ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs

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1 ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin, Reeived: November, 005 Aepted: April, 006 Communited by: N.S. Brnett 000 AMS Sub. Clss.: 6D5. Key words: Abstrt: Ostrowski s inequlity, Probbility density funtion, Differene of integrl mens. A new inequlity is presented, whih is used to obtin omplement of reently obtined inequlity onerning the differene of two integrl mens. Some pplitions for pdfs re lso given. Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge of 4 Go Bk

2 Introdution 3 Some Inequlities 5 3 Applitions for PDFs Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge of 4 Go Bk

3 . Introdution In 938, Ostrowski proved the following inequlity [5]. Theorem.. Let f : [, b] R be ontinuous on [, b] nd differentible on (, b) with f (x) M for ll x (, b), then, (.) f (x) [ ( ) f (t) dt b x +b ] 4 + (b ) (b ) M, for ll x [, b]. The onstnt is the best possible. 4 In [3] N.S. Brnett, P. Cerone, S.S. Drgomir nd A.M. Fink obtined the following inequlity for the differene of two integrl mens: Theorem.. Let f : [, b] R be n bsolutely ontinuous mpping with the property tht f L [, b], then for < d b, (.) b f (t) dt d f (t) dt b d (b + d) f, the onstnt being the best possible. For = d = x this n be seen s generliztion of (.). In reent ppers [], [], [4], [6] some generliztions of inequlity (.) re given. Note tht estimtions of the differene of two integrl mens re obtined lso in the se where < b d (see [], []), while in the se where (, b) (, d) =, there is no orresponding result. In this pper we present new inequlity whih is used to obtin some estimtions for the differene of two integrl mens in the se where (, b) (, d) =, whih in Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge 3 of 4 Go Bk

4 limiting ses redues to omplement of Ostrowski s inequlity (.). Inequlities for pdfs (Probbility density funtions) relted to some results in [3, p ] re lso given. Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge 4 of 4 Go Bk

5 . Some Inequlities The key result of the present pper is the following inequlity: Theorem.. Let f, g be two ontinuously differentible funtions on [, b] nd twie differentible on (, b) with the properties tht, (.) g > 0 on (, b), nd tht the funtion f is bounded on (, b). For < d < b the g following estimtion holds, f f(b) f(d) (x) (.) inf x (,b) g (x) b d g(b) g(d) b d f() f() g() g() f (x) sup x (,b) g (x). Proof. Let s be ny number suh tht < s < d < b. Consider the mppings f, g : [d, b] R defined s: (.3) f (x) = f (x) f (s) (x s) f (s), g (x) = g (x) g (s) (x s) g (s). Clerly f, g re ontinuous on [d, b] nd differentible on (d, b). Further, for ny x [d, b], by pplying the men vlue Theorem, g (x) = g (x) g (s) = (x s) g (σ) for some σ (s, x), whih, ombined with (.), gives g (x) 0, for ll x (d, b). Hene, we n pply Cuhy s men vlue theorem to f, g on the intervl [d, b] to obtin, f (b) f (d) g (b) g (d) = f (τ) g (τ) Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge 5 of 4 Go Bk

6 for some τ (d, b) whih n further be written s, (.4) f (b) f (d) (b d) f (s) g (b) g (d) (b d) g (s) = f (τ) f (s) g (τ) g (s). Applying Cuhy s men vlue theorem to f, g on the intervl [s, τ], we hve tht for some ξ (s, τ) (, b), (.5) Combining (.4) nd (.5) we hve, f (τ) f (s) g (τ) g (s) = f (ξ) g (ξ). (.6) m f (b) f (d) (b d) f (s) g (b) g (d) (b d) g (s) M f for ll s (, ), where m = inf (x) x (,b) nd M = sup f (x) g (x) x (,b). g (x) By further pplition of the men vlue Theorem nd using the ssumption (.) we redily get, (.7) g (b) g (d) (b d) g (s) > 0. Multiplying (.6) by (.7), (.8) m (g (b) g (d) (b d) g (s)) f (b) f (d) (b d) f (s) M (g (b) g (d) (b d) g (s)). Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge 6 of 4 Go Bk Integrting the inequlities (.7) nd (.8) with respet to s from to we obtin respetively, (.9) ( ) (g (b) g (d)) (b d) (g () g ()) > 0

7 nd (.0) m (( ) (g (b) g (d)) (b d) (g () g ())) ( ) (f (b) f (d)) (b d) (f () f ()) M (( ) (g (b) g (d)) (b d) (g () g ())). Finlly, dividing (.0) by (.9), s required. m ( ) (f (b) f (d)) (b d) (f () f ()) ( ) (g (b) g (d)) (b d) (g () g ()) M Remrk. It is obvious tht Theorem. holds lso in the se where g < 0 on (, b). Corollry.. Let < d < b nd F, G be two ontinuous funtions on [, b] tht re differentible on (, b). If G > 0 on (, b) or G < 0 on (, b) nd F is G bounded (, b), then, F (x) (.) inf x (,b) G (x) nd (.) b d d b d F (t) dt G (t) dt d (b + d ) inf F (x) x (,b) b d F (t) dt sup G (t) dt x (,b) d F (t) dt F (x) G (x) (b + d ) sup F (x). x (,b) F (t) dt Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge 7 of 4 Go Bk The onstnt in (.) is the best possible.

8 Proof. If we pply Theorem. for the funtions, f (x) := x F (t) dt, g (x) := x G (t) dt, x [, b], then we immeditely obtin (.). Choosing G (x) = x in (.) we get (.). Remrk. Substituting d = b in (.) of Theorem. we get, (.3) b F (x) dx F (x) dx b b ( ) F. Setting d = in (.) of Corollry. we get, (.4) Now, b b inf F (x) x (,b) b F (x) dx = = b F (x) dx ( b ( b F (x) dx sup F (x). x (,b) F (x) dx F (x) dx ) F (x) dx F (x) dx + F (x) dx ) F (x) dx Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge 8 of 4 Go Bk

9 = ( b b = b ( b Using this in (.4) we derive the inequlity, inf F (x) x (,b) b F (x) dx F (x) dx b F (x) dx b ) F (x) dx ) F (x) dx. F (x) dx sup F (x). x (,b) From this we lerly get gin inequlity (.3). Consequently, inequlity (.) n be seen s omplement of (.). Corollry.3. Let F, G be two ontinuous funtions on n intervl I R nd differentible on the interior I of I with the properties G > 0 on I or G < 0 on I nd F bounded on I. Let, b be ny numbers in I suh tht < b, then for ll G x I (, b), tht is, x I but x / (, b), we hve the estimtion: F (t) (.5) inf t ({,b,x}) G (t) b b F (t) dt F (x) G (t) dt G (x) sup where ({, b, x}) := (min {, x, }, mx {x, b}). F (t) t ({,b,x}) G (t), Proof. Let u, w, y, z be ny numbers in I suh tht u < w y < z. Aording to Corollry. we then hve the inequlity, F (t) (.6) inf t (u,z) G (t) z z y y z z y y F (t) dt w u G (t) dt w u w u w u F (t) dt sup G (t) dt t (u,z) F (t) G (t). Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge 9 of 4 Go Bk

10 We distinguish two ses: If x <, then by hoosing y =, z = b nd u = w = x in (.) nd ssuming w w tht F (t) dt = F (x) nd G (t) dt = G (x) s limiting ses, (.6) w u u w u u redues to, F (t) inf t (x,b) G (t) b b F (t) dt F (x) sup G (t) dt G (x) t (x,b) F (t) G (t). Hene (.5) holds for ll x <. If x > b, then by hoosing u =, w = b nd y = z = x, in (.6), similrly to the bove, we n prove tht for ll x > b the inequlity (.5) holds. Corollry.4. Let F be ontinuous funtion on n intervl I R. If F L I, then for ll, b I with b > nd ll x I (, b) we hve: (.7) F (x) F (t) dt b + x b F, (min{,x},mx{b,x}). The inequlity (.7) is shrp. Proof. Applying (.5) for G (x) = x we redily get (.7). Choosing F (x) = x in (.7) we see tht the equlity holds, so the onstnt is the best possible. (.7) is now used to obtin n extension of Ostrowski s inequlity (.). Proposition.5. Let F be s in Corollry.3, then for ll, b I with b > nd Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge 0 of 4 Go Bk

11 for ll x I, (.8) F (x) b [ 4 + F (t) dt ] ( x +b (b ) ) (b ) F,(min{,x},mx{b,x}). Proof. Clerly, the restrition of inequlity (.8) on [, b] is Ostrowski s inequlity (.). Moreover, simple lultion yields [ ( ) b + x x +b ] 4 + (b ) (b ) for ll x R. Combining this ltter inequlity with (.7) we onlude tht (.8) holds lso for x I (, b) nd so (.8) is vlid for ll x I. Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge of 4 Go Bk

12 3. Applitions for PDFs We now use inequlity (.) in Theorem. to obtin improvements of some results in [3, p ]. Assume tht f : [, b] R + is probbility density funtion (pdf) of ertin rndom vrible X, tht is f (x) dx =, nd Pr (X x) = x f (t) dt, x [, b] is its umultive distribution funtion. Working similrly to [3, p ] we n stte the following: Proposition 3.. With the previous ssumptions for f, we hve tht for ll x [, b], (3.) (b x) (x ) inf f (x) x Pr (X x) x (,b) b (b x) (x ) sup f (x), x (,b) provided tht f C [, b] nd f is differentible nd bounded on (, b). Proof. Apply Theorem. for f (x) = Pr (X x), g (x) = x, = d = x. Proposition 3.. Let f be s bove, then, (3.) (x ) (3b x) inf f (x) x (,b) (x ) (b ) x Pr (X x) + E x (X) Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge of 4 Go Bk

13 for ll x [, b], where E x (X) := (x ) (3b x) sup f (x), x (,b) x t Pr (X t) dt, x [, b]. Proof. Integrting (3.) from to x nd using, in the resulting estimtion, the following identity, (3.3) x Pr (X x) dx = x Pr (X x) we esily get the desired result. Remrk 3. Setting x = b in (3.) we get, x = x Pr (X x) E x (X) (b )3 inf f (x) E (X) + b x (,b) x (Pr (X x)) dx (b )3 sup x (,b) f (x). Proposition 3.3. Let f, Pr (X x) be s bove. If f L [, b], then we hve, (b x) (x ) inf f (x) x x [,b] b (b E (X)) x Pr (X x) + E x (X) for ll x [, b]. (b x) (x ) sup f (x) x [,b] Proof. Apply Theorem. for f (x) := x Pr (X t) dt, g (x) := x, x [, b], nd identity (3.3). Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge 3 of 4 Go Bk

14 Referenes [] A. AGLIC ALJINOVIĆ, J. PEČARIĆ AND I. PERIĆ, Estimtes of the differene between two weighted integrl mens vi weighted Montgomery identity, Mth. Inequl. Appl., 7(3) (004), [] A. AGLIC ALJINOVIĆ, J. PEČARIĆ AND A. VUKELIĆ, The extension of Montgomery identity vi Fink identity with pplitions, J. Inequl. Appl., 005(), [3] N.S. BARNETT, P. CERONE, S.S. DRAGOMIR AND A. M. FINK, Compring two integrl mens for bsolutely ontinuous mppings whose derivtives re in L [, b] nd pplitions, Comput. Mth. Appl., 44(-) (00), 4 5. [4] P. CERONE AND S.S. DRAGOMIR, Differenes between mens with bounds from Riemnn-Stieltjes integrl, Comp. nd Mth. Appl., 46 (003), [5] A. OSTROWSKI, Uber die Absolutbweihung einer differenzierbren funktion von ihren integrlmittelwert, Comment. Mth. Helv., 0 (938), 6 7 (Germn). [6] J. PEČARIĆ, I. PERIĆ AND A. VUKELIĆ, Estimtions of the differene between two integrl mens vi Euler-type identities, Mth. Inequl. Appl., 7(3) (004), Inequlity A.I. Kehriniotis nd N.D. Assimkis vol. 8, iss., rt. 0, 007 Title Pge Pge 4 of 4 Go Bk