ON SOME GRÜSS TYPE INEQUALITY IN 2-INNER PRODUCT SPACES AND APPLICATIONS. S.S. Kim, S.S. Dragomir, A. White and Y.J. Cho. 1.

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1 ON SOME GRÜSS TYPE INEQUALITY IN 2-INNER PRODUCT SPACES AND APPLICATIONS S.S. Kim, S.S. Drgomir, A. White nd Y.J. Cho Abstrct. In this pper, we shll give generliztion of the Grüss type inequlity nd obtin some pplictions of the Grüss type inequlity in terms of 2-inner product spces. 1. Introduction Let X be liner spce of dimension greter thn 1 nd (, ) be relvlued function on X X X stisfying the following conditions: (2I 1 ) (x, x z) 0, (x, x z) = 0 if nd only if x nd z re linerly dependent, (2I 2 ) (x, x z) = (z, z x), (2I 3 ) (x, y z) = (y, x z), (2I ) (αx, y z) = α(x, y z) for ny rel number α, (2I 5 ) (x + x, y z) = (x, y z) + (x, y z). (.,..) is clled 2-inner product nd (X, (, )) is clled 2-inner product spce (or 2-pre-Hilbert spce) ([3]). Some bsic properties of the 2-inner product (, ) re s follows ([3], []): 1991 AMS Subject Clssifiction Code : 26D15, 26D99. Key words nd Phrses : Grüss type inequlity, 2-inner product, Isotonic functionl. 1 Typeset by AMS-TEX

2 2 S.S. KIM, S.S. DRAGOMIR, A. WHITE AND Y.J. CHO (1) For ll x, y, z X, (x, y z) (x, x z) (y, y z). (2) For ll x, y X, (x, y y) = 0. (3) If (X, (, )) is n inner product spce, 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 ll x, y, z X. Under the sme ssumptions over X, the rel-vlued function, on X X stisfying the following conditions: (2N 1 ) x, y = 0 if nd only if x nd y re linerly dependent, (2N 2 ) x, y = y, x, (2N 3 ) αx, y = α x, y for ll rel number α, (2N ) x, y + z x, y + x, z., is clled 2-norm on X nd (X,, ) is clled liner 2-normed spce ([7]). Note tht it is esy to show tht the 2-norm, is non-negtive nd, for ll x, y X nd rel numbers α, x, y + αx = x, y. For ny non-zero x 1, x 2,..., x n in X, let V (x 1, x 2,..., x n ) denote the subspce of X generted by x 1, x 2,..., x n. Whenever the nottion V (x 1, x 2,..., x n ) is used, by it will understood x 1, x 2,..., x n to be linerly independent. Note tht, on ny 2-inner product spce (X, (, )), x, y = (x, x y) defines 2-norm for which we hve (1.1) (1.2) (x, y z) = 1 ( x + y, z 2 x y, z 2 ), x + y, z 2 + x y, z 2 = 2( x, z 2 + y, z 2 ) for ll x, y, z X. On the other hnd, if (X,, ) is liner 2-normed spce in which the condition (1.2) is stisfied for ll x, y, z X, then we cn define 2-inner product (, ) on X by the condition (1.1).

3 ON SOME GRÜSS TYPE INEQUALITY 3 For 2-inner product spce (X, (, )), Cuchy-Schwrz s inequlity (1.3) (x, y z) (x, x z) 1/2 (y, y z) 1/2 = x, z y, z, 2-dimensionl nlogue of Cuchy-Schwrz s inequlity, holds. For further detils on 2-inner product spces nd liner 2-normed spces, refer to the ppers ([2]-[5], [9], [10]). Y. J. Cho et l. ([1]), S. S. Drgomir et l. ([6]) studied the inequlities of 2-inner product spces nd obtined some relted results. In this pper, we shll give generliztion of the Grüss type inequlity nd obtin some pplictions of the Grüss type inequlity in terms of 2-inner product spces. 2. The Min Results In 1935, G. Grüss proved the integrl inequlity 1 f(x)g(x)dx 1 f(x)dx b b b b 1 (M m)(n n) 1 g(x)dx b b if f nd g re two integrble functions on [,b] stisfying the condition: m f(x) M, n g(x) N for ll x [, b]([8]). In this section, we shll give generliztion of the Grüss type inequlity in terms of 2-inner product spces. Theorem 2.1. Let (X, (, )) be 2-inner product spce nd x, y, z, e X with e, z = 1 nd z / V (x, e, y). If m, n, M, N re rel numbers such tht (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. CHO then we hve the inequlity (2.2) (x, y z) (x, e z)(e, y z) 1 M m N n. Proof. Note tht (x, y z) (x, e z)(e, y z) = (x (x, e z)e, y (e, y z)e z). By the Cuchy-Schwrz s inequlity (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 hnd, we hve (2.) (M (x, e z))((x, e z) m) (Me x, x me z) = x, z 2 (x, e z) 2 nd (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 hve (2.6) (M (x, e z))((x, e z) m) x, z 2 (x, e z) 2 nd (2.7) (N (e, y z))((e, y z) n) y, z 2 (e, y z) 2. Also, by the inequlity b ( + b) 2 for, b R, we hve (2.8) (M (x, e z))((x, e z) m) 1 (M m)2

5 ON SOME GRÜSS TYPE INEQUALITY 5 nd, similrly, (2.9) (N (e, y z))((e, y z) n) 1 (N n)2. Thus, using (2.3) (2.9), we hve the inequlity (x, y z) (x, e z)(e, y z) M m 2 N n 2 nd so we hve the desired inequlity (2.2). This completes the proof. The mpping (, ) p : R n R given by (x, y z) p = 1 n p i p j (x i z j x j z i )(y i z j y j z i ), 2 i,j=1 where x = (x 1, x 2,..., x n ), y = (y 1, y 2,..., y n ), z = (z 1, z 2,..., z n ) R n nd p = (p 1, p 2,..., p n ) > 0, tht is, p i > 0 for ll i = 1, 2,..., n, is obviously 2-inner product on R n generting the 2-norm on R n [ 1 n ] 1/2 x, y p = p i p j (x i z j x j z i ) 2. 2 i,j=1 Propsition 2.2. Let (R n, (, ) p ) be 2-inner product spce nd x, y, z, e R n such tht e, z = 1 nd z / V (x, y, e). If m, n, M, N re rel numbers such tht (Me x, x me z) p 0, (Ne y, y ne z) p 0, then we hve the inequlity 1 n p i p j (x i z j x j z i )(y i z j y j z i ) 2 i,j=1 ( 1 n ) p i p j (x i z j x j z i )(e i z j e j z i ) 2 i,j=1 ( 1 n p i p j (e i z j e j z i )(y i z j y j z i )) 2 i,j=1 1 M m N n.

6 6 S.S. KIM, S.S. DRAGOMIR, A. WHITE AND Y.J. CHO Next, let (, ) be 2-inner product nd {(, ) i } i N be sequence of 2-inner products stisfying the following condition: x, z 2 > x, z 2 i i=1 for ll x, z being linerly independent. Let p N. Define mpping p (x, y z) p = (x, y z) (x, y z) i, i=1 for x, y, z X nd z / V (x, y). properties: Then the mpping (, ) p stisfies the (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, () (x, x z) p = (z, z x) p for every x, x, y, z X nd α, β R. By Theorem 2.1, we hve the following: Proposition 2.3. If there exist rel numbers m, n, M, N re rel numbers such tht then we hve (Me x, x me z) p 0, (Ne y, y ne z) p 0, (x, y z) p (x, e z) p (e, y z) p 1 M m N n. 3. Applictions for Isotonic functionls Let E be nonempty set, F (E, R) be the rel lgebr of ll rel-vlued functions defined on E nd L be sublgebr of F (E, R). A functionl A

7 ON SOME GRÜSS TYPE INEQUALITY 7 is sid to be isotonic if f g, tht is, f(t) g(t) for every t E, implies A(f) A(g) for ll f, g L. A functionl A is sid to be normlized on L if 1 L, tht is, 1(t) = 1 for ll t E implies A(1) = 1. For some inequlities involving liner isotonic functionls is given in [8]. Suppose tht fgh 2, fh 2, gh 2 L for ll f, g L. For isotonic liner functionl A : L R, we define functionl (, ) A : L L L R by (f, g h) A = A(fgh 2 ) for every f, g, h L. Then we hve the following properties: (1) (f, f h) A = A(f 2 h 2 ) 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, () (f, f h) A = (h, h f) A. for every f, f, g, h L nd α, β R. Theorem 3.1. Let L be s bove, fgh 2, fh 2, gh 2, f, g, e, h L with e, h = 1 nd h / V (f, g, e). If m, n, M, N re rel numbers such tht (3.1) m f M, n g N nd A : L R is n isotonic liner functionl, then we hve the following inequlity A(fgh 2 ) A(fh 2 )A(gh 2 ) 1 (M m)(n n). Proof. Choose e=1. Then since e, h = 1, (e, e h) A = 1, A(e 2 h 2 ) = A(h 2 ) = 1 nd we hve (Me f, f me h) A = A((M f)(f m)h 2 ) 0 nd (Ne g, g ne h) A = A((N g)(g n)h 2 ) 0.

8 8 S.S. KIM, S.S. DRAGOMIR, A. WHITE AND Y.J. CHO Applying Theorem 2.1 for (, ) A, we hve (f, g h) A (f, e h)(e, g h) A 1 (M m)(n n). This completes the proof. Corollry 3.2. Let fg, f, g, e, h L. Suppose 1 L nd A : L R is normlized isotonic liner functionl. If m, n, M, N stisfy (3.1), then we hve the following inequlity A(fg) A(f)A(g) 1 (M m)(n n). Let L 2 [,b] be rel Hilbert spce of squre integrble mpping on [, b], tht is, b L 2 [,b] R by f 2 dm < if f L 2 [,b]. Define mpping (, ) : L2 [,b] L2 [,b] = 1 2 (f, g l) b b (f(x)l(y) f(y)l(x))(g(x)l(y) g(y)l(x))dm(x)dm(y). Then (, ) is 2-inner product on L 2 [,b] generting the 2-norm f, l = ( 1 b 2 b (f(x)l(y) f(y)l(x)) 2 dm(x)dm(y)) 1/2. Proposition 3.3. Let (L 2 [,b], (, ) be 2-inner product spce nd f, g, e, h L 2 [,b] with e, h = 1 nd h / V (f, g, e). There exist rel numbers m, n, M, N such tht m f M, n g N.

9 ON SOME GRÜSS TYPE INEQUALITY 9 Then we hve the inequlity b b (f(x)h(y) f(y)h(x))(g(x)h(y) g(y)h(x))dm(x)dm(y) where [ b b [ b ] (f(x)h(y) f(y)h(x))(h(y) h(x))dm(x)dm(y) b 1 M m N n. Proof. By Theorem 3.1, pplied to A(f, g h) = A(fgh 2 ) = the result follows. (h(y) h(x))(g(x)h(y) g(y)h(x))dm(x)dm(y)] b b f(x) det(f, h) = h(x) References det(f, h) det(g, h)dm(x)dm(y), f(y) h(y), [1] Y. J. Cho, S. S. Drgomir, A. White nd S. S. Kim, Some inequlities in 2-inner product spces, Demonstrtio Mth., 32(1999), [2] Y. J. Cho nd S. S. Kim, Gâteux derivtives nd 2-inner product spces, Glsnik Mt. 27(1992), [3] C. Dimininnie, S. Gähler nd A. White, 2-inner product spces, Demonstrtio Mth., 6(1973), [] C. Dimininnie, S. Gähler nd A. White, 2-inner product spces II, Demonstrtio Mth., 10(1977), [5] C. Dimininie nd A. White, 2-inner product spces nd Gâuteux prtil derivtives, Comment. Mth. Univ. Croline 16(1975), [6] S. S. Drgomir, Y. J. Cho nd S. S. Kim, Superdditivity nd monotonicity of 2-norms generted by inner products nd relted results, Soochow J. Mth. 2(1)(1998),

10 10 S.S. KIM, S.S. DRAGOMIR, A. WHITE AND Y.J. CHO [7] S. Gähler, Linere 2-normierte Räume, Mth. Nchr., 28(1965), 1-3. [8] D. S. Mitrinović, J. E. Pečrić nd A. M. Fink, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic, Dordrecht, Norwell, MA, [9] A. White, Y. J. Cho nd S. S. Kim, Heron s formul in inner product spces, Demonstrtio Mth., 31(1998), [10] A. White, Y. J. Cho nd S. S. Kim, Chrcteriztions of 2-inner product spces, Mth. Jpon. 7(1998), S. S. KIM School of Communictions nd Informtics Victori Unniversity of Techology Melbourne City, MC 8001 Austrli e-mil:kim@sci.vu.edu.u Deprtment of Mthemtics Dongeui University Pusn 61-71, KOREA e-mil:sskim@hyomin.dongeui.c.kr S. S. DRAGOMIR School of Communictions nd Informtics Victori Unniversity of Techology Melbourne City, MC 8001 Austrli e-mil:sever@mtild.vu.edu.u A. WHITE Deprtment of Mthemtics St. Bonventure University St. Bonventure, NY 1778, U.S.A. e-mil:white@sbu.edu Y. J. CHO Deprtment of Mthemtics Gyeongsng Ntionl University Chinju , KOREA e-mil:yjcho@nonge.gsnu.c.kr