Journal of Inequalities in Pure and Applied Mathematics

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1 Journl of Inequlities in Pure nd Applied Mthemtics Volume 3, Issue, Article 4, 00 ON AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL AND SOME RAMIFICATIONS P. CERONE SCHOOL OF COMMUNICATIONS AND INFORMATICS VICTORIA UNIVERSITY OF TECHNOLOGY PO BOX 448 MELBOURNE CITY MC VICTORIA 800, AUSTRALIA. URL: Received 5 April, 00; ccepted 7 July, 00. Communicted by R.P. Agrwl ABSTRACT. An identity for the Chebychev functionl is presented in which Riemnn-Stieltjes integrl is involved. This llows bounds for the functionl to be obtined for functions tht re of bounded vrition, Lipschitzin nd monotone. Some pplictions re presented to produce bounds for moments of functions bout generl point γ nd for moment generting functions. Key words nd phrses: Chebychev functionl, Bounds, Riemnn-Stieltjes, Moments, Moment Generting Function. 000 Mthemtics Subject Clssifiction. Primry 6D5, 6D0; Secondry 65Xxx.. INTRODUCTION For two mesurble functions f, g : [, b R, define the functionl, which is known in the literture s Chebychev s functionl, by (.) T (f, g) : M (fg) M (f) M (g), where the integrl men is given by (.) M (f) f (x) dx. The integrls in (.) re ssumed to exist. Further, the weighted Chebychev functionl is defined by (.3) T (f, g; p) : M (f, g; p) M (f; p) M (g; p), ISSN (electronic): c 00 Victori University. All rights reserved. The uthor undertook this work while on sbbticl t the Division of Mthemtics, L Trobe University, Bendigo. Both Victori University nd the host University re commended for giving the uthor the time nd opportunity to think

2 P. CERONE where the weighted integrl men is given by p (x) f (x) dx (.4) M (f; p) p (x) dx. We note tht, T (f, g; ) T (f, g) nd M (f; ) M (f). It is the im of this rticle to obtin bounds on the functionls (.) nd (.3) in terms of one of the functions, sy f, being of bounded vrition, Lipschitzin or monotonic nondecresing. This is ccomplished by developing identities involving Riemnn-Stieltjes integrl. These identities seem to be new. The min results re obtined in Section, while in Section 3 bounds for moments bout generl point γ re obtined for functions of bounded vrition, Lipschitzin nd monotonic. In previous rticle, Cerone nd Drgomir [ obtined bounds in terms of the f p, p where it necessitted the differentibility of the function f. There is no need for such ssumptions in the work covered by the current development. A further ppliction is given in Section 4 in which the moment generting function is pproximted.. AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL It is worthwhile noting tht number of identities relting to the Chebychev functionl lredy exist. The reder is referred to [7 Chpters IX nd X. Korkine s identity is well known, see [7, p. 96 nd is given by (.) T (f, g) () (f (x) f (y)) (g (x) g (y)) dxdy. It is identity (.) tht is often used to prove n inequlity of Grüss for functions bounded bove nd below, [7. The Grüss inequlity is given by (.) T (f, g) 4 (Φ f φ f ) (Φ g φ g ) where φ f f (x) Φ f for x [, b. If we let S (f) be n opertor defined by (.3) S (f) (x) : f (x) M (f), which shifts function by its integrl men, then the following identity holds. Nmely, (.4) T (f, g) T (S (f), g) T (f, S (g)) T (S (f), S (g)), nd so (.5) T (f, g) M (S (f) g) M (fs (g)) M (S (f) S (g)) since M (S (f)) M (S (g)) 0. For the lst term in (.4) or (.5) only one of the functions needs to be shifted by its integrl men. If the other were to be shifted by ny other quntity, the identities would still hold. A weighted version of (.5) relted to T (f, g) M ((f (x) κ) S (g)) for κ rbitrry ws given by Sonin [8 (see [7, p. 46). The interested reder is lso referred to Drgomir [5 nd Fink [6 for extensive tretments of the Grüss nd relted inequlities. J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

3 AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL 3 The following lemm presents n identity for the Chebychev functionl tht involves Riemnn- Stieltjes integrl. Lemm.. Let f, g : [, b R, where f is of bounded vrition nd g is continuous on [, b, then (.6) T (f, g) where () ψ (t) df (t), (.7) ψ (t) (t ) A (t, b) (b t) A (, t) with (.8) A (, b) g (x) dx. Proof. From (.6) integrting the Riemnn-Stieltjes integrl by prts produces { b b } b () ψ (t) df (t) () ψ (t) f (t) f (t) dψ (t) () { ψ (b) f (b) ψ () f () since ψ (t) is differentible. Thus, from (.7), ψ () ψ (b) 0 nd so () ψ (t) df (t) () } f (t) ψ (t) dt [() g (t) A (, b) f (t) dt [g (t) M (g) f (t) dt M (fs (g)) from which the result (.6) is obtined on noting identity (.5). The following well known lemms will prove useful nd re stted here for lucidity. Lemm.. Let g, v : [, b R be such tht g is continuous nd v is of bounded vrition on [, b. Then the Riemnn-Stieltjes integrl g (t) dv (t) exists nd is such tht b (.9) g (t) dv (t) sup g (t) (v), t [,b where b (v) is the totl vrition of v on [, b. Lemm.3. Let g, v : [, b R be such tht g is Riemnn-integrble on [, b nd v is L Lipschitzin on [, b. Then (.0) g (t) dv (t) L g (t) dt with v is L Lipschitzin if it stisfies for ll x, y [, b. v (x) v (y) L x y J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

4 4 P. CERONE Lemm.4. Let g, v : [, b R be such tht g is continuous on [, b nd v is monotonic nondecresing on [, b. Then (.) g (t) dv (t) g (t) dv (t). It should be noted tht if v is nonincresing then v is nondecresing. Theorem.5. Let f, g : [, b R, where f is of bounded vrition nd g is continuous on [, b. Then (.) () T (f, g) sup t [,b ψ (t) b (f), L ψ (t) dt, ψ (t) df (t), where b (f) is the totl vrition of f on [, b. for f L Lipschitzin, for f monotonic nondecresing, Proof. Follows directly from Lemms..4. Tht is, from the identity (.6) nd (.9) (.). The following lemm gives n identity for the weighted Chebychev functionl tht involves Riemnn-Stieltjes integrl. Lemm.6. Let f, g, p : [, b R, where f is of bounded vrition nd g, p re continuous on [, b. Further, let P (b) p (x) dx > 0, then (.3) T (f, g; p) P (b) where T (f, g; p) is s given in (.3), Ψ (t) df (t), (.4) Ψ (t) P (t) Ḡ (t) P (t) G (t) with (.5) nd P (t) t p (x) dx, P (t) P (b) P (t) G (t) t p (x) g (x) dx, Ḡ (t) G (b) G (t). Proof. The proof follows closely tht of Lemm.. We first note tht Ψ (t) my be represented in terms of only P ( ) nd G ( ). Nmely, (.6) Ψ (t) P (t) G (b) P (b) G (t). J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

5 AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL 5 It my further be noticed tht Ψ () Ψ (b) 0. Thus, integrting from (.3) nd using either (.4) or (.6) gives P (b) Ψ (t) df (t) where we hve used the fct tht P (b) P (b) P (b) f (t) dψ (t) [P (b) G (t) P (t) G (b) f (t) dt [ p (t) g (t) G (b) P (b) p (t) f (t) dt p (t) g (t) f (t) dt G (b) P (b) P (b) M (f, g; p) M (g; p) M (f; p) T (f, g; p), G (b) P (b) M (g; p). P (b) p (t) f (t) dt Theorem.7. Let the conditions of Lemm.6 on f, g nd p continue to hold. Then sup Ψ (t) b (f), t [,b (.7) P (b) T (f, g; p) L Ψ (t) dt, for f L Lipschitzin, Ψ (t) df (t), for f monotonic nondecresing. where T (f, g; p) is s given by (.3) nd Ψ (t) P (t) G (b) P (b) G (t), with P (t) t p (x) dx, G (t) t p (x) g (x) dx. Proof. The proof uses Lemms..4 nd follows closely tht of Theorem.5. Remrk.8. If we tke p (x) in the bove results involving the weighted Chebychev functionl, then the results obtined erlier for the unweighted Chebychev functionl re recptured. Grüss type inequlities obtined from bounds on the Chebychev functionl hve been pplied in vriety of res including in obtining perturbed rules in numericl integrtion, see for exmple [4. In the following section the bove work will be pplied to the pproximtion of moments. For other relted results see lso [ nd [3. Remrk.9. If f is differentible then the identity (.6) would become (.8) T (f, g) nd so () ψ (t) f (t) dt ψ f, f L [, b ; () T (f, g) ψ q f p, f L p [, b, p >, p + ; q ψ f, f L [, b ; J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

6 6 P. CERONE where the Lebesgue norms re defined in the usul wy s ( ) g p : g (t) p p dt, for g L p [, b, p, p + q nd g : ess sup g (t), for g L [, b. t [,b The identity for the weighted integrl mens (.3) nd the corresponding bounds (.7) will not be exmined further here. Theorem.0. Let g : [, b R be bsolutely continuous on [, b then for (.9) D (g;, t, b) : M (g; t, b) M (g;, t), ( ) g, g L [, b ; (.0) D (g;, t, b) [ (t ) q + (b t) q q g p, g L p [, b, q + p >, + ; p q g, g L [, b ; b (g), ( Proof. Let the kernel r (t, u) be defined by (.) r (t, u) : g of bounded vrition; ) L, g is L Lipschitzin. u, u [, t, t b u, u (t, b b t then stright forwrd integrtion by prts rgument of the Riemnn-Stieltjes integrl over ech of the intervls [, t nd (t, b gives the identity (.) Now for g bsolutely continuous then (.3) D (g;, t, b) nd so r (t, u) dg (u) D (g;, t, b). D (g;, t, b) ess sup r (t, u) u [,b where from (.) r (t, u) g (u) du (.4) ess sup r (t, u) u [,b g (u) du, for g L [, b, J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

7 AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL 7 nd so the third inequlity in (.0) results. Further, using the Hölder inequlity gives ( ) ( D (g;, t, b) r (t, u) q q b ) du g (t) p p (.5) dt where explicitly from (.) ( (.6) r (t, u) q du Also for p >, ) q p + q, [ t ( ) q u b ( ) q b u du + du q t t b t ( ) [(t ) q + (b t) q q u q q du [ (t ) q + (b t) q q + (.7) D (g;, t, b) ess sup g (u) u [,b q. 0 r (t, u) du, nd so from (.6) with q gives the first inequlity in (.0). Now, for g (u) of bounded vrition on [, b then from Lemm., eqution (.9) nd identity (.) gives D (g;, t, b) ess sup r (t, u) (g) u [,b producing the fourth inequlity in (.0) on using (.4). From (.0) nd (.) we hve, by ssociting g with v nd r (t, ) with g ( ), D (g;, t, b) L b r (t, u) du nd so from (.6) with q gives the finl inequlity in (.0). Remrk.. The results of Theorem.0 my be used to obtin bounds on ψ (t) since from (.7) nd (.9) ψ (t) (t ) (b t) D (g;, t, b). Hence, upper bounds on the Chebychev functionl my be obtined from (.) nd (.8) for generl functions g. The following two sections investigte the exct evlution (.) for specific functions for g ( ). 3. RESULTS INVOLVING MOMENTS In this section bounds on the n th moment bout point γ re investigted. Define for n nonnegtive integer, (3.) M n (γ) : (x γ) n h (x) dx, γ R. If γ 0 then M n (0) re the moments bout the origin while tking γ M (0) gives the centrl moments. Further the expecttion of continuous rndom vrible is given by (3.) E (X) h (x) dx, J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

8 8 P. CERONE where h (x) is the probbility density function of the rndom vrible X nd so E (X) M (0). Also, the vrince of the rndom vrible X, σ (X) is given by (3.3) σ (X) E [ (X E (X)) (x E (X)) h (x) dx, which my be seen to be the second moment bout the men, nmely The following corollry is vlid. σ (X) M (M (0)). Corollry 3.. Let f : [, b R be integrble on [, b, then (3.4) M n (γ) Bn+ A n+ M (f) n + b sup φ (t) (f), for f of bounded vrition on [, b, n+ t [,b L φ (t) dt, for f L Lipschitzin, n + n + φ (t) df (t), for f monotonic nondecresing. where M n (γ) is s given by (3.), M (f) is the integrl men of f s defined in (.), nd (3.5) φ (t) (t γ) n B b γ, A γ [( ) t (b γ) n+ + ( ) b t ( γ) n+. Proof. From (.) tking g (t) (t γ) n then using (.) nd (.) gives () T (f, (t γ) n ) M n (γ) Bn+ A n+ M (f) n +. The right hnd side is obtined on noting tht for g (t) (t γ) n, φ (t) ψ(t) b. Remrk 3.. It should be noted here tht Cerone nd Drgomir [ obtined bounds on the left hnd expression for f L p [, b, p. They obtined the following Lemms which will prove useful in procuring expressions for the bounds in (3.4) in more explicit form. Lemm 3.3. Let φ (t) be s defined by (3.5), then n odd, ny γ nd t (, b) < 0 { γ <, t (, b) n even (3.6) φ (t) < γ < b, t [c, b) > 0, n even { γ > b, t (, b) < γ < b, t (, c) J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

9 AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL 9 where φ (c) 0, < c < b nd c > γ, γ < +b γ, γ +b < γ, γ > +b. Lemm 3.4. For φ (t) s given by (3.5) then (3.7) φ (t) dt B A [B n+ A n+ Bn+ A n+ n+, { n odd nd ny γ n even nd γ < ; {[ C n+ B n+ A n+ + n+ (b ) () (c ) B n+ + [ (b c) () } A n+, n even nd < γ < b; B n+ A n+ B A [B n+ A n+, n even nd γ > b, n+ where B b γ, A γ, C c γ, (3.8) C c C (t) dt, C C (t) dt, c with C (t) ( ) t b B n+ + ( ) b t b A n+ nd φ (c) 0 with < c < b. Lemm 3.5. For φ (t) s defined by (3.5), then C (t ) Bn+ A n+, n odd, n even nd γ < ; (n+)(b A) (3.9) sup where t [,b φ (t) B n+ A n+ C (n+)(b A) (t ) n even nd γ > b; m +m + m m n even nd < γ < b, (3.0) (t γ) n Bn+ A n+ (n + ) (B A), C (t) is s defined in (3.8), m φ (t ), m φ (t ) nd t, t, t stisfy (3.0) with t < t. The following lemm is required to determine the bound in (3.4) when f is monotonic nondecresing. This ws not covered in Cerone nd Drgomir [ since they obtined bounds ssuming tht f were differentible. Lemm 3.6. The following result holds for φ (t) s defined by (3.5), χ n (, b), n odd or n even nd γ <, (3.) n + φ (t) df χ n (, b), n even nd γ > b, χ n (c, b) χ n (, c), n even nd < γ < b J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

10 0 P. CERONE nd for f : [, b R, monotonic nondecresing (3.) where (3.3) n + Proof. Let α, β [, b nd χ n (α, β) φ (t) df B (B n ) A (A n ) f (b), n odd or n even n + nd γ < ; A (A n ) B (B n ) f (b), n even nd γ > b; n + [ B n+ C n+ (Bn A n ) f (b) (b c) n even nd [ n + (B n A n ) f () + (c ) (C n+ A n+ ) n +, < γ < b, χ n (, b) [ (t γ) n (Bn A n ) f (t) dt, (n + ) () A γ, B b γ, C c γ. n + β α φ (t) df φ (α) f (α) φ (β) f (β) n + β α [ (t γ) n (Bn A n ) f (t) dt (n + ) () nd χ n (, b) is s given by (3.3) since φ () φ (b) 0. Further, using the results of Lemm 3.3 s represented in (3.6), nd, the fct tht β χ (α, β), φ (t) < 0, t [α, β φ (t) df n + α χ (α, β), φ (t) > 0, t [α, β gives the results s stted. We now use the fct tht f is monotonic nondecresing so tht from (3.3) χ n (, b) f (b) [(t γ) n Bn A n dt. (n + ) () Further, nd χ n (c, b) f (b) [(t γ) n Bn A n dt c (n + ) () [ B n+ C n+ f (b) (Bn A n ) (b c) n + (n + ) () c χ n (, c) f () [(t γ) n Bn A n dt (n + ) () [ C n+ A n+ (Bn A n ) (c ) f () n + (n + ) () so tht the proof of the lemm is now complete. J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

11 AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL The following corollry gives bounds for the expecttion. Corollry 3.7. Let f : [, b R + be probbility density function ssocited with rndom vrible X. Then the expecttion E (X) stisfies the inequlities () 3 b (f), f of bounded vrition, 6 (3.4) E (X) + b ( ) L, f L Lipschitzin, [ + b f (b), f monotonic nondecresing. Proof. Tking n in Corollry 3. nd using Lemms gives the results fter some strightforwrd lgebr. In prticulr, ( φ (t) t ( + b) t + b t + b ) ( ) + nd t the one solution of φ (t) 0 is t +b. The following corollry gives bounds for the vrince. We shll ssume tht < γ E [X < b. Corollry 3.8. Let f : [, b R + be p.d.f. ssocited with rndom vrible X. The vrince σ (X) is such tht (3.5) σ (X) S where nd γ E (X). b [m + m + m m (f), f of bounded vrition, 6 { C [ 4 b (c ) 3 B 3 (b c) A 3 + (B + A ) ( b ) } (AB) L, f is L Lipschitzin, 3 [B 3 C 3 ( + b) (b c) f(b) 3 + [( + b) (c ) (C 3 A 3 ) f(), f monotonic nondecresing. 3 S (b E (X))3 + (E (X) ) 3, 3 () ( ) ( ) m φ E (X) S, m φ E (X) + S, φ (t) ( ) ( ) b t t (t γ) 3 + (γ ) 3 (b γ) 3, A γ, B b γ, C c γ, φ (c) 0, < c < b Proof. Tking n in Corollry 3. gives from (3.5) ( b t φ (t) (t γ) 3 + where < γ E (X) < b. ) A 3 ( ) t B 3 J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

12 P. CERONE From Lemm 3.5 nd the third inequlity in (3.9) with n gives t E [X S, t E [X + S, nd hence the first inequlity is shown from the first inequlity in (3.4). Now, if f is Lipschitzin, then from the second inequlity in (3.4) nd since n nd < γ E (X) < b, the second identity in (3.7) produces the reported result given in (3.5) fter some simplifiction. The lst inequlity is obtined from (3.) of Lemm 3.6 with n nd hence the corollry is proved. 4. APPROXIMATIONS FOR THE MOMENT GENERATING FUNCTION Let X be rndom vrible on [, b with probbility density function h (x) then the moment generting function M X (p) is given by (4.) M X (p) E [ e px e px h (x) dx. The following lemm will prove useful, in the proof of the subsequent corollry, s it exmines the behviour of the function θ (t) (4.) () θ (t) ta p (, b) [A p (t, b) + ba p (, t), where (4.3) A p (, b) ebp e p. p Lemm 4.. Let θ (t) be s defined by (4.) nd (4.3) then for ny, b R, θ (t) hs the following chrcteristics: (i) θ () θ (b) 0, (ii) θ (t) is convex for p < 0 nd concve for ( p > ) 0, (iii) there is one turning point t t ln Ap(,b) nd t b. p b Proof. The result (i) is trivil from (4.) using stndrd properties of the definite integrl to give θ () θ (b) 0. Now, (4.4) θ (t) A p (, b) ept, θ (t) pe pt giving θ (t) > 0 for p < 0 nd θ (t) < 0 for p > 0 nd (ii) holds. Further, from (4.4) θ (t ) 0 where t ( ) p ln Ap (, b). To show tht t b it suffices to show tht θ () θ (b) < 0 since the exponentil is continuous. Here θ () is the right derivtive t nd θ (b) is the left derivtive t b. Now, ( ) ( ) θ () θ Ap (, b) (b) Ap (, b) ep ebp J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

13 AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL 3 but A p (, b) e pt dt, the integrl men over [, b so tht θ () > 0, nd θ (b) < 0 for p > 0 nd θ () < 0 nd θ (b) > 0 for p < 0, giving tht there is point t [, b where θ (t ) 0. Thus the lemm is now completely proved. Corollry 4.. Let f : [, b R be of bounded vrition on [, b then (4.5) where e pt f (t) dt A p (, b) M (f) ) (m (ln (m) ) + bep e bp b (f), [( ) L () m p for f L Lipschitzin on [, b, p () m [f (b) f (), f monotonic nondecresing, (4.6) m A p (, b) ebp e p p (). Proof. From (.) tking g (t) e pt nd using (.) nd (.) gives (4.7) () ( ) T f, e pt e pt f (t) dt A p (, b) M (f) sup θ (t) b (f), for f of bounded vrition on [, b, t [,b L θ (t) dt, for f L Lipschitzin on [, b, θ (t) df (t), f monotonic nondecresing on [, b, where the bounds re obtined from (.) on noting tht for g (t) e pt, θ (t) ψ(t) is s given b by (4.) (4.3). Now, using the properties of θ (t) s expounded in Lemm 4. will id in obtining explicit bounds from (4.7). Firstly, from (4.), (4.3) nd (4.6) sup θ (t) θ (t ) t [,b [ t m A p (t, b) + b A p (, t ) m p ln (m) ( ) e bp m b ( ) m e p p p m p (ln (m) ) + bep e bp p (). J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

14 4 P. CERONE In the bove we hve used the fct tht m 0 nd tht pt ln (m). Using from Lemm 4. the result tht θ (t) is positive or negtive for t [, b depending on whether p > 0 or p < 0 respectively, the first inequlity in (4.5) results. For the second inequlity we hve tht from (4.), (4.3) nd Lemm 4., θ (t) dt [ b pmt ( e bp e tp) + b (e tp e p ) dt [ ( ) b pm ( e bp be p) e pt dt [ ( ) b pm ( e bp be p) () m [ ( ) + b () m p ( e bp be p) [ ( ) e bp e p + b p p ( e bp be p) ( e bp e p) ( ). p Using (4.6) gives the second result in (4.5) s stted. For the finl inequlity in (4.5) we need to determine θ (t) df (t) for f monotonic nondecresing. Now, from (4.) nd (4.3) θ (t) df (t) [mt bep e bp ept df (t) p () p [pmt + bep e bp e pt df (t), where we hve used the fct tht sgn (θ (t)) sgn (p). Integrtion by prts of the Riemnn-Stieltjes integrl gives (4.8) Now, nd θ (t) df (t) p { ( pmt + bep e bp e tp f (t) dt f (b) ( e pt m ) f (t) dt. m ) b e pt f (t) p e tp dt ebp e p f (b) () mf (b) p f (t) dt m () f () [ } m e pt f (t) dt so tht combining with (4.8) gives the inequlities for f monotonic nondecresing. Remrk 4.3. If f is probbility density function then M (f) nd f is non-negtive. b J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00

15 AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL 5 REFERENCES [ N.S. BARNETT AND S.S. DRAGOMIR, Some elementry inequlities for the expecttion nd vrince of rndom vrible whose pdf is defined on finite intervl, RGMIA Res. Rep. Coll., (7), Article. [ONLINE [ P. CERONE AND S.S. DRAGOMIR, On some inequlities rising from Montgomery s identity, J. Comput. Anl. Applics., (ccepted). [3 P. CERONE AND S.S. DRAGOMIR, On some inequlities for the expecttion nd vrince, Koren J. Comp. & Appl. Mth., 8() (000), [4 P. CERONE AND S.S. DRAGOMIR, Three point qudrture rules, involving, t most, first derivtive, RGMIA Res. Rep. Coll., (4), Article 8. [ONLINE (999). [5 S.S. DRAGOMIR, Some integrl inequlities of Grüss type, Indin J. of Pure nd Appl. Mth., (ccepted). [6 A.M. FINK, A tretise on Grüss inequlity, T.M. Rssis (Ed.), Kluwer Acdemic Publishers, (999). [7 D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers, Dordrecht, 993. [8 N.J. SONIN, O nekotoryh nervenstvh otnosjšcihsjk opredelennym integrlm, Zp. Imp. Akd. Nuk po Fiziko-mtem, Otd.t., 6 (898), 54. J. Inequl. Pure nd Appl. Mth., 3() Art. 4, 00