In this paper some new inequalities for the Čebyšev functional are presented. They have applications in a variety of branches of applied mathematics

Cerone, Pietro

Dragomir, Sever S

English

Victoria University Eprints Repository

NEW BOUNDS FOR THE CˇEBYSˇEV FUNCTIONALP. CERONE AND S.S. DRAGOMIRAbstract. In this paper some new inequalities for the Cˇebysˇev functionalare presented. They have applications in a variety of branches of appliedmathematics.1. IntroductionOver the last five years, the development of Gru¨ss type inequalities has experi-enced a surge, having been stimulated by their applications in different branchesof Applied Mathematics including: in perturbed quadrature rules (see for example[13], [14]) and in the approximation of integral transforms (see [15], [16]) and thereferences therein.The main aim of the present paper is to point out other such results that maybe used as new tools in obtaining perturbed version of classical quadrature ornew approximations for different kinds of integral operators encountered in variousbranches of Applied Mathematics. Now for some preliminaries.For two Lebesgue integrable functions f, g : [a, b] → R, consider the Cˇebysˇevfunctional:(1.1) T (f, g) :=1b− a∫ baf (t) g (t) dt− 1b− a∫ baf (t) dt · 1b− a∫ bag (t) dt.In 1934, G. Gru¨ss [4] showed that(1.2) |T (f, g)| ≤ 14(M −m) (N − n) ,provided m,M,n,N are real numbers with the property(1.3) −∞ < m ≤ f ≤M <∞, −∞ < n ≤ g ≤ N <∞ a.e. on [a, b] .The constant 14 is best possible in (1.2) in the sense that it cannot be replacedby a smaller one. Another less well known inequality for T (f, g) was derived in1882 by Cˇebysˇev [3] under the assumption that f ′, g′ exist and are continuous in[a, b] and is given by(1.4) |T (f, g)| ≤ 112‖f ′‖∞ ‖g′‖∞ (b− a)2 ,where ‖f ′‖∞ := supt∈[a,b]|f ′ (t)| .The constant 112 cannot be improved in the general case.Cˇebysˇev’s inequality (1.4) also holds if f, g : [a, b]→ R are assumed to be abso-lutely continuous and f ′, g′ ∈ L∞ [a, b] .Date: March 18, 2003.1991 Mathematics Subject Classification. Primary 26D15; Secondary 26D10.Key words and phrases. Gru¨ss inequality, Cˇebysˇev functional.12 P. CERONE AND S.S. DRAGOMIRIn 1970, A.M. Ostrowski [5] proved, amongst others, the following result that isin a sense a combination of the Cˇebysˇev and Gru¨ss results(1.5) |T (f, g)| ≤ 18(b− a) (M −m) ‖g′‖∞ ,provided f is Lebesgue integrable on [a, b] and satisfying (1.3) with g : [a, b] → Rbeing absolutely continuous and g′ ∈ L∞ [a, b] . Here the constant 18 is also sharp.Finally, let us recall a result by Lupas¸ (see for example [1, p. 210]), which statesthat:(1.6) |T (f, g; a, b)| ≤ 1pi2‖f ′‖2 ‖g′‖2 (b− a) ,provided f, g are absolutely continuous and f ′, g′ ∈ L2 [a, b]. The constant 1pi2 isthe best possible here also.For other Gru¨ss type integral inequalities, see the books [1], [2], and the papers[6]-[11], where further references are given.2. Bounds for the Cˇebysˇev FunctionalThe following lemma holds and it will prove useful for procuring specifically aninequality for the Cˇebysˇev functional in terms of the Hilbertian norm. The lemmais also of intrinsic interest in its own right.Lemma 1. Assume that the function f : [a, b] → R is Lebesgue integrable and∫ baf (x) dx = 0. Define F (x) =∫ xaf (t) dt, e (x) = x, x ∈ [a, b] and assume thatF, f, ef ∈ L2 [a, b] . Then we have the inequality∫ baF 2 (x) dx ≤ 4 ·∫ baf2 (t) dt∫ bat2f2 (t) dt−(∫ batf2 (t) dt)2∫ baf2 (t) dt(2.1)≤ 4∫ bat2f2 (t) dt.Proof. For a given λ ∈ R we have, on integrating the Lebesgue integral by parts,∫ ba[F (x)]2 dx =∫ ba[F (x)]2 d (x− λ)(2.2)= F 2 (x) (x− λ)∣∣∣∣ba−∫ ba(x− λ) ddx(F 2 (x))dx= 2∫ ba(λ− x) f (x)F (x) dx.Using the Cauchy-Schwartz-Buniakowski inequality for integrals, we have(2.3)∫ ba(λ− x) f (x)F (x) dx ≤[∫ ba(λ− x)2 f2 (x) dx] 12(∫ ba[F (x)]2 dx) 12.Combining (2.2) with (2.3) and dividing by(∫ ba[F (x)]2 dx) 12 ≥ 0 (since we mayassume that f 6= 0), we deduce(∫ ba[F (x)]2 dx) 12≤ 2[∫ ba(λ− x)2 f2 (x) dx] 12BOUNDS FOR THE CˇEBYSˇEV FUNCTIONAL 3for any λ ∈ R, which is clearly equivalent to(2.4)∫ ba[F (x)]2 dx ≤ 4∫ ba(λ− x)2 f2 (x) dx, λ ∈ R.Taking the infimum in (2.4) for λ ∈ R, we deduce(2.5)∫ ba[F (x)]2 dx ≤ 4 infλ∈Rg (λ) ,whereg (λ) :=∫ ba(λ− x)2 f2 (x) dx.Now, observe thatg (λ) = λ2∫ baf2 (x) dx− 2λ∫ baxf2 (x) dx+∫ bax2f2 (x) dx.Sinceinfλ∈Rg (λ) =∫ baf2 (x) dx∫ bax2f2 (x) dx−(∫ baxf2 (x) dx)2∫ baf2 (x) dx,then by (2.5) we deduce the desired inequality (2.1). If a > 0, we may point out the following inequality that may be easier to applyin practice.Corollary 1. Assume that f satisfies the assumptions in Lemma 1 and 0 < a < b.Then we have the inequality(2.6)∫ baF 2 (x) dx ≤ (b− a)2ab·(∫ batf2 (t) dt)2∫ baf2 (t) dt≤ (b− a)2ab·∫ baf2 (t) t2dt.Proof. We use the following integral version of Cassels’ inequality (see for example[12])(2.7)∫ bap (t) l2 (t) dt∫ bap (t)h2 (t) dt(∫ bap (t) l (t)h (t) dt)2 ≤ (M +m)24mM ,provided0 < m ≤ h (t)l (t)≤M <∞ for a.e.t on [a, b]and p ≥ 0 a.e. on [a, b] .Applying (2.7) for p (t) = f2 (t) , l (t) = 1, h (t) = t, t ∈ [a, b] , we get(2.8)∫ baf2 (t) dt∫ baf2 (t) t2dt(∫ batf2 (t) dt)2 ≤ (a+ b)24abgiving(2.9)∫ baf2 (t) dt∫ bat2f2 (t) dt−(∫ batf2 (t) dt)2≤ (b− a)24ab(∫ batf2 (t) dt)2.Using (2.1) and (2.9), we deduce the first inequality in (2.6).The last inequality is obvious by Schwartz’s inequality. 4 P. CERONE AND S.S. DRAGOMIRThe following Gru¨ss type inequality holds for the Cˇebysˇev functional T (f, g).Theorem 1. Assume that f : [a, b]→ R is a measurable function on [a, b] and suchthat f¯ := f − 1b−a∫ baf (t) dt, ef¯ ∈ L2 [a, b] . If g : [a, b]→ R is absolutely continuousand g′ ∈ L2 [a, b] , then we have the inequality|T (f, g)| ≤ 2b− a ‖g′‖2∫ baf¯2 (t) dt∫ bat2f¯2 (t) dt−(∫ batf¯2 (t) dt)2∫ baf¯2 (t) dt12(2.10)≤ 2b− a ‖g′‖2∥∥ef¯∥∥2.Proof. Denote(2.11) F (x) =∫ xaf (t) dt− x− ab− a∫ baf (u) du =∫ xaf¯ (t) dt, x ∈ [a, b] ,where, as above,f¯ (t) := f (t)− 1b− a∫ baf (u) du, t ∈ [a, b] .We observe that, on integrating by parts, we have1b− a∫ baF (x) g′ (x) dx =1b− aF (x) g (x)∣∣∣∣ba− 1b− a∫ bag (x)F (t) dx(2.12)= −T (f, g) .Taking the modulus in (2.12) and using the Cauchy-Buniakowski-Schwartz inequal-ity, we have|T (f, g)| ≤ 1b− a∫ ba∣∣F (x)∣∣ |g′ (x)| dx(2.13)≤ 1b− a(∫ ba|g′ (x)|2 dx) 12(∫ ba∣∣F (x)∣∣2 dx) 12 .Applying Lemma 1, we deduce the desired inequality (2.10). The following corollary holds.Corollary 2. With the assumptions of Theorem 1 and if 0 < a < b, then we havethe inequality|T (f, g)| ≤ 1√ab‖g′‖2∫ batf¯2 (t) dt(∫ baf¯2 (t) dt) 12(2.14)≤ 1√ab‖g′‖2∥∥ef¯∥∥2.BOUNDS FOR THE CˇEBYSˇEV FUNCTIONAL 53. Further BoundsWe observe that, indeed, for a Lebesgue integrable function f, F as defined by(2.11) is an absolutely continuous function and(3.1) F (a) = F (b) = 0.We also must note thatF (t) =(b− t) ∫ taf (s) ds− (t− a) ∫ btf (s) dsb− a(3.2)=(b− t) ∫ baf (s) ds− (b− a) ∫ btf (s) dsb− a .The following properties for F may also be stated.(1) If f ∈ L [a, b] , then F ∈ BV [a, b] and(3.3)b∨a(F)=∫ ba∣∣f (t)∣∣ dt.(2) If f ∈ L∞ [a, b] , then F ∈ LipK [a, b] , where(3.4) K :=∥∥f∥∥[a,b],∞ .The following identity is useful in the sequel.Lemma 2. Assume that f ∈ L [a, b] and g ∈ C [a, b] . Then for any γ ∈ R, one hasthe identity(3.5) T (f, g) =1b− a∫ ba(g (t)− γ) d (F (t)) .Proof. If f ∈ L [a, b] , then obviously F ∈ AC [a, b] and since g ∈ C [a, b] , theStieltjes integral in the right hand side of (3.5) exists.Reducing to a Lebesgue integral, we have1b− a∫ ba(g (t)− γ) d (F (t)) = 1b− a∫ ba(g (t)− γ) ddt(F (t))dt=1b− a∫ ba(g (t)− γ) f (t) dt= T (f, g) .The following corollary containing some particular cases of interest also holds.6 P. CERONE AND S.S. DRAGOMIRCorollary 3. Under the assumptions of Lemma 2, we have specificallyT (f, g) =1b− a∫ bag (t) d(F (t))(3.6)=1b− a∫ ba(g (t)− m+M2)d(F (t))(m,M ∈ R)=1b− a∫ ba[g (t)− g(a+ b2)]d(F (t))=1b− a∫ bag (t) d(F (t)).We may now state the first result relating to Lemma 2.Theorem 2. Let f ∈ L [a, b] and g ∈ C [a, b] . Then one has the inequality(3.7) |T (f, g)| ≤ infγ∈R‖g − γ‖∞1b− a∫ ba∣∣f (t)∣∣ dt.Proof. If f ∈ L [a, b] , then obviously F ∈ AC [a, b] and(3.8)dF (t)dt= f (t) for a.e. t ∈ [a, b] .Since any absolutely continuous function is of bounded variation, it follows thatF ∈ BV [a, b] and its total variation is(3.9)b∨a(F)=∫ ba∣∣∣∣dF (t)dt∣∣∣∣ dt = ∫ ba∣∣f (t)∣∣ dt.It is known that if p ∈ C [c, d] and v ∈ BV [c, d] , then(3.10)∣∣∣∣∣∫ bap (s) dv (s)∣∣∣∣∣ ≤ ‖p‖∞b∨a(v) .Using the property, we deduce that|T (f, g)| = 1b− a∣∣∣∣∣∫ ba(g (t)− γ) d (F (t))∣∣∣∣∣(3.11)≤ 1b− a ‖g − γ‖∞ ·b∨a(F)= ‖g − γ‖∞1b− a∫ ba∣∣f (t)∣∣ dtfor any γ ∈ R.Taking the infimum over γ ∈ R, we deduce the desired inequality (3.7). There are a number of bounds that are coarser than (3.7) but may prove to bemore useful in practical applications. They arise from taking particular choices ofγ in the identity (3.5) and the result (3.7).Corollary 4. If f ∈ L [a, b] and g ∈ C [a, b] , then(3.12) |T (f, g)| ≤ ‖g‖∞1b− a∫ ba∣∣f (t)∣∣ dt.BOUNDS FOR THE CˇEBYSˇEV FUNCTIONAL 7The constant 1 cannot be replaced by a smaller constant.Proof. The inequality is obvious from (3.7).To prove the sharpness of the constant C = 1, assume that (3.12) holds with aconstant C > 0. That is,(3.13) |T (f, g)| ≤ C ‖g‖∞1b− a∫ ba∣∣f (t)∣∣ dt.Consider [a, b] = [−1, 1] and define the functions(3.14) f0 (x) =−1 if x ∈ [−1, 0);0 if x = 0;1 if x ∈ (0, 1],, gn (x) =−1 if x ∈ [−1,− 1n) ;nx if x ∈ [− 1n , 1n] ;1 if x ∈ ( 1n , 1] .Obviously, gn ∈ C [−1, 1] . We also have∫ 1−1f0 (x) gn (x) = 2∫ 10gn (x) dx = 2(1− 12n)and thusT (f0, gn) = 1− 12n, ‖gn‖∞ = 1,12∫ 1−1∣∣∣∣f0 (t)− 12∫ 1−1f0 (s) ds∣∣∣∣ dt = 1,giving from (3.13) that1− 12n≤ C.Letting n→∞, we deduce C ≥ 1, and the sharpness of the constant is proved. Corollary 5. If f ∈ L [a, b] , g ∈ C [a, b] and there exists the real constants m,Msuch that(3.15) m ≤ g (x) ≤M for any x ∈ [a, b] ,then(3.16) |T (f, g)| ≤ 12(M −m) 1b− a∫ ba∣∣f (t)∣∣ dt.The constant 12 is best possible.Proof. The inequality follows by (3.7) on taking into account that if g satisfies(3.15), then ∥∥∥∥g − m+M2∥∥∥∥∞≤ 12(M −m) .The sharpness of the constant may be proved in a similar way as in the proof of thatsharpness of (3.12) by selecting the same examples (3.14). We omit the details. Remark 1. The inequality (3.16) was obtained in a different way in [13]. Forgeneralisations, best constants and discrete versions, see also [14].8 P. CERONE AND S.S. DRAGOMIRCorollary 6. Let f ∈ L [a, b] . If g : [a, b] → R is of r − H−Ho¨lder type withr ∈ (0, 1], H > 0, so that(3.17) |g (t)− g (s)| ≤ H |t− s|r for each t, s ∈ [a, b] ,then we have the bound:(3.18) |T (f, g)| ≤ 12rH (b− a)r−1∫ ba∣∣f (t)∣∣ dt.Proof. Since g is of r −H−Ho¨lder type, thensupt∈[a,b]∣∣∣∣g (t)− g(a+ b2)∣∣∣∣ ≤ H supt∈[a,b]∣∣∣∣t− a+ b2∣∣∣∣r = 12rH (b− a)r .Using (3.7) with γ = g(a+b2), we deduce (3.18). The following corollary also holds.Corollary 7. If f ∈ L [a, b] and g ∈ C [a, b] , then(3.19) |T (f, g)| ≤ ‖g‖∞1b− a∫ ba∣∣f (t)∣∣ dt.The proof is obvious by (3.7) with γ = 1b−a∫ bag (s) ds.The following theorem also holds.Theorem 3. Let f ∈ Lp [a, b] and g ∈ Lq [a, b] , with p > 1, 1p + 1q = 1 (for p =∞we choose q = 1). Then we have the inequality(3.20) |T (f, g)| ≤ 1b− a infγ∈R ‖g − γ‖q∥∥f∥∥p.Proof. Follows by Lemma 2 and by Ho¨lder’s inequality. Similar particular inequalities for different choices of γ may be stated. We omitany further details.The application of these bounds to problems in applied mathematics for theCˇebysˇev functional is left to future work and the pursuit of the interested readers.References[1] J. PECˇARIC´, F. PROCHAN and Y. TONG, Convex Functions, Partial Orderings and Sta-tistical Applications, Academic Press, San Diego, 1992.[2] D.S. MITRINOVIC´, J.E. PECˇARIC´ and A.M. FINK, Classical and New Inequalities inAnalysis, Kluwer Academic Publishers, Dordrecht, 1993.[3] P.L. CHEBYSHEV, Sue les expressions approximatives des inte´grales de´finies par les autresprises entre les meˆme limites, Proc. Math. Soc. Charkov, 2 (1982), 93-98.[4] G. GRU¨SS, U¨ber das Maximum des absoluten Betrages von 1b−a∫ ba f (x) g (x) dx −1(b−a)2∫ ba f (x) dx∫ ba g (x) dx, Math. Z., 39 (1934), 215-226.[5] A.M. OSTROWSKI, On an integral inequality, Aequat. Math., 4 (1970), 358-373.[6] S.S. DRAGOMIR, A generalization of Gru¨ss’ inequality in inner product spaces and appli-cations, J. Math. Anal. Appl., 237 (1999), 74-82.[7] P. CERONE, On an identity for the Chebychev Functional and some ram-ifications, J. Ineq. Pure & Appl. Math., 3(1) (2002), Article 4. [ONLINEhttp://jipam.vu.edu.au/v3n1/034 01.html][8] P. CERONE and S.S. DRAGOMIR, New upper and lower bounds for the Ceby-sev functional, J. Ineq. Pure and Appl. Math., 3(5) (2002), Article 77. [ONLINE:http://jipam.vu.edu.au/v3n5/048 02.html]BOUNDS FOR THE CˇEBYSˇEV FUNCTIONAL 9[9] S.S. DRAGOMIR and I. FEDOTOV, An inequality of Gru¨ss’ type for Riemann-Stieltjesintegral and applications for special means, Tamkang J. of Math., 29(4)(1998), 286-292.[10] S.S. DRAGOMIR, A Gru¨ss type integral inequality for mappings of r-Ho¨lder’s type andapplications for trapezoid formula, Tamkang Journal of Mathematics, 31(1) (2000), 43-47.[11] S.S. DRAGOMIR, Some integral inequalities of Gru¨ss type, Indian J. Pure Appl. Math.,31(4) (2000), 397-415.[12] G.S. WATSON, Serial correlation in regression analysis, I, Biometrika,42(1955), 327-341.[13] X.-L. CHENG and J. SUN, Note on the perturbed trapezoid inequal-ity, J. Inequal. Pure & Appl. Math. 3(2002). No. 2, Article 29 [ON LINE:http://jipam.vu.edu.au/v3n2/046 01.html][14] P. CERONE and S.S. DRAGOMIR, A refinement of the Gruss inequalityand applications, RGMIA Res. Rep. Coll., 5(2)(2002). Article 14 [ON LINE:http://rgmia.vu.edu.au/v5n2.html][15] S.S. DRAGOMIR and A. KALAM, An approximation of the Fourier Sine Transform via Gru¨sstype inequalities and applications for electrical circuits, J. KSIAM, 63(1) (2002), 33-45.[16] G. HANNA, S.S. DRAGOMIR and J. ROUMELIOTIS, An approximation for the Finite-Fourier transform of two independent variables, Proc. 4th Int. Conf. on Modelling and Sim-ulation, Victoria University, Melbourne, 2002, 375-380School of Computer Science and Mathematics, Victoria University of Technology,PO Box 14428, Melbourne City MC 8001, Victoria, Australia.E-mail address: pc@matilda.vu.edu.auURL: http://rgmia.vu.edu.au/ceroneE-mail address: sever.dragomir@vu.edu.auURL: http://rgmia.vu.edu.au/SSDragomirWeb.html

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New Bounds for the Čebyšev Functional

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