A GRÜSS TYPE INEQUALITY FOR SEQUENCES OF VECTORS IN NORMED LINEAR SPACES AND APPLICATIONS

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1 A GRÜSS TYPE INEQUALITY FOR SEQUENCES OF VECTORS IN NORMED LINEAR SPACES AND APPLICATIONS S. S. DRAGOMIR Abstract. A discrete iequality of Grüss type i ormed liear spaces ad applicatios for the discrete Fourier trasform, Melli trasform of sequeces, for polyomials with coefficiets i ormed spaces ad for vector valued Lipschitzia mappigs are give. 1. Itroductio I 1935, G. Grüss [9] proved the followig itegral iequality which gives a approximatio of the itegral of the product i terms of the product of the itegrals as follows (1.1) 1 b a b 1 (Φ φ) (Γ γ), 4 a f (x) g (x) dx 1 b a b a f (x) dx 1 b a where f, g : [a, b] R are itegrable o [a, b] ad satisfy the coditio (1.) φ f (x) Φ, γ g (x) Γ b a g (x) dx for each x [a, b], where φ, Φ, γ, Γ are give real costats. Moreover, the costat 1 4 is sharp i the sese that it caot be replaced by a smaller oe. For a simple proof of (1.1) as well as for some other itegral iequalities of Grüss type, see Chapter X of the recet book [11] ad the papers [1]-[8] ad [10]. I 1950, M. Bieracki, H. Pidek ad C. Ryll-Nardjewski [11, Chapter X] established the followig discrete versio of Grüss iequality: Theorem 1. Let a (a 1,..., a ), b (b 1,..., b ) be two tuples of real umbers such that r a i R ad s b i S for i 1,...,. The oe has 1 (1.3) a i b i 1 a i 1 b i 1 [ ] ( 1 1 [ ] ) (R r) (S s), where [x] deotes the iteger part of x, x R. A weighted versio of the discrete Grüss iequality was proved by J. E. Pečarić i 1979 [11, Chapter X]: Date: May, Mathematics Subject Classificatio. Primary 6D15, 6D99; Secodary 46Bxx. Key words ad phrases. Grüss Iequality, Normed liear spaces, Fourier Trasforms, Melli Trasform, Polyomials. 1

2 S. S. DRAGOMIR Theorem. Let a ad b be two mootoic tuples ad p a positive oe. The 1 p i a i b i 1 1 (1.4) p i a i p i b i P P P Pk Pk+1 a a 1 b b 1 max, 1 k where P : p i, ad P k+1 P P k+1. I 1981, A. Lupaş, [11, Chapter X] proved some similar results for the first differece of a as follows. Theorem 3. Let a, b be two mootoic tuples i the same sese ad p a positive -tuple. The ( ) mi a i mi b i 1 i 1 (1.5) p i ip i 1 i 1 i P P 1 P p i a i b i 1 P p i a i 1 P max a i max b i 1 1 i 1 i P p i b i P ( i 1 p i P ) ip i, where a i : a i+1 a i is the forward first differece. If there exist the umbers ā, ā 1, r, r 1 (rr 1 > 0) such that a k ā + kr ad b k ā 1 + kr 1, the equality holds i (1.5). I the recet paper [6], the authors obtaied the followig related result Theorem 4. Let (X, ) be a ormed liear space over K, K R, C, x i X, α i K ad p i 0 (i 1,..., ) such that p i 1. The we have the iequality (1.6) p i α i x i p i α i p i x i ( ) max α j max x j i p i ip i. 1 j 1 j The iequality (1.6) is sharp i the sese that the costat c 1 i the right had side caot be replaced by a smaller oe. I this paper we poit out aother iequality of Grüss type ad apply it i approximatig the discrete Fourier trasform, the Melli trasform of sequeces, for polyomials with coefficiets i ormed liear spaces ad for vector valued Lipschitzia mappigs.. A New Discrete Iequality of Grüss Type The followig iequality of Grüss type holds.

3 A GRÜSS TYPE INEQUALITY 3 Theorem 5. Let (X, ) be a ormed liear space over K, K R, C, x i X, α i K ad p i 0 (i 1,..., ) ( ) such that p i 1. The we have the iequality (.1) p i α i x i p i α i p i x i 1 p i (1 p i ) α i x i, where α i : α i+1 α i (i 1,..., 1) ad x i : x i+1 x i (i 1,..., 1) are the usual forward differeces. The costat 1 is sharp i the sese that it caot be replaced by a smaller costat. Proof. Let us start with the followig idetity i ormed liear spaces which ca be proved by direct computatio [6] p i α i x i p i α i p i x i 1 p i p j (α j α i ) (x j x i ) As i < j, we ca write ad i, j 1 j 1 α j α i (α k+1 α k ) α k p i p j (α j α i ) (x j x i ). j 1 j 1 x j x i (x l+1 x l ) x l. Usig the geeralized triagle iequality, we have successively: (.) p i α i x i p i α i p i x i j 1 j 1 p i p j α k x l li li li j 1 j 1 p i p j α k x l li j 1 j 1 p i p j α k x l : A. li It is obvious for all 1 i < j 1, we have that ad j 1 α k α k j 1 x l x l li l1

4 4 S. S. DRAGOMIR ad the (.3) A α k x l Now, let us observe that (.4) l1 p i p j. p i p j 1 p i p j p i p j i, ij 1 p i p j p i 1 p i (1 p i ). Usig (.) (.4), we deduce the desired iequality (.1). To prove the sharpess of the costat 1, let us assume that (.1) holds with a costat c > 0. That is, (.5) p i α i x i p i α i p i x i c p i (1 p i ) α i x i for all α i, x i, p i (i 1,..., ) as above ad. Choose i (.1) ad compute p i α i x i p i α i p i x i 1 Also, p i (1 p i ) 1 α i i, 1 i<j p i p j (α i α j ) (x i x j ) p i p j (α i α j ) (x i x j ) p 1 p (α 1 α ) (x 1 x ). 1 x i (p 1 p + p 1 p ) α 1 α x 1 x. Substitutig i (.5), we obtai p 1 p α 1 α x 1 x cp 1 p α 1 α x 1 x. If we assume that p 1, p > 0, α 1 α, x 1 x, the we obtai c 1, which proves the sharpess of the costat 1. The followig corollary holds. Corollary 1. Uder the above assumptios for x i, α i (i 1,..., ), we have the iequality 1 (.6) α i x i 1 α i 1 x i 1 ( 1 1 ) α i x i, ad the costat 1 is sharp.

5 A GRÜSS TYPE INEQUALITY 5 Cosiderig the case of real or complex umbers is importat i practical applicatios. Corollary. Let α i, β i K, p i 0 (i 1,..., ) with p i 1. The we have the iequality (.7) p i α i β i p i α i p i β i 1 p i (1 p i ) α i β i, ad the costat 1 is sharp. Remark 1. If i the above iequality we choose β i ᾱ i (i 1,..., ), the we get (.8) 0 1 p i α i p i α i p i (1 p i ) ( α i ), ad the costat 1 is sharp. 3. Applicatios for the Discrete Fourier Trasform Let (X, ) be a ormed liear space over K, K C, R, ad x (x 1,..., x ) be a sequece of vectors i X. For a give w K, defie the discrete Fourier trasform (3.1) F w ( x) (m) : exp (wimk), m 1,...,. The followig approximatio result for the Fourier trasform (3.1) holds. Theorem 6. Let (X, ) ad x X be as above. The we have the iequality F si (wm) w ( x) (m) si (wm) exp [( + 1) im] 1 (3.) for all m {1,..., }. ( 1) si (wm) x i, Proof. Usig the iequality (.6), we ca state that a k a k 1 (3.3) 1 a k

6 6 S. S. DRAGOMIR for all a k K, X, k 1,...,. Now, choose i (3.3), a k exp (wimk) to obtai (3.4) F w ( x) (m) exp (wimk) 1 1 exp (wim (k + 1)) exp (wimk), for all m {1,..., }. However, exp (wim) 1 exp (wimk) exp (wim) exp (wim) 1 cos (wm) + i si (wm) 1 exp (wim) cos (wm) + i si (wm) 1 [ si ] (wm) + i si (wm) cos (wm) exp (wim) si (wm) + i si (wm) cos (wm) si (wm) si (wm) i cos (wm) exp (wim) si (wm) si (wm) i cos (wm) si (wm) cos (wm) + i si (wm) exp (wim) si (wm) cos (wm) + i si (wm) We observe that si (wm) si (wm) si (wm) si (wm) si (wm) si (wm) exp (wim) exp (iwm) exp (iwm) exp [wim + iwm iwm] exp [( + 1) mi]. exp (wim (k + 1)) exp (wimk) cos (wm (k + 1)) + i si (wm (k + 1)) cos (wmk) i si (wmk) cos (wm (k + 1)) cos (wmk) + i [si (wm (k + 1)) si (wmk)] wm (k + 1) + wmk wm (k + 1) wmk si si wm (k + 1) + wmk wm (k + 1) wmk +i cos si si ((k + 1) wm) si (wm) + i cos ((k + 1) wm) si (wm) i si (wm) [cos [(k + 1) mw] + i si [(k + 1) mw]] i si (wm) exp [(k + 1) mwi], ad the exp (wim (k + 1)) exp (wimk) si (wm)

7 A GRÜSS TYPE INEQUALITY 7 for all k 1,..., 1. Cosequetly, exp (wim (k + 1)) exp (wimk) si (wm) ( 1) ad by (3.4), we get the desired iequality (3.). 4. Applicatios for the Discrete Melli Trasform Let (X, ) be a ormed liear space over K, K C, R, ad x (x 1,..., x ) be a sequece of vectors i X. Defie the Melli trasform (4.1) M ( x) (m) : k m 1, m 1,..., ; of the sequece x X. The followig result holds. Theorem 7. Let (X, ) ad x X be as above. The we have the iequality M ( x) (m) S m 1 () 1 (4.) ( 1) ( m 1 1 ), where S p (), p R, N is the p powers sum of the first atural umbers, i.e., S p () k p. Proof. Usig the iequality (3.3), we ca state that k m 1 k m (k + 1) m 1 k m 1 ad the iequality (4.) is proved. ad ( 1) ( m 1 1 ), Cosider the followig particular values of the Melli trasform µ 1 ( x) : k The followig corollary holds. µ ( x) : k.

8 8 S. S. DRAGOMIR Corollary 3. Let X ad x be as specified above. The we have the iequalities: (4.3) µ 1 ( x) + 1 ( 1) ad (4.4) µ ( x) ( + 1) ( + 1) 6 ( 1) ( + 1). Remark. If we assume that p (p 1,..., p ) is a probability distributio, i.e., p k 0 (k 1,..., ) ad p k 1, the, by (4.3) ad (4.4), we get the iequalities (4.5) kp k + 1 ( 1) p k+1 p k ad (4.6) k p k ( + 1) ( + 1) 6 ( 1) ( + 1) p k+1 p k, which have bee obtaied i [4] ad applied for the estimatio of the 1 ad - momets of a guessig mappig. 5. Applicatios for Polyomials Let (X, ) be a ormed liear space over K, K C, R, ad c (c 0,..., c ) be a sequece of vectors i X. Defie the polyomial P : C X with the coefficiets c by P (z) c 0 + zc z c, z C, c 0. The followig approximatio result for the polyomial P holds. Theorem 8. Let X, c ad P be as above. The we have the iequality: (5.1) P (z) z+1 1 c c z 1 z ( z 1) ( z 1) c k for all z C, z 1. Proof. Usig the iequality (3.3), we ca state that z k c k z k 1 (5.) c k + 1 ad, as z k+1 z k c k zk z+1 1 z 1 z k z 1 c k z 1 ( z 1) ( z 1) c k,, z 1, the iequality (5.1) is proved.

9 A GRÜSS TYPE INEQUALITY 9 The followig result for the complex roots of the uity also holds: ( ) ( ) Theorem 9. Let z k : cos +1 + i si +1, k {0,..., } be the complex ( + 1) roots of the uity. The we have the iequality (5.3) P (z k ) si c k ( + 1) for all k {1,..., }. Proof. As i the proof of Theorem 8, we have P (z) z (5.4) c k z m0 z m z 1 c k. If we choose z z k, k {1,..., }, we have z k m 1, z +1 k 1 ad the, by (5.4) we deduce (5.5) P (z k ) z k 1 c k. However, z k 1 cos ( ) + 1 ( si ( + 1) ( i si ( + 1) ( + i si + 1 ) ) [ cos ) 1 + i si ( + 1) + i si ( + 1) [ cos [ ad the z k 1 si, for all k {1,..., }. ( + 1) Usig (5.5), we deduce the desired iequality (5.3). ] ( + 1) ]] ( + 1) Corollary 4. Let P (z) c kz k be a polyomial with real coefficiets satisfyig the coditio c 0 c 1... c. The we have the iequality [ (5.6) P (z k ) si ( + 1) for all k {1,..., }, where z k are as i Theorem 9. ] (c c 0 ), 6. Applicatios for Lipschitzia Mappigs Let (X, ) be as above ad F : X Y a mappig defied o the ormed liear space X with values i the ormed liear space Y which satisfies the Lipschitzia coditio (6.1) F (x) F (y) L x y for all x, y X, where deotes the orm o Y. The followig theorem holds.

10 10 S. S. DRAGOMIR Theorem 10. Let F : X Y be as above ad x i X, p i 0 (i 1,..., ) ad p i 1. The we have the iequality: (6.) ( ) p i F (x i ) F p i x i L p i (1 p i ). Proof. As F is Lipschitzia, we have (6.1) for all x, y X. Choose x p ix i ad y x j (j 1,..., ) to get ( ) (6.3) F p i x i F (x j ) L p i x i x j for all j {1,..., }. If we multiply (6.3) by p j 0 ad sum over j from 1 to, we obtai ( ) (6.4) p j F p i x i F (x j ) L p j p i x i x j. Usig the geeralized triagle iequality, we have ( ) ( (6.5) p j F p i x i F (x j ) ) F p i x i p j F (x j ). By the geeralized triagle iequality i the ormed space X, we also have (6.6) p j p i x i x j p j p i (x i x j ) p i p j x i x j i, As i the proof of Theorem 3, we have, for i < j j 1 j 1 x i x j x k ad the Sice B p i p j 1 p i p j x i x j : B. j 1 p i p j p i p j. p i (1 p i ) the we get, by (6.4) (6.6), the desired iequality (6.). The followig corollary is a atural cosequece of the above results.

11 A GRÜSS TYPE INEQUALITY 11 Corollary 5. Let x i X ad p i be as above. The we have the iequality: 0 p i x i p i x i p i (1 p i ). Refereces [1] S.S. Dragomir, Grüss iequality i ier product spaces, The Australia Math Soc. Gazette, 6 (1999), No., [] S.S. Dragomir, A geeralizatio of Grüss iequality i ier product spaces ad applicatios, J. Math. Aal. Appl., 37 (1999), [3] S.S. Dragomir, A Grüss type itegral iequality for mappigs of r-hölder s type ad applicatios for trapezoid formula, Tamkag J. of Math., 31(1) (000), [4] S.S. Dragomir, Some discrete iequalities of Grüss type ad applicatios i guessig theory, Hoam Math. J., 1(1) (1999), [5] S.S. Dragomir, Some itegral iequalities of Grüss type, Idia J. of Pure ad Appl. Math., 31(4) (000), [6] S.S. Dragomir ad G.L. Booth, O a Grüss-Lupaş type iequality ad its applicatios for the estimatio of p-momets of guessig mappigs, Mathematical Commuicatios, 5 (000), [7] S.S. Dragomir ad I. Fedotov, A iequality of Grüss type for Riema-Stieltjes itegral ad applicatios for special meas, Tamkag J. of Math., 9(4)(1998), [8] S.S. Dragomir ad J.E. Pečarić, O some iequalities for the momets of guessig mappig, submitted. [9] G. Grüss, Über das Maximum des absolute Betrages vo 1 b b a a f(x)g(x)dx 1 b (b a) a f(x)dx b a g(x)dx, Math. Z., 39(1935), [10] A.M. Fik, A treatise o Grüss iequality, Aalytic ad Geometric Iequalities, , Math. Appl. 478, Kluwer Academic Publ., [11] D.S. Mitriović, J.E. Pečarić ad A.M. Fik, Classical ad New Iequalities i Aalysis, Kluwer Academic Publishers, Dordrecht, School of Commuicatios ad Iformatics, Victoria Uiversity of Techology, PO Box 1448, Melboure City, MC 8001 Australia address: sever@matilda.vu.edu.au URL: