative such thatf" L1 (a, b), the inequality proved in theabove theorem holds with V’(f’) replaced by Ilf"ll . Therefore thisinequality can be regarded as a double correction and, in the same time,as an extension of the analogous result from [15] for a class of functionsf with f" E L (a,b). The first correction is related to the expressionwithin the absolute value sign at the left hand side (the same as inRemark 3), while the second one is related to the obviously incorrectequality IlK(x, ")ll suPt[a,b] Ig(x, t)[ ((b a)2/4) which is statedand used in [15].OSTROWSKI INEQUALITY 803THEOREM 6 Let f:[a, b]-R be such that f(n) is R-integrable andf(n) Lp[a, b] for some n > and < p < c. Thenf(x)f(x)b a f(t) dt Tn_ (x) "Cn(X)(If -a dr)< (b- a)n-l+l/q IPn(t) en(_ a)lq 1/qb a f(t) dt- Tn(x)<_ (b a)n-l+l/q IP,(OI q dtfor every x [a, b], where 1/p + 1/q 1.Proof Since f(n) is R-integrable, the integrals which occur in the ex-pressions for/(x) and/2n(x) are the usual Riemann integrals withdf(n-l)(t) replaced withf(n)(t) dr. So by applying the H61der inequalitywe get from (2.6)f(x)which proves the first stated inequality. The second one follows from(2.5) by the similar argument.Remark 6 This first inequality of the theorem above for n wasproved by A. M. Fink [18] (see also [8] and [17]).804 L.J. DEDIt et al.THEOREM 7thenIff" is R-integrable andf" E Lp[a, b], for < p < cxz,b-l[f(x)+f(a)f(b)] -1(x a+b/’(x)2aY(t)dt (b a) q-l(b-a) 2+(l/q)< Ill’lip-2 2[B(q q- 1, q + 1) + Bx,(q + 1, q + 1) + Wx2(q + 1, q q- 1)](l/q)forx a, 2[B(q + 1, q + 1) + Bx3(q + 1, q + 1) + Bx4(q + 1, q + 1)](l/q)[a+bb],forx6 2where (l/p) + (l/q) 1, p > 1, q > 1, and B(., .) and Br(’, ") are theBeta and the incomplete Beta function ofEuler given byB(I, s) t- (1 t)s- dt, l, s > O, B,.(l, s) = t- (1 t)s- dt.,.(1, s) tt-(1 + t)s- dtis a real positive valued integral, x (2(x a)/(b a)), X2 x,x3 Xl 1, x4 2 -Xl.Proof Assuming the correction of the expression at the left hand sideas in the Remark 3, this inequality was proved in 15].Remark 7 For n and c (x a/b a) we get the inequalitywhich is a result from [18].References[1 Abramowitz, M. and Stegun, I. A. (Eds) (1965) Handbook of Mathematical Functionswith Formulae, Graphs and Mathematical Tables, 4th printing, Applied Math.Series 55(National Bureau of Standards, Washington).[2] Berezin, I. S. and Zhidkov, N. P. (1965) Computing methods, Vol. (Pergamon press,Oxford).OSTROWSKI INEQUALITY 805[3] Krylov, V. I. (1962) Approximate Calculation of Integrals (Macmillan, New York-London).[4] Dedi6, L. J., Mati6, M. and Pe6ari6, J. (2000) "On generalizations of Ostrowski inequalityvia some Euler-type identities", Math. lnequal. & Appl. 3(3), 337-353.[5] Dedi6, L. J., Mati6, M. and Pe6ari6, J. (2001) "On Euler trapezoid formulae", AppliedMathematics and Computation 123, 37-62.[6] Dedi6, L. J., Mati6, M. and Pe6ari6, J. "On Euler mid-point formulae", ANZIAM J.(accepted).[7] Dragomir, S. S. and Wang, S. (1997) "A new inequality of Ostrowski’s type in L1 normand applications to some special means and to some numerical quadrature roles",Tamkang Journal ofMathematics 3(28), 239-244.[8] Dragomir, S. S. and Wang, S. (1998) "A new inequality of Ostrowski’s type in Lp normand applications to some special means and to some numerical quadrature rules", IndianJournal ofMathematics 40(3), 299-304.[9] Dragomir, S. S. (1999) "On the Ostrowski inequality for mappings with bounded variationand applications", Bull. Austral. Math. Soc. 60, 495-508.[10] Dragomir, S. S. (1999) "On the Ostrowski inequality for Lipschitzian mappings andapplications", Computers and Mathematics with Applications 38, 33-37.11] Cerone, P., Dragomir, S. S. and Roumeliotis, J. (1999) "An inequality of Ostrowski typefor mappings whose second derivatives are bounded and applications", East Asian Math.J 15(1), 1-9.[12] Cerone, P., Dragomir, S. S. and Roumeliotis, J. (1999) "An inequality of Ostrowski typefor mappings whose second derivatives belong to L (a, b) and applications", HonamMathematical J. 1(1), 127-137.13] Dragomir, S. S. and Barnett, N. S. (1999) "On the Ostrowski inequality for mappings withbounded variation and applications", J. Indian Math. Soc. (N.S.) 66(1-4), 237-245.[14] Dragomir, S. S. and Sofo, A. (1999) "An integral inequality for twice differentiablemappings and applications", RGMIA Res. Rep. Coll. 2(2).15] Dragomir, S. S. and Sofo, A. (2000) "An inequality of Ostrowski for twice differentiablemappings in terms of the Lp norm and applications", RGMIA Res. Rep. Coll. 3(1).[16] Dragomir, S. S., Agarwal, R. P. and Cerone, P. (2000) "On Simpson’s inequalityand applications", Journal Inequalities and Applications , 533-579.[17] Dragomir, S. S., Agarwal, R. P. and Barnett, N. S. (2000) "Inequalities for Beta andGamma functions via some classical and new inequalities", Journal ofInequalities andApplications 5, 103-165.[18] Fink, A. M. (1992) "Bounds of the deviation of a function from its averages",Czechoslovak Math. J. 42(117), 289-310.

https://digital.library.adelaide.edu.au/dspace/bitstream/2440/55914/1/hdl_55914.pdf

On generalizations of Ostrowski inequality via Euler harmonic identities

Abstract

Similar works

Full text

Available Versions

Crossref

Adelaide Research & Scholarship