WEIGHTED INTEGRAL INEQUALITIES OF OSTROWSKI, 1 (b a) 2. f(t)g(t)dt. provided that there exists the real numbers m; M; n; N such that

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1 Preprints ( NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v WEIGHTED INTEGRAL INEQUALITIES OF OSTROWSKI, µcebyšev AND LUPAŞ TYPE WITH APPLICATIONS SILVESTRU SEVER DRAGOMIR Abstrct. In this pper we estblish some weighted integrl inequlities of Ostrowski, µcebyšev nd Lupş type. Applictions for continuous probbility density functions supported on in nite intervls with two exmples re lso given.. Introduction For two Lebesgue integrble functions f g [ b] R, consider the Cebyšev µ functionl (.) C (f g) = b In 935, Grüss [7] showed tht f(t)g(t)dt (b ) f(t)dt (.) jc (f g)j (M m) (N n) 4 provided tht there exists the rel numbers m M n N such tht g(t)dt (.3) m f (t) M nd n g (t) N for.e. t [ b] The constnt 4 is best possible in (.) in the sense tht it cnnot be replced by smller quntity. Another, however less known result, even though it ws obtined by µcebyšev in 88, [4], sttes tht (.4) jc (f g)j kf k k k (b ) provided tht f exist nd re continuous on [ b] nd kf k = sup t[b] jf (t)j The constnt cnnot be improved in the generl cse. The µcebyšev inequlity (.4) lso holds if f g [ b] R re ssumed to be bsolutely continuous nd f L [ b] while kf k = essup t[b] jf (t)j A mixture between Grüss result (.) nd µcebyšev s one (.4) is the following inequlity obtined by Ostrowski in 97, [4] (.5) jc (f g)j 8 (b ) (M m) kg k provided tht f is Lebesgue integrble nd stis es (.3) while g is bsolutely continuous nd L [ b] The constnt 8 is best possible in (.5). 99 Mthemtics Subject Clssi ction. 6D5 6D. Key words nd phrses. Ostrowski s inequlity, µcebyšev inequlity, Lupş inequlity, Weighted integrls, Probbility density functions, Cumultive probbility function. 8 by the uthor(s). Distributed under Cretive Commons CC BY license.

2 Preprints ( NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v S. S. DRAGOMIR The cse of eucliden norms of the derivtive ws considered by A. Lupş in [] in which he proved tht (.6) jc (f g)j kf k k k (b ) provided tht f g re bsolutely continuous nd f L [ b] The constnt is the best possible. Consider now the weighted µ Cebyšev functionl Z b (.7) C w (f g) = w (t) dt w (t) f (t) g (t) dt Z b Z b w (t) dt w (t) f (t) dt w (t) dt w (t) g (t) dt where f g w [ b] R nd w (t) for.e. t [ b] re mesurble functions such tht the involved integrls exist nd w (t) dt > In [6], Cerone nd Drgomir obtined, mong others, the following inequlities (.8) jc w (f g)j Z (M m) b Z w (t) dt w (t) g (t) b w (s) ds w (s) g (s) ds dt " Z (M m) b Z p # w (t) dt w (t) g (t) b p w (s) ds w (s) g (s) ds dt Z (M m) essup g (t) b w (s) ds w (s) g (s) ds t[b] for p > provided < m f (t) M < for.e. t [ b] nd the corresponding integrls re nite. The constnt is shrp in ll the inequlities in (.8) in the sense tht it cnnot be replced by smller constnt. In ddition, if < n g (t) N < for.e. t [ b] then the following re nement of the celebrted Grüss inequlity is obtined (.9) jc w (f g)j Z (M m) b Z w (t) dt w (t) g (t) b w (s) ds w (s) g (s) ds dt Z (M m) 4 b Z 3 R b w (t) dt w (t) g (t) b w (s) ds w (s) g (s) ds dt5 (M m) (N n) 4 Here, the constnts nd 4 re lso shrp in the sense mentioned bove. For other inequlity of Grüss type see []-[5], [7]-[6], [8]-[3] nd [5]-[8]. Motivted by the bove results, in this pper we estblish some weighted integrl inequlities of Ostrowski, µcebyšev nd Lupş type. Applictions for continuous

3 Preprints ( NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v INEQUALITIES OF OSTROWSKI, µcebyšev AND LUPAS TYPE 3 probbility density functions supported on in nite intervls with two exmples re lso given. We cn de ne, s bove (.) C h (f g) = h (b) h (). Weighted Inequlities h (b) h () f (t) g (t) h (t) dt f (t) h (t) dt h (b) h () g (t) h (t) dt where h is bsolutely continuous nd f g re Lebesgue mesurble on [ b] nd such tht the bove integrls exist. The following weighted version of Ostrowski s inequlity holds Theorem. Let h [ b] [h () h (b)] be continuous strictly incresing function tht is di erentible on ( b) If f is Lebesgue integrble nd stis es the condition m f (t) M for t [ b] nd g [ b] R is bsolutely continuous on [ b] nd g h is essentilly bounded, nmely g h L [ b] then we hve (.) jc h (f g)j 8 [h (b) h ()] (M m) [b] The constnt 8 Proof. Assume tht [c d] [ b] If g [c d] C is bsolutely continuous on [c d] then g h [h (c) h (d)] C is bsolutely continuous on [h (c) h (d)] nd using the chin rule nd the derivtive of inverse functions we hve (.3) g h (z) = h (z) h (z) = h (z) (h h ) (z) for lmost every (.e.) z [h (c) h (d)] If x [c d] then by tking z = h (x) we get g h (z) = h h (h (x)) (h h ) (h (x)) = g (x) h (x) Therefore, since g h L [c d], hence g h L [h (c) h (d)] Also g h [h(c)h(d)] = [cd] Now, if we use the Ostrowski s inequlity (.5) for the functions f h nd g h on the intervl [h () h (b)] then we get Z h(b) (.4) f h (u)g h (u)du h (b) h () h() Z h(b) [h (b) h ()] f h (u)du g h (u)du h() h h() 8 [h (b) h ()] (M m) g h [h()h(b)]

4 Preprints ( NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v 4 S. S. DRAGOMIR since m f h (u) M for ll u [h () h (b)] Observe lso tht, by the chnge of vrible t = h (u) u [g () g (b)] we hve u = h (t) tht gives du = h (t) dt nd nd h() h() h() f h (u) du = g h (u)du = f h (u)g h (u)du = g h [h()h(b)] = f (t) h (t) dt g (t) h (t) dt f (t) g (t) h (t) dt h [b] By mking use of (.4) we then get the desired result (.). The best constnt follows by Ostrowski s inequlity (.5). If w [ b] R is continuous nd positive on the intervl [ b] then the function W [ b] [ ) W (x) = R x w (s) ds is strictly incresing nd di erentible on ( b) We hve W (x) = w (x) for ny x ( b) Corollry. Assume tht w [ b] ( ) is continuous on [ b] f is Lebesgue integrble nd stis es the condition m f (t) M for t [ b] nd g [ b] R is bsolutely continuous on [ b] with g g w is essentilly bounded, nmely w L [ b] then we hve (.5) jc w (f g)j 8 (M m) w w (s) ds [b] The constnt 8 Remrk. Under the ssumptions of Corollry nd if there exists constnt K > such tht j (t)j Kw (t) for.e. t [ b] then by (.5) we get (.6) jc w (f g)j 8 (M m) K w (s) ds We hve the following weighted version of µcebyšev inequlity Theorem. Let h [ b] [h () h (b)] be continuous strictly incresing function tht is di erentible on ( b) If f, g [ b] R re bsolutely continuous on [ b] nd f h g h L [ b] then we hve (.7) jc h (f g)j [h (b) h f ()] [b] [b] The constnt The proof follows by the use of µcebyšev inequlity (.4) for the functions f h nd g h on the intervl [h () h (b)] h h

5 Preprints ( NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v INEQUALITIES OF OSTROWSKI, µcebyšev AND LUPAS TYPE 5 Corollry. Assume tht w [ b] ( ) is continuous on [ b] If f, g [ b] R re bsolutely continuous on [ b] nd f w g w L [ b] then we hve (.8) jc w (f g)j f w [b] w The constnt [b] w (s) ds Remrk. Under the ssumptions of Corollry nd if there exists the constnts K L > such tht jf (t)j Lw (t), j (t)j Kw (t) for.e. t [ b] then by (.8) we get (.9) jc w (f g)j LK w (s) ds We lso hve the following version of Lupş inequlity Theorem 3. Let h [ b] [h () h (b)] be continuous strictly incresing function tht is di erentible on ( b) If f, g [ b] R re bsolutely continuous on f [ b] nd g L (h ) = (h ) = [ b] then we hve (.) jc h (f g)j f (h ) = The constnt [b] Proof. Using the identity (.3) bove, we hve h() (h ) = [h (b) h ()] [b] g h Z h(b) h (u) (u) du = (h h du ) (u) By the chnge of vrible t = h (u) u [h () h (b)] we hve u = h (t) tht gives du = h (t) dt Therefore h() h (u) (h h du = ) (u) In similr wy, we lso hve h() = b b h() (t) h (t) h (t) dt (t) [h (t)] = dt = (h ) = f h (u) f (h h du = ) (u) (h ) = [b] [b]

6 Preprints ( NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v 6 S. S. DRAGOMIR By mking use of Lupş inequlity (.6) for the functions f h nd g h on the intervl [h () h (b)] we get Z h(b) f h (u)g h (u)du h (b) h () h() [h (b) h ()] h() f h (u)du h() g h (u)du f h [h()h(b)] g h [h()h(b)] [h (b) h ()] which together with the bove clcultions produces the desired result (.). Corollry 3. Assume tht w [ b] ( ) is continuous on [ b] If f, g f [ b] R re bsolutely continuous on [ b] nd g L w = w = [ b] then we hve (.) jc w (f g)j f w (s) ds [b] [b] The constnt w = w = We cn give some exmples of interest for severl function h [ b] [h () h (b)] tht re continuous strictly incresing functions nd di erentible on ( b) ). If we tke h [ b] ( ) R, h (t) = ln t in (.), then we get for ` (t) = t tht (.) jc` (f g)j b 8 (M m) k`g k [b] ln where (.3) C` (f g) = ln b f (t) g (t) dt t Z b ln b f (t) t dt ln b g (t) dt t nd provided tht f is Lebesgue integrble nd stis es the condition m f (t) M for t [ b] nd g [ b] R is bsolutely continuous on [ b] nd ` L [ b] If f, g [ b] R re bsolutely continuous on [ b] nd `f ` L [ b] then by (.7) we hve (.4) jc` (f g)j k`f k [b] k` k [b] ln b Also, if f, g [ b] R re bsolutely continuous on [ b] nd `= f `= L [ b] then we hve by (.) (.5) jc` (f g)j `= f [b] `= [b] b ln b). If we tke h [ b] R ( ), h (t) = exp t in (.), then we get (.6) jc exp (f g)j 8 (M m) exp (exp b exp ) [b]

7 Preprints ( NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v INEQUALITIES OF OSTROWSKI, µcebyšev AND LUPAS TYPE 7 where (.7) C exp (f g) = exp b exp exp b exp f (t) g (t) exp tdt f (t) exp tdt exp b exp g (t) exp tdt nd provided tht f is Lebesgue integrble nd stis es the condition m f (t) M for t [ b] nd g [ b] R is bsolutely continuous on [ b] nd g exp L [ b] If f, g [ b] R re bsolutely continuous on [ b] nd f exp exp L [ b] then by (.7) we hve (.8) jc exp (f g)j f exp [b] exp (exp b exp ) [b] f exp = Also, if f, g [ b] R re bsolutely continuous on [ b] nd L [ b] then we hve by (.) tht (.9) jc exp (f g)j f (exp b exp ) [b] [b] exp = exp = c). If we tke h [ b] ( ) R, h (t) = t r r > in (.), then we get (.) jc r`r (f g)j 8r (br r ) (M m) ` r [b] where (.) C r`r (f g) = b r r r f (t) g (t) t r b r r r dt f (t) t r dt b r r r g (t) t r exp = nd provided tht f is Lebesgue integrble nd stis es the condition m f (t) M for t [ b] nd g [ b] R is bsolutely continuous on [ b] nd ` r L [ b] If f, g [ b] R re bsolutely continuous on [ b] nd ` r f ` r L [ b] then by (.7) we hve (.) jc r`r (f g)j ` r f ` [b] r [b] (b r r ) Also, if f, g [ b] R re bsolutely continuous on [ b] nd ` r f ` r L [ b] then we hve by (.) tht (.3) jc r`r (f g)j r ` f r ` g (b r r ) [b] [b] dt 3. Applictions for Probbility Density Functions The bove result cn be extended for in nite intervls I by ssuming tht the function f I C is loclly bsolutely continuous on I. For instnce, if I = [ ), w (s) > for s [ ) with R w (s) ds = nmely w is probbility density function on [ ), f is Lebesgue mesurble nd stis es

8 Preprints ( NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v 8 S. S. DRAGOMIR the condition m f (t) M for t [ ) nd g [ ) R is loclly bsolutely continuous on [ ) with g w L [ ) then by considering the functionl C w (f g) = Z w (t) f (t) g (t) dt Z w (t) f (t) dt we hve from (.) tht (3.) jc w (f g)j 8 (M m) w [) Moreover, if f w L [ ) then lso by (.7) (3.) jc w (f g)j f w [) w [) f w = If L w = [ ) then we hve by (.) (3.3) jc w (f g)j f [) w = w = Z [) w (t) g (t) dt In probbility theory nd sttistics, the bet prime distribution (lso known s inverted bet distribution or bet distribution of the second kind) is n bsolutely continuous probbility distribution de ned for x > with two prmeters nd, hving the probbility density function where B is Bet function B ( ) = w (x) = x ( + x) B ( ) Z The cumultive distribution function is t ( t) > W (x) = I x ( ) +x where I is the regulrized incomplete bet function de ned by I z ( ) = B (z ) B ( ) Here B ( ) is the incomplete bet function de ned by B (z ) = Consider the functionl C B (f g) = B ( ) Z Z Z z t ( t) z > t ( + t) f (t) g (t) dt t ( + t) f (t) dt Z t ( + t) g (t) dt where > Therefore, by (3.)-(3.3) we hve for ` (t) = t tht (3.4) jc B (f g)j 8 (M m) B3 ( ) ` ( + `) + [)

9 Preprints ( NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v INEQUALITIES OF OSTROWSKI, µcebyšev AND LUPAS TYPE 9 provided m f (t) M for t [ ) nd ` ( + `) + L [ ) (3.5) jc B (f g)j B4 ( ) f ` ( + `) + g` [) ( + `) + [) provided f ` ( + `) + ` ( + `) + L [ ) nd (3.6) jc B (f g)j B3 ( ) f ` ( + `) + g` ( + `) + [) [) provided f ` ( + `) + ` ( + `) + L [ ) Similr results my be stted for the probbility distributions tht re supported on the whole xis R = ( ). Nmely, if I = ( ), f R C is loclly bsolutely continuous on R nd w (s) > for s R with R w (s) ds = nmely w is probbility density function on ( ), f is Lebesgue mesurble nd stis es the condition m f (t) M for t ( ) nd g ( ) R is loclly bsolutely continuous on ( ) with g w L ( ) then, by considering the functionl Z Z Z C w (f g) = w (t) f (t) g (t) dt w (t) f (t) dt w (t) g (t) dt we hve (3.7) jc w (f g)j 8 (M m) w ( ) Moreover, if f w L ( (3.8) jc w (f g)j f w = ) then lso f w ( ) w ( ) If L w = ( ) then we hve (3.9) jc w (f g)j f ( ) w = w = ( ) In wht follows we give n exmple. The probbility density of the norml distribution on ( ) is w (x) = p (x ) exp x R, where is the men or expecttion of the distribution (nd lso its medin nd mode), is the stndrd devition, nd is the vrince. The cumultive distribution function is W (x) = + x erf p where the error function erf is de ned by erf (x) = p Z x exp t dt

10 Preprints ( NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v S. S. DRAGOMIR Consider the functionl C N (f g) = p Z (t ) exp f (t) g (t) dt Z (t ) Z exp f (t) dt exp with the prmeters nd s bove. Therefore, by (3.7)-(3.9) we hve (3.) jc N (f g)j 8 (M m) p 3 exp provided m f (t) M for t ( Moreover, if f exp (` ) (3.) jc N (f g)j p 4 f exp If f exp (` ) (t ) g (t) dt (` ) ( ) ) nd exp (` ) L ( ) L ( ) then lso (` ) ( ) g exp exp (` ) L ( ) then we hve (` ) ( ) (3.) jc N (f g)j p 3 f exp (` ) ( ) References g exp (` ) ( ) [] M. W. Alomri, A compnion of Grüss type inequlity for Riemnn-Stieltjes integrl nd pplictions. Mt. Vesnik 66 (4), no.,. [] D. Andric nd C. Bde, Grüss inequlity for positive liner functionls. Period. Mth. Hungr. 9 (988), no., [3] D. Blenu, S. D. Purohit nd F. Uçr, On Grüss type integrl inequlity involving the Sigo s frctionl integrl opertors. J. Comput. Anl. Appl. 9 (5), no. 3, [4] P. L. Chebyshev, Sur les expressions pproximtives des intègrls dè nis pr les outres prises entre les même limites, Proc. Mth. Soc. Chrkov, (88), [5] P. Cerone, On µcebyšev-type functionl nd Grüss-like bounds. Mth. Inequl. Appl. 9 (6), no., 87. [6] P. Cerone nd S. S. Drgomir, A re nement of the Grüss inequlity nd pplictions, Tmkng J. Mth., 38() (7), Preprint RGMIA Res. Rep. Coll., 5() (), Article 4. [ONLINE http//rgmi.vu.edu.u/v5n.html]. [7] P. Cerone nd S. S. Drgomir, Some new Ostrowski-type bounds for the µcebyšev functionl nd pplictions. J. Mth. Inequl. 8 (4), no., [8] P. Cerone, S. S. Drgomir nd J. Roumeliotis, Grüss inequlity in terms of -seminorms nd pplictions. Integrl Trnsforms Spec. Funct. 4 (3), no. 3, 5 6. [9] S. S. Drgomir, A generliztion of Grüss s inequlity in inner product spces nd pplictions. J. Mth. Anl. Appl. 37 (999), no., [] S. S. Drgomir, A Grüss type integrl inequlity for mppings of r-hölder s type nd pplictions for trpezoid formul. Tmkng J. Mth. 3 (), no., [] S. S. Drgomir, Some integrl inequlities of Grüss type. Indin J. Pure Appl. Mth. 3 (), no. 4,

11 Preprints ( NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v INEQUALITIES OF OSTROWSKI, µcebyšev AND LUPAS TYPE [] S. S. Drgomir, Integrl Grüss inequlity for mppings with vlues in Hilbert spces nd pplictions. J. Koren Mth. Soc. 38 (), no. 6, [3] S. S. Drgomir, A Grüss relted integrl inequlity nd pplictions. Nonliner Anl. Forum 8 (3), no., [4] S. S. Drgomir nd I. A. Fedotov, An inequlity of Grüss type for Riemnn-Stieltjes integrl nd pplictions for specil mens. Tmkng J. Mth. 9 (998), no. 4, [5] S. S. Drgomir nd I. Gomm, Some integrl nd discrete versions of the Grüss inequlity for rel nd complex functions nd sequences. Tmsui Oxf. J. Mth. Sci. 9 (3), no., [6] A. M. Fink, A tretise on Grüss inequlity. Anlytic nd Geometric Inequlities nd Applictions, 93 3, Mth. Appl., 478, Kluwer Acd. Publ., Dordrecht, 999. R [7] G. Grüss, Über ds Mximum des bsoluten Betrges von b b R f(x)g(x)dx b (b ) f(x)dx g(x)dx, Mth. Z., 39(935), 5-6. [8] D. Jnkov Mširević nd T. K. Pogány, Bounds on µcebyšev functionl for C '[ ] function clss. J. Anl. (4), 7 7. [9] Z. Liu, Re nement of n inequlity of Grüss type for Riemnn-Stieltjes integrl. Soochow J. Mth. 3 (4), no. 4, [] Z. Liu, Notes on Grüss type inequlity nd its ppliction. Vietnm J. Mth. 35 (7), no., 7. [] A. Lupş, The best constnt in n integrl inequlity, Mthemtic (Cluj, Romni), 5(38)() (973), 9-. [] A. Mc.D. Mercer nd P. R. Mercer, New proofs of the Grüss inequlity. Aust. J. Mth. Anl. Appl. (4), no., Art., 6 pp. [3] N. Minculete nd L. Ciurdriu, A generlized form of Grüss type inequlity nd other integrl inequlities. J. Inequl. Appl. 4, 49, 8 pp. [4] A. M. Ostrowski, On n integrl inequlity, Aequt. Mth., 4 (97), [5] B. G. Pchptte, A note on some inequlities nlogous to Grüss inequlity. Octogon Mth. Mg. 5 (997), no., 6 66 [6] J. Peµcrić nd Š. Ungr, On inequlity of Grüss type. Mth. Commun. (6), no., [7] M. Z. Sriky nd H. Budk, An inequlity of Grüss like vi vrint of Pompeiu s men vlue theorem. Konurlp J. Mth. 3 (5), no., [8] N. Ujević, A generliztion of the pre-grüss inequlity nd pplictions to some qudrture formule. J. Inequl. Pure Appl. Mth. 3 (), no., Article 3, 9 pp. Mthemtics, College of Engineering & Science, Victori University, PO Box 448, Melbourne City, MC 8, Austrli. E-mil ddress sever.drgomir@vu.edu.u URL http//rgmi.org/drgomir DST-NRF Centre of Excellence in the Mthemticl, nd Sttisticl Sciences, School of Computer Science, & Applied Mthemtics, University of the Witwtersrnd,, Privte Bg 3, Johnnesbur5, South Afric