1. Introduction. 1 b b

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1 Journl of Mhemicl Inequliies Volume, Number 3 (007), SOME IMPROVEMENTS OF GRÜSS TYPE INEQUALITY N. ELEZOVIĆ, LJ. MARANGUNIĆ AND J. PEČARIĆ (communiced b A. Čižmešij) Absrc. In his pper some inequliies reled o Chebshev s funcionl re proved. The underling funcion spces re L p spces wih weigh funcion nd exponens which need no be conjuge. Vrious bounds obined in his nd previous ppers re compred.. Inroducion Severl ppers on vrious improvemens of Grüss pe inequliies ppered in recen ime. For wo mesurble funcions f, g : [, b] R denoe b T(f, g) Chebshev s funcionl: T(f, g) := b f (x)g(x) b S. S. Drgomir ([]) proved he following resul: f (x) b g(x). () THEOREM A. Le f, g : [, b] R be wo mppings, differenible on (, b).if f L (, b) nd g L (, b), wih > nd + =, hen we hve he inequli ( T(f, g) b (b ) x f () d d) x ( b ) (b ) x g () d d b 6 f g. () J. Pečrić nd B. Tepeš ([4]) improved his resul nd showed h T(f, g) b 8 f g. (3) Mhemics subjec clssificion (000): 6D5. Ke words nd phrses: Grüss inequli, Chebshev s funcionl. x c D l,zgreb Pper JMI

2 46 N. ELEZOVIĆ, LJ.MARANGUNIĆ AND J. PEČARIĆ We shll furher generlize his improvemen b considering weighed version of his Grüss pe inequli. Le p(x) be nonnegive inegrble funcion defined on [, b]. Iiswellknown h Chebshev s funcionl T(f, g; p) := p(x) cn be wrien in he Korkine s form f (x)g(x)p(x) f (x)p(x) g(x)p(x) (4) T(f, g; p) = p(x)p()[f (x) f ()][g(x) g()] d (5) The following heorem is proved in []. THEOREM B. Le f, g : [, b] R be wo mppings, differenible on (, b) nd p : [, b] [0, ) is inegrble on [, b].iff L (, b),g L (, b) wih > nd + =, hen we hve he inequli ( T(f, g; p) b p(x)p() x f () d d) x ( ) p(x)p() x g () d d x ( b ) b x p(x)p()d f g. (6) Le us denoe J (f ) := p(x)p() x x f () d d (7) Furher improvemens cn be obined b nlzing his inegrl. We hve Le us denoe J (f )= = F() := x f () d (x ) x This funcion cn be wrien in he following form: F() = xp(x) p()d f () d p(x)p()d (x )p(x)p()d (8) (x )p(x)p()d (9) xp(x) p()d. (0)

3 SOME IMPROVEMENTS OF GRÜSS TYPE INEQUALITY 47 From (8) nd (9) i follows immediel: Hence, we m se he following heorem. J (f ) mx b F() f. () THEOREM. Le f, g : [, b] R be wo mppings, bsoluel coninuous nd differenible on (, b) nd p : [, b] [0, ) is inegrble on [, b].iff L (, b), g L (, b) wih > nd + =, hen we hve he inequli T(f, g; p) J (f ) / J (g) / ( ) mx F() f g. () b Therefore, we shll serch for he mximum of F.From F () = ( x)p()p(x) + (x )p()p(x) [ ] = p() xp(x) p(x) i follows h he exremum of his funcion is obined he firs momen of weigh funcion: 0 = xp(x). (3) p(x) From he sign of he firs derivive i is obvious h his exremum is he locl mximum.. Smmeric cse Le us suppose h he weigh funcion is smmeric, i.e. Then we hve Hence, i follows h p(x) =p( + b x), x [, b]. (4) xp(x) = =( + b) ( + b u)p( + b u)du xp(x) = + b p(x) p(x) xp(x).

4 48 N. ELEZOVIĆ, LJ.MARANGUNIĆ AND J. PEČARIĆ nd herefore, he bsciss (3) of he poin of mximum of funcion F() is 0 = + b. (5) In his cse, using (0), i is es o obin: ( )( b F( 0 )= p(x) +b ) +b xp(x) xp(x) In he specil cse p(x) =/(b ), i follows from (6): F( 0 )= ([ ] [ ]) b ( + b) ( + b) = b b 8 8 4, nd (3) follows. (6) 3. Generl cse We shll improve Theorem b removing he ssumpion h nd re conjuge exponens. In his secion we ssume onl >, > ndγ >. Le us denoe b, nd γ he corresponding conjuge exponens. From Hölder s inequli i follows x / f (x) f () x / f () d, x / g(x) g() x / g () d. (7) Therefore, we hve he following esimion for Chebshev s funcionl: T(f, g; p) ( ( p(x)p() f (x) f () g(x) g() d p(x)p() x + p(x)p() x + p(x)p() x + p(x)p() x + d x f () x d g () d d x γ ) γ f () d d x g () d γ d γ ) f g. (8)

5 SOME IMPROVEMENTS OF GRÜSS TYPE INEQUALITY 49 In he specil cse p(x) = b,wehve: (b ) + T(f, g) ( )( ) f g (b ) = ( )( ) f g (9) 3 4 From (8) we cn obin, b king n pproprie limi, some esimions which re no covered erlier. Le us noe onl he following hree: CORROLARY. In he cse = = we hve: ( T(f, g; p) b f g p(x)). (0) If = =,hen T(f, g; p) f g ( p(x)p()(x ) d ) () nd, in he cse =, = we hve: T(f, g; p) f g p(x)p() x d. () COROLLARY. Le p(x) = b. Then in he cse of = = we hve: If = =,hen T(f, g) f g. (3) T(f, g) (b ) f g, (4) nd, in he cse =, =, we hve: T(f, g) 6 (b ) f g. (5) Le us noe h he bound in (8) is no so ccure s in (). We cn improve i in he cse when + <. Assume firs h p(x) = b.wehve:

6 430 N. ELEZOVIĆ, LJ.MARANGUNIĆ AND J. PEČARIĆ THEOREM. Le f, g : [, b] R be wo mppings, bsoluel coninuous nd differenible on (, b).iff L (, b) nd g L (, b), wih >, > nd, hen he following inequli holds: + T(f, g) (b ) ( )( ) 3 4 [ + 3] + f g. (6) Proof. Le us denoe b, nd γ he conjuge vlues of, nd γ. From (8) we hve: T(f, g) (b ) [ x + x + ) γ ] γ f () d d ) γ g () d d γ. (7) If γ, γ re conjuge vlues such h γ, γ hen + γ + γ =. Thus, he condiion + cn be relized b seing such vlues for γ, γ which sisf γ γ 0, 0. In his cse, le us consider: I = = [ x + f () d) γ d { x + } γ ] γ x + d { x + [ x f () d} γ d x + x f () d d ] γ = J γ J γ (8) where we denoe J = J = x + d, x + x f () d d.

7 SOME IMPROVEMENTS OF GRÜSS TYPE INEQUALITY 43 We hve for J nd J : J = J = = x x f () d (x ) + d = (x ) + d x (b ) + + ( )( ), (9) f () d (x ) + d = ( )( ) f () [ (b ) + + ( ) + + (b ) + +] d. B compuing exreml vlues of he funcion ϕ() =(b ) r ( ) r (b ) r, [, b], r = + + we obin: ( mx ϕ = ϕ 0 = + b ) =(b ) r (b )r b r =(b ) r [ r ] nd so we hve: J ( + (b ) + )( ) + Thus we obin he following esime for I : [ ] f. (30) I J γ J γ (b ) + + ( )( ) Using (7) we finll obin: T(f, g) (b ) + + ( )( ) (b ) + + ( )( ) (b ) [ ] γ f γ. [ [ ] γ ] γ f γ g γ γ γ

8 43 N. ELEZOVIĆ, LJ.MARANGUNIĆ AND J. PEČARIĆ (b ) ( (b ) + + )( ) [ ] + f g wherefrom i follows (6). REMARK. For comprison, he esime (6) improves (9), since [ + 3] + = =, (9) nd (6) coincide. < for 0 < +. In he cse of + A beer esimion is described in he following heorem. = 0, i.e. THEOREM 3. Le f, g : [, b] R be wo mpings, bsoluel coninuous nd differenible on (, b) nd p : [, b] [0, ) is inegrble on [, b].iff L (, b), g L (, b) wih >, > nd +, hen we hve he inequli: T(f, g; p) M mx F() + f g (3) where [ M = p(x)p()(x ) d ], F() = p(x) (x )p() d. (3) Proof. Denoe δ :=. (33) Then + = + = + δ.using(7), i is es o obin he following esime: f (x) f () g(x) g() x = x d ) f () d x ) δ ) g () d ) f () d g () d). (34)

9 SOME IMPROVEMENTS OF GRÜSS TYPE INEQUALITY 433 Appling gin Hölder s inequli, we obin s before: T(f, g; p) ) δ p(x)p() x d f () d) g () d) d = ) ( δ x p(x)p() x d p(x)p() x f () d) ( x p(x)p() x g () d) d ( b ) b δ p(x)p()(x ) d (J (f )) ( J (g) ). Using () we finll obin: T(f, g; p) [ ] [ M mx F() f b In he cse p(x) = b = M [ mx b F() + we hve: mx F() b ] f g. mx F() = (b ), b 4 [ ] M = (b ), 6 ] g T(f, g) (b ) 4 (b ) + f g = ( ) 3 + (b ) f g. Thus we showed: COROLLARY 3. For >, >, + we hve: T(f, g) ( ) 3 + (b ) f g. (35)

10 434 N. ELEZOVIĆ, LJ.MARANGUNIĆ AND J. PEČARIĆ ( REMARK. B elemenr clculus we cn show h 3 ) x < (3 x)(4 x) for x (0, ], hus he esime (35) improves (9). In he cse x = + = 0, i.e. = =, (9) nd (35) coincide. REMARK 3. The esime (35) is beer hn (6), since i holds h: ( ) x 3 < ( x 3 ) x, x (0, ). (3 x)(4 x) On he following picure grphs of funcions ppering on he lef-hnd nd he righ-hnd side of his inequli re ploed. In cses = = nd =, = (6) nd (35) coincide. 4. Comprison wih previous resuls We shll compre esime (3) from [4] wih he following one which ws proved much erlier in [3]: T(f, g) b [ ] / [ ] / f g. (36) 4 ( + ) ( + )

11 SOME IMPROVEMENTS OF GRÜSS TYPE INEQUALITY 435 THEOREM 4. I holds [ ] / [ ] /. ( + ) ( + ) Therefore, inequli (3) is sronger hn (36). Proof. Le us clcule firs he following inegrl I := 0 (x ) d = In priculr, for = i follows h I = B using Hölder s inequli, we obin = I = = 0 ( 0 0 (x )d 0 [x (x ) ] = (x )d =. (x )(x ) d x ) / ( (x ) d x =(I ) / (I ) / [ ] / [ ] / =, ( + ) ( + ) which proves he heorem. 0 ( + ). ) / (x ) d x Acknowledgemen. suggesions. We wish o hnks o he referee for his observions nd REFERENCES [] S. S. DRAGOMIR, Some inegrl inequliies of Grüss pe, Indin J. Pure Appl. Mh 3, 4 (000), [] P. CERONE, S. S. DRAGOMIR AND J. ROUMELIOTIS, Grüss Inequli in erms of Δ -seminorms nd pplicions, RGMIA Reserch Repor Collecion 3, 3 (000), Aricle 3. [3] D. S. MITRINOVIĆ, J. E. PEČARIĆ AND A. M. FINK, Clssicl nd New inequliies in Anlsis, Kluwer Acdemic Publishers, Dordrech, 993. [4] J. PEČARIĆ, B. TEPEŠ, Improvemens of some inegrl inequliies of Grüss pe, Tmkng Journl of Mh. 36, (005), 39 4.

12 436 N. ELEZOVIĆ, LJ. MARANGUNIĆ AND J. PEČARIĆ [5] J. PEČARIĆ, B. TEPEŠ, A noe on Grüss pe inequli in erms of Δ -seminorms, Mcedonin Acdem of Sciences nd Ars, Conribuions (Prilozi MANU) 3-4 (00-003), N. Elezović Deprmen of Applied Mhemics Fcul of Elecricl Engineering nd Compuing Universi of Zgreb Unsk 3, 0000 Zgreb Croi e-mil: neven.elez@fer.hr Lj. Mrngunić Fcul of Elecricl Engineering nd Compuing Universi of Zgreb Unsk 3, 0000 Zgreb Croi e-mil: ljubo.mrngunic@fer.hr J. Pečrić Fcul of Texile Technolog Universi of Zgreb Pieroijev 6, 0000 Zgreb Croi e-mil: pecric@elemen.hr Journl of Mhemicl Inequliies jmi@ele-mh.com