In this paper, we shall give a generalization of the Grüss type inequality and obtain some applications of the Grüss type inequality in terms of 2-inner product spaces

Cho, Yeol Je

Dragomir, Sever S

Kim, Seong Sik

White, A

English

Victoria University Eprints Repository

ON SOME GRU¨SS TYPE INEQUALITY IN2-INNER PRODUCT SPACES AND APPLICATIONSS.S. Kim, S.S. Dragomir, A. White and Y.J. ChoAbstract. In this paper, we shall give a generalization of the Gru¨ss typeinequality and obtain some applications of the Gru¨ss type inequality in termsof 2-inner product spaces.1. IntroductionLet X be a linear space of dimension greater than 1 and (·, ·|·) be a real-valued function on X ×X ×X satisfying the following conditions:(2I1) (x, x|z) ≥ 0,(x, x|z) = 0 if and only if x and z are linearly dependent,(2I2) (x, x|z) = (z, z|x),(2I3) (x, y|z) = (y, x|z),(2I4) (αx, y|z) = α(x, y|z) for any real number α,(2I5) (x+ x′, y|z) = (x, y|z) + (x′, y|z).(., .|.) is called a 2-inner product and (X, (·, ·|·)) is called a 2-inner productspace (or a 2-pre-Hilbert space) ([3]).Some basic properties of the 2-inner product (·, ·|·) are as follows ([3], [4]):1991 AMS Subject Classification Code : 26D15, 26D99.Key words and Phrases : Gru¨ss type inequality, 2-inner product, Isotonic functional.Typeset by AMS-TEX12 S.S. KIM, S.S. DRAGOMIR, A. WHITE AND Y.J. CHO(1) For all x, y, z ∈ X,|(x, y|z)| ≤√(x, x|z)√(y, y|z).(2) For all x, y ∈ X, (x, y|y) = 0.(3) If (X, (·, ·)) is an inner product space, then the 2-inner product (·, ·|·)is defined on X by(x, y|z) =∣∣∣∣ (x|y) (x|z)(y|z) (z|z) ∣∣∣∣ = (x|y)‖z‖2 − (x|z)(y|z)for all x, y, z ∈ X.Under the same assumptions over X, the real-valued function ‖·, ·‖ onX ×X satisfying the following conditions:(2N1) ‖x, y‖ = 0 if and only if x and y are linearly dependent,(2N2) ‖x, y‖ = ‖y, x‖,(2N3) ‖αx, y‖ = |α|‖x, y‖ for all real number α,(2N4) ‖x, y + z‖ ≤ ‖x, y‖+ ‖x, z‖.‖·, ·‖ is called a 2-norm on X and (X, ‖·, ·‖) is called a linear 2-normedspace ([7]).Note that it is easy to show that the 2-norm ‖·, ·‖ is non-negative and,for all x, y ∈ X and real numbers α, ‖x, y + αx‖ = ‖x, y‖.For any non-zero x1, x2, ..., xn in X, let V (x1, x2, ..., xn) denote the sub-space ofX generated by x1, x2, ..., xn. Whenever the notation V (x1, x2, ..., xn)is used, by it will understood x1, x2, ..., xn to be linearly independent.Note that, on any 2-inner product space (X, (·, ·|·)), ‖x, y‖ = √(x, x|y)defines a 2-norm for which we have(x, y|z) = 14(‖x+ y, z‖2 − ‖x− y, z‖2),(1.1)‖x+ y, z‖2 + ‖x− y, z‖2 = 2(‖x, z‖2 + ‖y, z‖2)(1.2)for all x, y, z ∈ X. On the other hand, if (X, ‖·, ·‖) is a linear 2-normed spacein which the condition (1.2) is satisfied for all x, y, z ∈ X, then we can definea 2-inner product (·, ·|·) on X by the condition (1.1).ON SOME GRU¨SS TYPE INEQUALITY 3For a 2-inner product space (X, (·, ·|·)), Cauchy-Schwarz’s inequality(1.3) |(x, y|z)| ≤ (x, x|z)1/2(y, y|z)1/2 = ‖x, z‖‖y, z‖,a 2-dimensional analogue of Cauchy-Schwarz’s inequality, holds.For further details on 2-inner product spaces and linear 2-normed spaces,refer to the papers ([2]-[5], [9], [10]).Y. J. Cho et al. ([1]), S. S. Dragomir et al. ([6]) studied the inequalitiesof 2-inner product spaces and obtained some related results.In this paper, we shall give a generalization of the Gru¨ss type inequalityand obtain some applications of the Gru¨ss type inequality in terms of 2-innerproduct spaces.2. The Main ResultsIn 1935, G. Gru¨ss proved the integral inequality∣∣∣∣ 1b− a ∫ ab f(x)g(x)dx− 1b− a∫ abf(x)dx · 1b− a∫ abg(x)dx∣∣∣∣≤ 14(M −m)(N − n)if f and g are two integrable functions on [a,b] satisfying the condition:m ≤ f(x) ≤M, n ≤ g(x) ≤ Nfor all x ∈ [a, b]([8]).In this section, we shall give a generalization of the Gru¨ss type inequalityin terms of 2-inner product spaces.Theorem 2.1. Let (X, (·, ·|·)) be a 2-inner product space and x, y, z, e ∈X with ‖e, z‖ = 1 and z /∈ V (x, e, y). If m,n,M,N are real numbers suchthat(2.1) (Me− x, x−me|z) ≥ 0, (Ne− y, y − ne|z) ≥ 0,4 S.S. KIM, S.S. DRAGOMIR, A. WHITE AND Y.J. CHOthen we have the inequality(2.2) |(x, y|z)− (x, e|z)(e, y|z)| ≤ 14|M −m||N − n|.Proof. Note that(x, y|z)− (x, e|z)(e, y|z) = (x− (x, e|z)e, y − (e, y|z)e|z).By the Cauchy-Schwarz’s inequality (1.3),(2.3)|(x− (x, e|z)e, y − (e, y|z)e|z)|2≤ ‖x− (x, e|z)e, z‖2‖y − (e, y|z)e, z‖2= (‖x, z‖2 − |(x, e|z)|2)(‖y, z‖2 − |(e, y|z)|2).On the other hand, we have(2.4) (M−(x, e|z))((x, e|z)−m)−(Me−x, x−me|z) = ‖x, z‖2−|(x, e|z)|2and(2.5) (N − (e, y|z))((e, y|z)−n)− (Ne− y, y−ne|z) = ‖y, z‖2− |(e, y|z)|2.Since (Me− x, x−me|z) ≥ 0, (Ne− y, y − ne|z) ≥ 0, we have(2.6) (M − (x, e|z))((x, e|z)−m) ≥ ‖x, z‖2 − |(x, e|z)|2and(2.7) (N − (e, y|z))((e, y|z)− n) ≥ ‖y, z‖2 − |(e, y|z)|2.Also, by the inequality 4ab ≤ (a+ b)2 for a, b ∈ R, we have(2.8) (M − (x, e|z))((x, e|z)−m) ≤ 14(M −m)2ON SOME GRU¨SS TYPE INEQUALITY 5and, similarly,(2.9) (N − (e, y|z))((e, y|z)− n) ≤ 14(N − n)2.Thus, using (2.3)∼(2.9), we have the inequality|(x, y|z)− (x, e|z)(e, y|z)|2 ≤ 116|M −m|2|N − n|2and so we have the desired inequality (2.2). This completes the proof.The mapping (·, ·|·)p : Rn → R given by(x, y|z)p = 12n∑i,j=1pipj(xizj − xjzi)(yizj − yjzi),where x = (x1, x2, . . . , xn), y = (y1, y2, . . . , yn), z = (z1, z2, . . . , zn) ∈ Rnand p = (p1, p2, . . . , pn) > 0, that is, pi > 0 for all i = 1, 2, . . . , n, isobviously a 2-inner product on Rn generating the 2-norm on Rn‖x, y‖p =[12n∑i,j=1pipj(xizj − xjzi)2]1/2.Propsition 2.2. Let (Rn, (·, ·|·)p) be a 2-inner product space and x, y, z, e ∈Rn such that ‖e, z‖ = 1 and z /∈ V (x, y, e). If m,n,M,N are real numberssuch that(Me− x, x−me|z)p ≥ 0, (Ne− y, y − ne|z)p ≥ 0,then we have the inequality∣∣∣∣12 n∑i,j=1pipj(xizj − xjzi)(yizj − yjzi)−(12n∑i,j=1pipj(xizj − xjzi)(eizj − ejzi))×(12n∑i,j=1pipj(eizj − ejzi)(yizj − yjzi))∣∣∣∣≤ 14|M −m||N − n|.6 S.S. KIM, S.S. DRAGOMIR, A. WHITE AND Y.J. CHONext, let (·, ·|·) be a 2-inner product and {(·, ·|·)i}i∈N be a sequence of2-inner products satisfying the following condition:‖x, z‖2 >∞∑i=1‖x, z‖2ifor all x, z being linearly independent. Let p ∈ N . Define a mapping(x, y|z)p = (x, y|z)−p∑i=1(x, y|z)i,for x, y, z ∈ X and z /∈ V (x, y). Then the mapping (·, ·|·)p satisfies theproperties:(1) (x, x|z)p ≥ 0,(2) (αx+ βx′, y|z)p = α(x, y|z)p + β(x′ + y|z)p,(3) (x, y|z)p + (y, x|z)p,(4) (x, x|z)p = (z, z|x)pfor every x, x′, y, z ∈ X and α, β ∈ R.By Theorem 2.1, we have the following:Proposition 2.3. If there exist real numbersm,n,M,N are real numberssuch that(Me− x, x−me|z)p ≥ 0, (Ne− y, y − ne|z)p ≥ 0,then we have|(x, y|z)p − (x, e|z)p(e, y|z)p| ≤ 14 |M −m||N − n|.3. Applications for Isotonic functionalsLet E be a nonempty set, F (E,R) be the real algebra of all real-valuedfunctions defined on E and L be a subalgebra of F (E,R). A functional AON SOME GRU¨SS TYPE INEQUALITY 7is said to be isotonic if f ≥ g, that is, f(t) ≥ g(t) for every t ∈ E, impliesA(f) ≥ A(g) for all f, g ∈ L. A functional A is said to be normalized on Lif 1∈ L, that is, 1(t) = 1 for all t ∈ E implies A(1) = 1.For some inequalities involving linear isotonic functionals is given in [8].Suppose that fgh2, fh2, gh2 ∈ L for all f, g ∈ L. For a isotonic linearfunctional A : L→ R, we define a functional (·, ·|·)A : L× L× L→ R by(f, g|h)A = A(fgh2)for every f, g, h ∈ L. Then we have the following properties:(1) (f, f |h)A = A(f2h2) ≥ 0,(2) (αf + βf ′, g|h)A = α(f, g|h)A + β(f ′, g|h)A,(3) (f, g|h)A = (g, f |h)A,(4) (f, f |h)A = (h, h|f)A.for every f, f ′, g, h ∈ L and α, β ∈ R.Theorem 3.1. Let L be as above, fgh2, fh2, gh2, f, g, e, h ∈ L with‖e, h‖ = 1 and h /∈ V (f, g, e). If m,n,M,N are real numbers such that(3.1) m ≤ f ≤M, n ≤ g ≤ Nand A : L → R is an isotonic linear functional, then we have the followinginequality|A(fgh2)−A(fh2)A(gh2)| ≤ 14(M −m)(N − n).Proof. Choose e=1. Then since ‖e, h‖ = 1, (e, e|h)A = 1, A(e2h2) =A(h2) = 1 and we have(Me− f, f −me|h)A = A((M − f)(f −m)h2) ≥ 0and(Ne− g, g − ne|h)A = A((N − g)(g − n)h2) ≥ 0.8 S.S. KIM, S.S. DRAGOMIR, A. WHITE AND Y.J. CHOApplying Theorem 2.1 for (·, ·|·)A, we have|(f, g|h)A − (f, e|h)(e, g|h)A| ≤ 14(M −m)(N − n).This completes the proof.Corollary 3.2. Let fg, f, g, e, h ∈ L. Suppose 1 ∈ L and A : L → R isa normalized isotonic linear functional. If m,n,M,N satisfy (3.1), then wehave the following inequality|A(fg)−A(f)A(g)| ≤ 14(M −m)(N − n).Let L2[a,b] be a real Hilbert space of square integrable mapping on [a, b],that is,b∫a|f2|dm <∞ if f ∈ L2[a,b]. Define a mapping (·, ·|·) : L2[a,b]×L2[a,b]×L2[a,b] → R by(f, g|l)=12∫ ba∫ ba(f(x)l(y)− f(y)l(x))(g(x)l(y)− g(y)l(x))dm(x)dm(y).Then (·, ·|·) is a 2-inner product on L2[a,b] generating the 2-norm‖f, l‖ =(12∫ ba∫ ba(f(x)l(y)− f(y)l(x))2dm(x)dm(y))1/2.Proposition 3.3. Let (L2[a,b], (·, ·|·) be a 2-inner product space andf, g, e, h ∈ L2[a,b] with ‖e, h‖ = 1 and h /∈ V (f, g, e). There exist real numbersm,n,M,N such thatm ≤ f ≤M, n ≤ g ≤ N.ON SOME GRU¨SS TYPE INEQUALITY 9Then we have the inequality∣∣∣∣ ∫ ba∫ ba(f(x)h(y)− f(y)h(x))(g(x)h(y)− g(y)h(x))dm(x)dm(y)−[∫ ba∫ ba(f(x)h(y)− f(y)h(x))(h(y)− h(x))dm(x)dm(y)]×[∫ ba∫ ba(h(y)− h(x))(g(x)h(y)− g(y)h(x))dm(x)dm(y)]∣∣∣∣≤ 14|M −m||N − n|.Proof. By Theorem 3.1, applied toA(f, g|h) = A(fgh2) =∫ ba∫ badet(f, h) det(g, h)dm(x)dm(y),wheredet(f, h) =∣∣∣∣ f(x) f(y)h(x) h(y) ∣∣∣∣ ,the result follows.References[1] Y. J. Cho, S. S. Dragomir, A. White and S. S. Kim, Some inequalities in2-inner product spaces, Demonstratio Math., 32(1999), 485-493.[2] Y. J. Cho and S. S. Kim, Gaˆteaux derivatives and 2-inner product spaces,Glasnik Mat. 27(1992), 271-282.[3] C. Dimininnie, S. Ga¨hler and A. White, 2-inner product spaces, Demon-stratio Math., 6(1973), 525-536.[4] C. Dimininnie, S. Ga¨hler and A. White, 2-inner product spaces II, Demon-stratio Math., 10(1977), 169-188.[5] C. Dimininie and A. White, 2-inner product spaces and Gaˆuteaux partialderivatives, Comment. Math. Univ. Carolinae 16(1975), 115-119.[6] S. S. Dragomir, Y. J. Cho and S. S. Kim, Superadditivity and monotonicityof 2-norms generated by inner products and related results, Soochow J.Math. 24(1)(1998), 13-32.10 S.S. KIM, S.S. DRAGOMIR, A. WHITE AND Y.J. CHO[7] S. Ga¨hler, Lineare 2-normierte Ra¨ume, Math. Nachr., 28(1965), 1-43.[8] D. S. Mitrinovic´, J. E. Pecˇaric´ and A. M. Fink, Classical and New In-equalities in Analysis, Kluwer Academic, Dordrecht, Norwell, MA, 1993.[9] A. White, Y. J. Cho and S. S. Kim, Heron’s formula in inner productspaces, Demonstratio Math., 31(1998), 97-102.[10] A. White, Y. J. Cho and S. S. Kim, Characterizations of 2-inner productspaces, Math. Japon. 47(1998), 239-244.S. S. KIMSchool of Communications and InformaticsVictoria Unniversity of TechologyMelbourne City, MC 8001 Australiae-mail:kim@sci.vu.edu.auDepartment of MathematicsDongeui UniversityPusan 614-714, KOREAe-mail:sskim@hyomin.dongeui.ac.krS. S. DRAGOMIRSchool of Communications and InformaticsVictoria Unniversity of TechologyMelbourne City, MC 8001 Australiae-mail:sever@matilda.vu.edu.auA. WHITEDepartment of MathematicsSt. Bonaventure UniversitySt. Bonaventure, NY 14778, U.S.A.e-mail:awhite@sbu.eduY. J. CHODepartment of MathematicsGyeongsang National UniversityChinju 660-701, KOREAe-mail:yjcho@nongae.gsnu.ac.kr

Characterizations of 2-inner product spaces,

Gˆ ateaux derivatives and 2-inner product spaces,

Heron’s formula in inner product spaces,

Peˇ cari´ c

product spaces and Gˆ auteaux partial derivatives,

product spaces II,

product spaces,

Some inequalities in 2-inner product spaces,

Superadditivity and monotonicity of 2-norms generated by inner products and related results,

On Some Grüss Type Inequality in 2-Inner Product Spaces and Applications

Victoria University Institutional Repository (VUIR)

On Some Grüss Type Inequality in 2-Inner Product Spaces and Applications

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