Journal of Inequalities in Pure and Applied Mathematics
|
|
- Kathleen McGee
- 6 years ago
- Views:
Transcription
1 Journl of Inequlities in Pure nd Applied Mthemtics SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARI- ABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NEIL S. BARNETT, PIETRO CERONE, SEVER S. DRAGOMIR AND JOHN ROUMELIOTIS School of Communictions nd Informtics Victori University of Technology PO Box 1448, Melbourne City MC 8001 Victori, Austrli EMil: neil@mtild.vu.edu.u EMil: pc@mtild.vu.edu.u EMil: sever@mtild.vu.edu.u EMil: johnr@mtild.vu.edu.u volume, issue 1, rticle 1, 001. Received 7 Jnury, 000; ccepted 16 June 000. Communicted by: C.E.M. Perce Abstrct Home Pge Go Bck c 000 School of Communictions nd Informtics,Victori University of Technology ISSN (electronic):
2 Abstrct Some inequlities for the dispersion of rndom vrible whose pdf is defined on finite intervl nd pplictions re given. 000 Mthemtics Subject Clssifiction: 60E15, 6D15 Key words: Rndom vrible, Expecttion, Vrince, Dispersion, Grüss Inequlity, Chebychev s Inequlity, Lupş Inequlity. 1 Introduction Some Inequlities for Dispersion Perturbed Results Using Grüss Type inequlities Perturbed Results Using Premture Inequlities Alternte Grüss Type Results for Inequlities Involving the Vrince Some Inequlities for Absolutely Continuous P.D.F s References Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
3 1. Introduction In this note we obtin some inequlities for the dispersion of continuous rndom vrible X hving the probbility density function (p.d.f.) f defined on finite intervl [, b]. Tools used include: Korkine s identity, which plys centrl role in the proof of Chebychev s integrl inequlity for synchronous mppings [4], Hölder s weighted inequlity for double integrls nd n integrl identity connecting the vrince σ (X) nd the expecttion E (X). Perturbed results re lso obtined by using Grüss, Chebyshev nd Lupş inequlities. In Section 4, results from n identity involving double integrl re obtined for vriety of norms. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 3 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
4 . Some Inequlities for Dispersion Let f : [, b] R R + be the p.d.f. of the rndom vrible X nd E (X) : its expecttion nd [ ] 1 σ (X) (t E (X)) f (t) dt its dispersion or stndrd devition. The following theorem holds. tf (t) dt [ Theorem.1. With the bove ssumptions, we hve (.1) 0 σ (X) 3(b ) t f (t) dt [E (X)] ] 1 6 f, provided f L, [, b] ; (b ) 1+ 1 q [(q+1)(q+1)] q (b ). Proof. Korkine s identity [4], is (.) p (t) dt 1 p (t) g (t) h (t) dt f p, provided f L p [, b] nd p > 1, 1 p + 1 q 1; p (t) g (t) dt p (t) h (t) dt p (t) p (s) (g (t) g (s)) (h (t) h (s)) dtds, Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 4 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
5 which holds for the mesurble mppings p, g, h : [, b] R for which the integrls involved in (.) exist nd re finite. Choose in (.) p (t) f (t), g (t) h (t) t E (X), t [, b] to get (.3) σ (X) 1 It is obvious tht (.4) f (t) f (s) (t s) dtds f (t) f (s) (t s) dtds. sup f (t) f (s) (t,s) [,b] (t s) dtds (b )4 f 6 nd then, by (.3), we obtin the first prt of (.1). For the second prt, we pply Hölder s integrl inequlity for double integrls to obtin f (t) f (s) (t s) dtds ( f p ) 1 ( f p (t) f p p b (s) dtds [ ] (b ) q+ 1 q, (q + 1) (q + 1) ) 1 (t s) q q dtds Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 5 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
6 where p > 1 nd , nd the second inequlity in (.1) is proved. p q For the lst prt, observe tht f (t) f (s) (t s) dtds sup (t,s) [,b] (t s) (b ) f (t) f (s) dtds s f (t) f (s) dtds f (t) dt f (s) ds 1. Using finer rgument, the lst inequlity in (.1) cn be improved s follows. Theorem.. Under the bove ssumptions, we hve (.5) 0 σ (X) 1 (b ). Proof. We use the following Grüss type inequlity: (.6) 0 p (t) ( ) g (t) dt p (t) g (t) dt 1 p (t) dt p (t) dt 4 (M m), provided tht p, g re mesurble on [, b] nd ll the integrls in (.6) exist nd re finite, p (t) dt > 0 nd m g M.e. on [, b]. For proof of this inequlity see [19]. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 6 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
7 Choose in (.6), p (t) f (t), g (t) t E (X), t [, b]. Observe tht in this cse m E (X), M b E (X) nd then, by (.6) we deduce (.5). Remrk.1. The sme conclusion cn be obtined for the choice p (t) f (t) nd g (t) t, t [, b]. The following result holds. Theorem.3. Let X be rndom vrible hving the p.d.f. given by f : [, b] R R +. Then for ny x [, b] we hve the inequlity: (.7) σ (X) + (x E (X)) [ (b ) (b ) + ( ) ] x +b f 1, provided f L [, b] ; [ (b x) q+1 +(x ) q+1 ( b q+1 + x +b Proof. We observe tht (.8) ). (x t) f (t) dt ] 1 q f p, provided f L p [, b], p > 1, nd 1 p + 1 q 1; ( x xt + t ) f (t) dt x xe (X) + t f (t) dt Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 7 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
8 nd s (.9) σ (X) we get, by (.8) nd (.9), t f (t) dt [E (X)], (.10) [x E (X)] + σ (X) which is of interest in itself too. We observe tht (x t) f (t) dt ess sup f (t) t [,b] (x t) f (t) dt, (x t) dt f (b x) 3 + (x ) 3 (b ) f [ 3 (b ) 1 + ( x + b ) ] nd the first inequlity in (.7) is proved. For the second inequlity, observe tht by Hölder s integrl inequlity, ( ) 1 ( (x t) f (t) dt f p p b ) 1 (t) dt (x t) q q dt [ ] (b x) q+1 + (x ) q+1 1 q f p, q + 1 Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 8 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
9 nd the second inequlity in (.7) is estblished. Finlly, observe tht, nd the theorem is proved. (x t) f (t) dt sup (x t) f (t) dt t [,b] mx { (x ), (b x) } The following corollries re esily deduced. (mx {x, b x}) ( b + x + b ), Corollry.4. With the bove ssumptions, we hve (.11) 0 σ (X) (b ) 1 [ (b ) + ( E (X) +b 1 [ (b E(X)) q+1 +(E(X) ) q+1 q+1 ) ] 1 f 1, provided f L [, b] ; ] 1 q f 1 p, if f L p [, b], p > 1 nd 1 p + 1 q 1; + E (X) +b. b Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 9 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
10 Remrk.. The lst inequlity in (.1) is worse thn the inequlity (.5), obtined by technique bsed on Grüss inequlity. The best inequlity we cn get from (.7) is tht one for which x +b, nd this pplies for ll the bounds since nd min x [,b] [ (b ) 1 + ( x + b ) ] (b x) q+1 + (x ) q+1 min x [,b] q + 1 [ b min + x [,b] x + b ] (b ) 1 (b )q+1 q (q + 1), b. Consequently, we cn stte the following corollry s well. Corollry.5. With the bove ssumptions, we hve the inequlity: (.1) 0 σ (X) + [ E (X) + b ] (b ) 3 1 f, provided f L [, b] ; (b ) q+1 4(q+1) 1 q (b ) 4. f p, if f L p [, b], p > 1, nd 1 p + 1 q 1;, Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 10 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
11 Remrk.3. From the lst inequlity in (.1), we obtin (.13) 0 σ (X) (b E (X)) (E (X) ) 1 4 (b ), which is n improvement on (.5). Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 11 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
12 3. Perturbed Results Using Grüss Type inequlities In 1935, G. Grüss (see for exmple [6]) proved the following integrl inequlity which gives n pproximtion for the integrl of product in terms of the product of the integrls. Theorem 3.1. Let h, g : [, b] R be two integrble mppings such tht φ h (x) Φ nd γ g (x) Γ for ll x [, b], where φ, Φ, γ, Γ re rel numbers. Then, (3.1) T (h, g) 1 (Φ φ) (Γ γ), 4 where (3.) T (h, g) 1 b 1 b h (x) g (x) dx h (x) dx 1 b g (x) dx nd the inequlity is shrp, in the sense tht the constnt 1 cnnot be replced 4 by smller one. For simple proof of this s well s for extensions, generlistions, discrete vrints nd other ssocited mteril, see [5], nd [1]-[1] where further references re given. A premture Grüss inequlity is embodied in the following theorem which ws proved in [3]. It provides shrper bound thn the bove Grüss inequlity. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 1 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
13 Theorem 3.. Let h, g be integrble functions defined on [, b] nd let d g (t) D. Then (3.3) T (h, g) D d where T (h, g) is s defined in (3.). T (h, h) 1, Theorem 3. will now be used to provide perturbed rule involving the vrince nd men of p.d.f Perturbed Results Using Premture Inequlities In this subsection we develop some perturbed results. Theorem 3.3. Let X be rndom vrible hving the p.d.f. given by f : [, b] R R +. Then for ny x [, b] nd m f (x) M we hve the inequlity (3.4) P V (x) : σ (X) + (x E (X)) M m (M m) (b ) 45 (b )3 45. [ (b (b ) 1 ( x + b ) ) ( + 15 x + b ) ] 1 Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 13 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
14 Proof. Applying the premture Grüss result (3.3) by ssociting g (t) with f (t) nd h (t) (x t), gives, from (3.1)-(3.3) (3.5) where from (3.) (x t) f (t) dt 1 b (3.6) T (h, h) 1 b Now, (3.7) nd 1 b 1 b (x t) 4 dt (x t) dt f (t) dt (b ) M m [ 1 b [T (h, h)] 1, (x t) dt]. (x t) dt (x )3 + (b x) 3 3 (b ) 1 ( ) ( b + x + b 3 (x t) 4 dt (x )5 + (b x) 5 5 (b ) giving, for (3.6), [ ] [ ] (x ) 5 + (b x) 5 (x ) 3 + (b x) 3 (3.8) 45T (h, h) 9 5. b b ) Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 14 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
15 Let A x nd B b x in (3.8) to give ( ) ( A 5 + B 5 A 3 + B 3 45T (h, h) 9 5 A + B A + B 9 [ A 4 A 3 B + A B AB 3 + B 4] 5 [ A AB + B ] ) ( 4A 7AB + 4B ) (A + B) [ (A ) ( ) ] + B A B + 15 (A + B). Using the fcts tht A + B b nd A B x ( + b) gives [ (b ) ( (b ) (3.9) T (h, h) + 15 x + b ) ] 45 nd from (3.7) giving (3.10) 1 b 1 b (x t) dt A3 + B 3 3 (A + B) 1 [ A AB + B ] 3 [ (A 1 ) ( ) ] + B A B + 3, 3 (x t) dt ( (b ) + x + b ). 1 Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 15 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
16 Hence, from (3.5), (3.9) (3.10) nd (.10), the first inequlity in (3.4) results. The corsest uniform bound is obtined by tking x t either end point. Thus the theorem is completely proved. Remrk 3.1. The best inequlity obtinble from (3.4) is t x +b giving [ (3.11) σ (X) + E (X) + b ] (b ) 1 M m (b ) The result (3.11) is tighter bound thn tht obtined in the first inequlity of (.1) since 0 < M m < f. For symmetric p.d.f. E (X) +b nd so the bove results would give bounds on the vrince. The following results hold if the p.d.f f (x) is differentible, tht is, for f (x) bsolutely continuous. Theorem 3.4. Let the conditions on Theorem 3.1 be stisfied. Further, suppose tht f is differentible nd is such tht Then f : sup f (t) <. t [,b] (3.1) P V (x) b 1 f I (x), where P V (x) is given by the left hnd side of (3.4) nd, [ (b ) ( (b ) (3.13) I (x) + 15 x + b ) ] Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 16 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
17 Proof. Let h, g : [, b] R be bsolutely continuous nd h, g be bounded. Then Chebychev s inequlity holds (see [3]) T (h, g) (b ) 1 sup h (t) sup g (t). t [,b] t [,b] Mtić, Pečrić nd Ujević [3] using premture Grüss type rgument proved tht (3.14) T (h, g) (b ) 1 sup g (t) T (h, h). t [,b] Associting f ( ) with g ( ) nd (x ) with h ( ) in (3.13) gives, from (3.5) nd (3.9), I (x) (b ) [T (h, h)] 1, which simplifies to (3.13) nd the theorem is proved. Theorem 3.5. Let the conditions of Theorem 3.3 be stisfied. Further, suppose tht f is loclly bsolutely continuous on (, b) nd let f L (, b). Then (3.15) P V (x) b π f I (x), where P V (x) is the left hnd side of (3.4) nd I (x) is s given in (3.13). Proof. The following result ws obtined by Lupş (see [3]). For h, g : (, b) R loclly bsolutely continuous on (, b) nd h, g L (, b), then T (h, g) (b ) π h g, Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 17 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
18 where k : ( 1 b k (t) ) 1 Mtić, Pečrić nd Ujević [3] further show tht for k L (, b). (3.16) T (h, g) b π g T (h, h). Associting f ( ) with g ( ) nd (x ) with h in (3.16) gives (3.15), where I (x) is s found in (3.13), since from (3.5) nd (3.9), I (x) (b ) [T (h, h)] Alternte Grüss Type Results for Inequlities Involving the Vrince Let (3.17) S (h (x)) h (x) M (h) where (3.18) M (h) 1 b Then from (3.), h (u) du. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 18 of 41 (3.19) T (h, g) M (hg) M (h) M (g). J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
19 Drgomir nd McAndrew [19] hve shown, tht (3.0) T (h, g) T (S (h), S (g)) nd proceeded to obtin bounds for trpezoidl rule. Identity (3.0) is now pplied to obtin bounds for the vrince. Theorem 3.6. Let X be rndom vrible hving the p.d.f. f : [, b] R R +. Then for ny x [, b] the following inequlity holds, nmely, (3.1) P V (x) 8 3 ν3 (x) f ( ) 1 b if f L [, b], where P V (x) is s defined by the left hnd side of (3.4), nd ν ν (x) ( 1 b ) ( ) 3 + x +b. Proof. Using identity (3.0), ssocite with h ( ), (x ) nd f ( ) with g ( ). Then (3.) where from (3.18), (x t) f (t) dt M ( (x ) ) M ( (x ) ) 1 b [ (x t) M ( (x ) )] [ f (t) 1 ] dt, b (x t) dt 1 [ (x ) 3 + (b x) 3] 3 (b ) Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 19 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
20 nd so (3.3) 3M ( (x ) ) ( ) ( b + 3 x + b ). Further, from (3.17), nd so, on using (3.3) S ( (x ) ) (x t) M ( (x ) ) (3.4) S ( (x ) ) (x t) 1 3 ( b ) ( x + b ). Now, from (3.) nd using (.10), (3.3) nd (3.4), the following identity is obtined [ (b (3.5) σ (X) + [x E (X)] 1 ) ( + 3 x + b ) ] 3 S ( (x t) ) ( f (t) 1 b where S ( ) is s given by (3.4). Tking the modulus of (3.5) gives (3.6) P V (x) S ( (x t) ) ( f (t) 1 ) dt b. ) dt, Observe tht under different ssumptions with regrd to the norms of the p.d.f. f (x) we my obtin vriety of bounds. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 0 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
21 For f L [, b] then (3.7) P V (x) f ( ) 1 ( b S (x t) ) dt. Now, let (3.8) S ( (x t) ) (t x) ν (t X ) (t X + ), where (3.9) nd ν M ( (x ) ) (x )3 + (b x) 3 3 (b ) 1 ( ) ( b + x + b ), 3 (3.30) X x ν, X + x + ν. Then, (3.31) H (t) S ( (x t) ) dt Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck [(t x) ν ] dt (t x)3 3 ν t + k Pge 1 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
22 nd so from (3.31) nd using (3.8) - (3.9) gives, (3.3) ( S (x t) ) dt H (X ) H () [H (X + ) H (X )] + [H (b) H (X + )] [H (X ) H (X + )] + H (b) H () } { ν3 3 ν X ν3 3 + ν X + (b x)3 + ν (x )3 b + + ν 3 3 [ν 3 3 ] ν3 + (b x)3 + (x ) 3 ν (b ) ν3. Thus, substituting into (3.7), (3.6) nd using (3.9) redily produces the result (3.1) nd the theorem is proved. Remrk 3.. Other bounds my be obtined for f L p [, b], p 1 however obtining explicit expressions for these bounds is somewht intricte nd will not be considered further here. They involve the clcultion of sup t [,b] for f L 1 [, b] nd (t x) ν mx { (x ) ν, ν, (b x) ν } ( (t x) ν q dt ) 1 q Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
23 for f L p [, b], 1 p + 1 q 1, p > 1, where ν is given by (3.9). Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 3 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
24 4. Some Inequlities for Absolutely Continuous P.D.F s We strt with the following lemm which is interesting in itself. Lemm 4.1. Let X be rndom vrible whose probbility density function f : [, b] R + is bsolutely continuous on [, b]. Then we hve the identity (4.1) σ (X) + [E (X) x] (b ) b + ( x + b where the kernel p : [, b] R is given by s, if s t b, p (t, s) : s b, if t < s b, for ll x [, b]. Proof. We use the identity (see (.10)) (4.) σ (X) + [E (X) x] for ll x [, b]. ) (t x) p (t, s) f (s) dsdt, (x t) f (t) dt Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 4 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
25 On the other hnd, we know tht (see for exmple [] for simple proof using integrtion by prts) (4.3) f (t) 1 b f (s) ds + 1 b p (t, s) f (s) ds for ll t [, b]. Substituting (4.3) in (4.) we obtin (4.4) σ (X) + [E (X) x] [ 1 b (t x) f (s) ds + 1 ] p (t, s) f (s) ds dt b b 1 b 1 [ (x ) 3 + (b x) 3] (t x) p (t, s) f (s) dsdt. b Tking into ccount the fct tht 1 [ (x ) 3 + (b x) 3] 3 (b ) 1 then, by (4.4) we deduce the desired result (4.1). + ( x + b ), x [, b], The following inequlity for P.D.F.s which re bsolutely continuous nd hve the derivtives essentilly bounded holds. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 5 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
26 Theorem 4.. If f : [, b] R + is bsolutely continuous on [, b] nd f L [, b], i.e., f : ess sup f (t) <, then we hve the inequlity: t [,b] ( (4.5) σ (X) + [E (X) x] (b ) x + b ) 1 for ll x [, b]. (b ) 3 [ (b ) 10 + ( x + b ) ] f Proof. Using Lemm 4.1, we hve ( σ (X) + [E (X) x] (b ) x + b ) 1 1 b (t x) p (t, s) f (s) dsdt 1 b f b (t x) p (t, s) f (s) dsdt (t x) p (t, s) dsdt. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 6 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
27 We hve I : (t x) p (t, s) dsdt [ t (t x) (s ) ds + t [ ] (t x) (t ) + (b t) dt 1 [ (t x) (t ) dt + I + I b. Let A x, B b x then I 0 (t x) (t ) dt (b )3 3 ] (b s) ds dt ] (t x) (b t) dt ( u Au + A ) u du [A 3 A (b ) + 35 ] (b ) Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 7 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
28 nd Now, I + I b I b 0 (b )3 3 (b )3 3 (b )3 3 (t x) (b t) dt (b )3 3 nd the theorem is proved. ( u Bu + B ) u du [B 3 B (b ) + 35 ] (b ) [ A + B 3 4 (A + B) (b ) + 3 ] 5 (b ) [ (b [ (b ) 10 ) ( + x + b + ( x + b ) 3 ) ] ] (b ) 0 The best inequlity we cn get from (4.5) is embodied in the following corollry. Corollry 4.3. If f is s in Theorem 4., then we hve [ (4.6) σ (X) + E (X) + b ] (b ) (b )4 f Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 8 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
29 We now nlyze the cse where f is Lebesgue p integrble mpping with p (1, ). Remrk 4.1. The results of Theorem 4. my be compred with those of Theorem 3.4. It my be shown tht both bounds re convex nd symmetric bout x +b. Further, the bound given by the premture Chebychev pproch, nmely from (3.1)-(3.13) is tighter thn tht obtined by the current pproch (4.5) which my be shown from the following. Let these bounds be described by B p nd B c so tht, neglecting the common terms nd where B p b 15 B c Y [ (b (b ) 100 ) + 15Y ] 1 + Y, ( x + b ). It my be shown through some strightforwrd lgebr tht B c B p > 0 for ll x [, b] so tht B c > B p. The current development does however hve the dvntge tht the identity (4.1) is stisfied, thus llowing bounds for L p [, b], p 1 rther thn the infinity norm. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 9 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
30 Theorem 4.4. If f : [, b] R + is bsolutely continuous on [, b] nd f L p, i.e., ( ) 1 f p : f (t) p p dt <, p (1, ) then we hve the inequlity (4.7) σ (X) + [E (X) x] f p (b ) 1 p (q + 1) 1 q (b ) 1 [ (x ) 3q+ B ( x + b ( b + (b x) 3q+ B ) ), q + 1, q + x ( )] b, q + 1, q + b x for ll x [, b], when nd B (,, ) is the qusi incomplete Euler s p q Bet mpping: B (z; α, β) : z 0 (u 1) α 1 u β 1 du, α, β > 0, z 1. Proof. Using Lemm 4.1, we hve, s in Theorem 4., tht ( (4.8) σ (X) + [E (X) x] (b ) x + b ) 1 1 b (t x) p (t, s) f (s) dsdt. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 30 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
31 Using Hölder s integrl inequlity for double integrls, we hve (4.9) (t x) p (t, s) f (s) dsdt ( (b ) 1 p f p ( ) 1 ( f (s) p p b dsdt where p > 1, 1 p + 1 q 1. We hve to compute the integrl ) 1 (t x) q p (t, s) q q dsdt (t x) q p (t, s) q dsdt ) 1 q, Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd (4.10) D : 1 q + 1 (t x) q p (t, s) q dsdt ] (b s) q ds [ t (t x) q (s ) q ds + t [ ] (t x) q (t ) q+1 + (b t) q+1 dt q + 1 [ (t x) q (t ) q+1 dt + ] (t x) q (b t) q+1 dt. dt Title Pge Go Bck Pge 31 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
32 Define (4.11) E : (t x) q (t ) q+1 dt. If we consider the chnge of vrible t (1 u) +ux, we hve t implies u 0 nd t b implies u b, dt (x ) du nd then x (4.1) E Define x (4.13) F : 0 [(1 u) + ux x] q [(1 u) + ux ] (x ) du (x ) 3q+ x (u 1) q u q+1 du 0( ) b (x ) 3q+ B, q + 1, q +. x (t x) q (b t) q+1 dt. If we consider the chnge of vrible t (1 v) b + vx, we hve t b implies Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 3 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
33 v 0, nd t implies v b, dt (x b) dv nd then b x (4.14) F 0 b b x [(1 v) b + vx x] q [b (1 v) b vx] q+1 (x b) dv (b x) 3q+ b x (v 1) q v q+1 dv 0( ) b (b x) 3q+ B, q + 1, q +. b x Now, using the inequlities (4.8)-(4.9) nd the reltions (4.10)-(4.14), since D 1 (E + F ), we deduce the desired estimte (4.7). q+1 The following corollry is nturl to be considered. Corollry 4.5. Let f be s in Theorem 4.4. Then, we hve the inequlity: (4.15) σ (X) + [ E (X) + b ] f p (b ) + 3 q (q + 1) 1 q 3+ q (b ) 1 [B (q + 1, q + 1) + Ψ (q + 1, q + )] 1 q, where , p > 1 nd B (, ) is Euler s Bet mpping nd Ψ (α, β) : p q 1 0 uα 1 (u + 1) β 1 du, α, β > 0. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 33 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
34 Proof. In (4.7) put x +b B (, q + 1, q + ). The left side is cler. Now 0 1 The right hnd side of (4.7) is thus: ( f b ) 3q+ q p (b ) 1 1 p (q + 1) q 0 (u 1) q u q+1 du (u 1) q u q+1 du + 1 (u 1) q u q+1 du B (q + 1, q + ) + Ψ (q + 1, q + ). [B (q + 1, q + ) + Ψ (q + 1, q + )] 1 q f p (b ) + 3 q (q + 1) 1 q 3+ q nd the corollry is proved. [B (q + 1, q + ) + Ψ (q + 1, q + )] 1 q Finlly, if f is bsolutely continuous, f L 1 [, b] nd f 1 f (t) dt, then we cn stte the following theorem. Theorem 4.6. If the p.d.f., f : [, b] R + is bsolutely continuous on [, b], then ( (4.16) σ (X) + [E (X) x] (b ) x + b ) 1 [ 1 f 1 (b ) (b ) + x + b ] Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 34 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
35 for ll x [, b]. Proof. As bove, we cn stte tht σ (X) + [E (X) x] (b ) 1 where 1 b ( x + b ) (t x) p (t, s) f (s) dsdt [ sup (t x) p (t, s) ] 1 (t,s) [,b] b f 1 G G : nd the theorem is proved. [ sup (t x) p (t, s) ] (t,s) [,b] (b ) sup (t x) t [,b] (b ) [mx (x, b x)] [ 1 (b ) (b ) + x + b ], f (s) dsdt It is cler tht the best inequlity we cn get from (4.16) is the one when x +b, giving the following corollry. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 35 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
36 Corollry 4.7. With the ssumptions of Theorem 4.6, we hve: [ (4.17) σ (X) + E (X) + b ] (b ) (b )3 f Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 36 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
37 References [1] P. CERONE AND S.S. DRAGOMIR, Three point qudrture rules involving, t most, first derivtive, submitted, RGMIA Res. Rep. Coll., (4) (1999), Article 8. [ONLINE] [] P. CERONE AND S.S. DRAGOMIR, Trpezoidl type rules from n inequlities point of view, Accepted for publiction in Anlytic- Computtionl Methods in Applied Mthemtics, G.A. Anstssiou (Ed), CRC Press, New York (000), [3] P. CERONE AND S.S. DRAGOMIR, Midpoint type rules from n inequlities point of view, Accepted for publiction in Anlytic-Computtionl Methods in Applied Mthemtics, G.A. Anstssiou (Ed), CRC Press, New York (000), [4] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, An inequlity of Ostrowski type for mppings whose second derivtives re bounded nd pplictions, Est Asin Mth. J., 15(1) (1999), 1 9. Preprint. RGMIA Res. Rep Coll., 1(1) (1998), Article 4, [ONLINE] [5] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, An inequlity of Ostrowski-Grüss type for twice differentible mppings nd pplictions, Kyungpook Mth. J., 39() (1999), Preprint. RGMIA Res. Rep Coll., 1() (1998), Article 8, [ONLINE] Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 37 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
38 [6] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, An Ostrowski type inequlity for mppings whose second derivtives belong to L p (, b) nd pplictions, Preprint. RGMIA Res. Rep Coll., 1(1) (1998), Article 5. [ONLINE] [7] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, On Ostrowski type for mppings whose second derivtives belong to L 1 (, b) nd pplictions, Honm Mth. J., 1(1) (1999), Preprint. RGMIA Res. Rep Coll., 1() (1998), Article 7. [ONLINE] [8] P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS, Some Ostrowski type inequlities for n-time differentible mppings nd pplictions, Demonstrtio Mth., 3() (1999), Preprint. RGMIA Res. Rep Coll., 1() (1998), [ONLINE] [9] P. CERONE, S.S. DRAGOMIR, J. ROUMELIOTIS AND J. SUNDE, A new generliztion of the trpezoid formul for n-time differentible mppings nd pplictions, Demonstrtio Mth., 33(4) (000), RGMIA Res. Rep. Coll., (5) (1999), Article 7. [ONLINE] [10] S.S. DRAGOMIR, Grüss type integrl inequlity for mppings of r- Hölder s type nd pplictions for trpezoid formul, Tmkng J. Mth., 31(1) (000), [11] S.S. DRAGOMIR, A Tylor like formul nd ppliction in numericl integrtion, submitted. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 38 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
39 [1] S.S. DRAGOMIR, Grüss inequlity in inner product spces, Austrl. Mth. Soc. Gz., 6() (1999), [13] S.S. DRAGOMIR, New estimtion of the reminder in Tylor s formul using Grüss type inequlities nd pplictions, Mth. Inequl. Appl., () (1999), [14] S.S. DRAGOMIR, Some integrl inequlities of Grüss type, Indin J. of Pure nd Appl. Mth., 31(4) (000), [15] S.S. DRAGOMIR AND N. S. BARNETT, An Ostrowski type inequlity for mppings whose second derivtives re bounded nd pplictions, J. Indin Mth. Soc., 66(1-4) (1999), Preprint. RGMIA Res. Rep Coll., 1() (1998), Article 9. [ONLINE] [16] S.S. DRAGOMIR, P. CERONE AND A. SOFO, Some remrks on the midpoint rule in numericl integrtion, Studi Mth. Bbeş-Bolyi Univ., (in press). [17] S.S. DRAGOMIR, P. CERONE AND A. SOFO, Some remrks on the trpezoid rule in numericl integrtion, Indin J. Pure Appl. Mth., 31(5) (000), Preprint: RGMIA Res. Rep. Coll., (5) (1999), Article 1. [ONLINE] [18] S.S. DRAGOMIR, Y.J. CHO AND S.S. KIM, Some remrks on the Milovnović-Pečrić Inequlity nd in Applictions for specil mens nd numericl integrtion, Tmkng J. Mth., 30(3) (1999), Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 39 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
40 [19] S.S. DRAGOMIR AND A. McANDREW, On Trpezoid inequlity vi Grüss type result nd pplictions, Tmkng J. Mth., 31(3) (000), RGMIA Res. Rep. Coll., () (1999), Article 6. [ONLINE] [0] S.S. DRAGOMIR, J.E. PEČARIĆ AND S. WANG, The unified tretment of trpezoid, Simpson nd Ostrowski type inequlity for monotonic mppings nd pplictions, Mth. nd Comp. Modelling, 31 (000), Preprint: RGMIA Res. Rep. Coll., (4) (1999), Article 3. [ONLINE] [1] S.S. DRAGOMIR AND A. SOFO, An integrl inequlity for twice differentible mppings nd pplictions, Preprint: RGMIA Res. Rep. Coll., () (1999), Article 9. [ONLINE] [] S.S. DRAGOMIR AND S. WANG, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtion of error bounds for some specil mens nd for some numericl qudrture rules, Comput. Mth. Appl., 33 (1997), 15. [3] M. MATIĆ, J.E. PEČARIĆ AND N. UJEVIĆ, On New estimtion of the reminder in Generlised Tylor s Formul, Mth. Inequl. Appl., (3) (1999), [4] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers, Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 40 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
41 [5] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequlities for Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, [6] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex Functions, Prtil Orderings, nd Sttisticl Applictions, Acdemic Press, 199. Some Inequlities for the Dispersion of Rndom Vrible whose PDF is Defined on Finite Intervl Neil S. Brnett, Pietro Cerone, Sever S. Drgomir nd Title Pge Go Bck Pge 41 of 41 J. Ineq. Pure nd Appl. Mth. (1) Art. 1, 001
Journal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics http://jipmvueduu/ Volume, Issue, Article, 00 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NS BARNETT,
More informationSOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL
SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS Abstrct Some ineulities for the dispersion of rndom
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.
More informationON PERTURBED TRAPEZOIDAL AND MIDPOINT RULES. f (t) dt
ON PERTURBED TRAPEZOIDAL AND MIDPOINT RULES P. CERONE Abstrct. Explicit bounds re obtined for the perturbed or corrected trpezoidl nd midpoint rules in terms of the Lebesque norms of the second derivtive
More informationNew Integral Inequalities for n-time Differentiable Functions with Applications for pdfs
Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353-362 New Integrl Inequlities for n-time Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute
More informationON THE WEIGHTED OSTROWSKI INEQUALITY
ON THE WEIGHTED OSTROWSKI INEQUALITY N.S. BARNETT AND S.S. DRAGOMIR School of Computer Science nd Mthemtics Victori University, PO Bo 14428 Melbourne City, VIC 8001, Austrli. EMil: {neil.brnett, sever.drgomir}@vu.edu.u
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics MOMENTS INEQUALITIES OF A RANDOM VARIABLE DEFINED OVER A FINITE INTERVAL PRANESH KUMAR Deprtment of Mthemtics & Computer Science University of Northern
More informationNEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS. := f (4) (x) <. The following inequality. 2 b a
NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS MOHAMMAD ALOMARI A MASLINA DARUS A AND SEVER S DRAGOMIR B Abstrct In terms of the first derivtive some ineulities of Simpson
More informationS. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:
FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy
More informationS. S. Dragomir. 2, we have the inequality. b a
Bull Koren Mth Soc 005 No pp 3 30 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Abstrct Compnions of Ostrowski s integrl ineulity for bsolutely
More informationTRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS
TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS S.S. DRAGOMIR AND A. SOFO Abstrct. In this pper by utilising result given by Fink we obtin some new results relting to the trpezoidl inequlity
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics http://jipm.vu.edu.u/ Volume 3, Issue, Article 4, 00 ON AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL AND SOME RAMIFICATIONS P. CERONE SCHOOL OF COMMUNICATIONS
More informationAN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir
RGMIA Reserch Report Collection, Vol., No., 999 http://sci.vu.edu.u/ rgmi AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS I. Fedotov nd S. S. Drgomir Astrct. An
More informationGENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS. (b a)3 [f(a) + f(b)] f x (a,b)
GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Abstrct. In this pper, we estblish some generliztions
More information0 N. S. BARNETT AND S. S. DRAGOMIR Using Gruss' integrl inequlity, the following pertured trpezoid inequlity in terms of the upper nd lower ounds of t
TAMKANG JOURNAL OF MATHEMATICS Volume 33, Numer, Summer 00 ON THE PERTURBED TRAPEZOID FORMULA N. S. BARNETT AND S. S. DRAGOMIR Astrct. Some inequlities relted to the pertured trpezoid formul re given.
More informationKeywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.
Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. Lmi-Athens Lmi 3500 Greece Abstrct Using
More informationA unified generalization of perturbed mid-point and trapezoid inequalities and asymptotic expressions for its error term
An. Ştiinţ. Univ. Al. I. Cuz Işi. Mt. (N.S. Tomul LXIII, 07, f. A unified generliztion of perturbed mid-point nd trpezoid inequlities nd symptotic expressions for its error term Wenjun Liu Received: 7.XI.0
More informationAN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
Applied Mthemtics E-Notes, 5(005), 53-60 c ISSN 1607-510 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
More informationf (a) + f (b) f (λx + (1 λ)y) max {f (x),f (y)}, x, y [a, b]. (1.1)
TAMKANG JOURNAL OF MATHEMATICS Volume 41, Number 4, 353-359, Winter 1 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI, M. DARUS
More informationON AN INTEGRATION-BY-PARTS FORMULA FOR MEASURES
Volume 8 (2007), Issue 4, Article 93, 13 pp. ON AN INTEGRATION-BY-PARTS FORMULA FOR MEASURES A. ČIVLJAK, LJ. DEDIĆ, AND M. MATIĆ AMERICAN COLLEGE OF MANAGEMENT AND TECHNOLOGY ROCHESTER INSTITUTE OF TECHNOLOGY
More informationSOME INTEGRAL INEQUALITIES OF GRÜSS TYPE
RGMIA Reserch Report Collection, Vol., No., 998 http://sci.vut.edu.u/ rgmi SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE S.S. DRAGOMIR Astrct. Some clssicl nd new integrl inequlities of Grüss type re presented.
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics ON LANDAU TYPE INEQUALITIES FOR FUNCTIONS WIT ÖLDER CONTINUOUS DERIVATIVES LJ. MARANGUNIĆ AND J. PEČARIĆ Deprtment of Applied Mthemtics Fculty of Electricl
More informationImprovement of Ostrowski Integral Type Inequalities with Application
Filomt 30:6 06), 56 DOI 098/FIL606Q Published by Fculty of Sciences nd Mthemtics, University of Niš, Serbi Avilble t: http://wwwpmfnicrs/filomt Improvement of Ostrowski Integrl Type Ineulities with Appliction
More informationWEIGHTED 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
Preprints (www.preprints.org) 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.
More informationRGMIA Research Report Collection, Vol. 1, No. 1, SOME OSTROWSKI TYPE INEQUALITIES FOR N-TIME DIFFERENTIA
ttp//sci.vut.edu.u/rgmi/reports.tml SOME OSTROWSKI TYPE INEQUALITIES FOR N-TIME DIFFERENTIABLE MAPPINGS AND APPLICATIONS P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS Astrct. Some generliztions of te Ostrowski
More informationSome estimates on the Hermite-Hadamard inequality through quasi-convex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper
More informationAn optimal 3-point quadrature formula of closed type and error bounds
Revist Colombin de Mtemátics Volumen 8), págins 9- An optiml 3-point qudrture formul of closed type nd error bounds Un fórmul de cudrtur óptim de 3 puntos de tipo cerrdo y error de fronter Nend Ujević,
More informationNew Integral Inequalities of the Type of Hermite-Hadamard Through Quasi Convexity
Punjb University Journl of Mthemtics (ISSN 116-56) Vol. 45 (13) pp. 33-38 New Integrl Inequlities of the Type of Hermite-Hdmrd Through Qusi Convexity S. Hussin Deprtment of Mthemtics, College of Science,
More informationA Companion of Ostrowski Type Integral Inequality Using a 5-Step Kernel with Some Applications
Filomt 30:3 06, 360 36 DOI 0.9/FIL6360Q Pulished y Fculty of Sciences nd Mthemtics, University of Niš, Seri Aville t: http://www.pmf.ni.c.rs/filomt A Compnion of Ostrowski Type Integrl Inequlity Using
More informationA Generalized Inequality of Ostrowski Type for Twice Differentiable Bounded Mappings and Applications
Applied Mthemticl Sciences, Vol. 8, 04, no. 38, 889-90 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.988/ms.04.4 A Generlized Inequlity of Ostrowski Type for Twice Differentile Bounded Mppings nd Applictions
More informationNew general integral inequalities for quasiconvex functions
NTMSCI 6, No 1, 1-7 18 1 New Trends in Mthemticl Sciences http://dxdoiorg/185/ntmsci1739 New generl integrl ineulities for usiconvex functions Cetin Yildiz Atturk University, K K Eduction Fculty, Deprtment
More informationON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs
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,
More informationINEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV FUNCTIONAL. Mohammad Masjed-Jamei
Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: http://www.pmf.ni.c.rs/filomt Filomt 25:4 20) 53 63 DOI: 0.2298/FIL0453M INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV
More informationRevista Colombiana de Matemáticas Volumen 41 (2007), páginas 1 13
Revist Colombin de Mtemátics Volumen 4 7, págins 3 Ostrowski, Grüss, Čebyšev type inequlities for functions whose second derivtives belong to Lp,b nd whose modulus of second derivtives re convex Arif Rfiq
More informationINEQUALITIES FOR BETA AND GAMMA FUNCTIONS VIA SOME CLASSICAL AND NEW INTEGRAL INEQUALITIES
INEQUALITIES FOR BETA AND GAMMA FUNCTIONS VIA SOME CLASSICAL AND NEW INTEGRAL INEQUALITIES S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT Abstrct. In this survey pper we present the nturl ppliction of
More informationResearch Article On New Inequalities via Riemann-Liouville Fractional Integration
Abstrct nd Applied Anlysis Volume 202, Article ID 428983, 0 pges doi:0.55/202/428983 Reserch Article On New Inequlities vi Riemnn-Liouville Frctionl Integrtion Mehmet Zeki Sriky nd Hsn Ogunmez 2 Deprtment
More informationImprovements of some Integral Inequalities of H. Gauchman involving Taylor s Remainder
Divulgciones Mtemátics Vol. 11 No. 2(2003), pp. 115 120 Improvements of some Integrl Inequlities of H. Guchmn involving Tylor s Reminder Mejor de lguns Desigulddes Integrles de H. Guchmn que involucrn
More informationThe Hadamard s inequality for quasi-convex functions via fractional integrals
Annls of the University of Criov, Mthemtics nd Computer Science Series Volume (), 3, Pges 67 73 ISSN: 5-563 The Hdmrd s ineulity for usi-convex functions vi frctionl integrls M E Özdemir nd Çetin Yildiz
More informationBulletin of the. Iranian Mathematical Society
ISSN: 07-060X Print ISSN: 735-855 Online Bulletin of the Irnin Mthemticl Society Vol 3 07, No, pp 09 5 Title: Some extended Simpson-type ineulities nd pplictions Authors: K-C Hsu, S-R Hwng nd K-L Tseng
More informationOstrowski Grüss Čebyšev type inequalities for functions whose modulus of second derivatives are convex 1
Generl Mthemtics Vol. 6, No. (28), 7 97 Ostrowski Grüss Čebyšev type inequlities for functions whose modulus of second derivtives re convex Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn Abstrct In this pper,
More informationResearch Article On The Hadamard s Inequality for Log-Convex Functions on the Coordinates
Hindwi Publishing Corportion Journl of Inequlities nd Applictions Volume 29, Article ID 28347, 3 pges doi:.55/29/28347 Reserch Article On The Hdmrd s Inequlity for Log-Convex Functions on the Coordintes
More informationDIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS
Krgujev Journl of Mthemtis Volume 38() (204), Pges 35 49. DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS MOHAMMAD W. ALOMARI Abstrt. In this pper, severl bouns for the ifferene between two Riemn-
More informationParametrized inequality of Hermite Hadamard type for functions whose third derivative absolute values are quasi convex
Wu et l. SpringerPlus (5) 4:83 DOI.8/s44-5-33-z RESEARCH Prmetrized inequlity of Hermite Hdmrd type for functions whose third derivtive bsolute vlues re qusi convex Shn He Wu, Bnyt Sroysng, Jin Shn Xie
More informationAn inequality related to η-convex functions (II)
Int. J. Nonliner Anl. Appl. 6 (15) No., 7-33 ISSN: 8-68 (electronic) http://d.doi.org/1.75/ijn.15.51 An inequlity relted to η-conve functions (II) M. Eshghi Gordji, S. S. Drgomir b, M. Rostmin Delvr, Deprtment
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued
More informationON COMPANION OF OSTROWSKI INEQUALITY FOR MAPPINGS WHOSE FIRST DERIVATIVES ABSOLUTE VALUE ARE CONVEX WITH APPLICATIONS
Miskolc Mthemticl Notes HU ISSN 787-5 Vol. 3 (), No., pp. 33 8 ON OMPANION OF OSTROWSKI INEQUALITY FOR MAPPINGS WHOSE FIRST DERIVATIVES ABSOLUTE VALUE ARE ONVEX WITH APPLIATIONS MOHAMMAD W. ALOMARI, M.
More informationWENJUN LIU AND QUÔ C ANH NGÔ
AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of Ostrowski-Grüss type on time scles nd thus unify corresponding continuous
More informationCommunications inmathematicalanalysis Volume 6, Number 2, pp (2009) ISSN
Communictions inmthemticlanlysis Volume 6, Number, pp. 33 41 009) ISSN 1938-9787 www.commun-mth-nl.org A SHARP GRÜSS TYPE INEQUALITY ON TIME SCALES AND APPLICATION TO THE SHARP OSTROWSKI-GRÜSS INEQUALITY
More informationImprovement of Grüss and Ostrowski Type Inequalities
Filomt 9:9 (05), 07 035 DOI 098/FIL50907A Pulished y Fculty of Sciences nd Mthemtics, University of Niš, Seri Aville t: http://wwwpmfnicrs/filomt Improvement of Grüss nd Ostrowski Type Inequlities An Mri
More informationSome Hermite-Hadamard type inequalities for functions whose exponentials are convex
Stud. Univ. Beş-Bolyi Mth. 6005, No. 4, 57 534 Some Hermite-Hdmrd type inequlities for functions whose exponentils re convex Silvestru Sever Drgomir nd In Gomm Astrct. Some inequlities of Hermite-Hdmrd
More informationarxiv: v1 [math.ca] 28 Jan 2013
ON NEW APPROACH HADAMARD-TYPE INEQUALITIES FOR s-geometrically CONVEX FUNCTIONS rxiv:3.9v [mth.ca 8 Jn 3 MEVLÜT TUNÇ AND İBRAHİM KARABAYIR Astrct. In this pper we chieve some new Hdmrd type ineulities
More informationIntegral inequalities for n times differentiable mappings
JACM 3, No, 36-45 8 36 Journl of Abstrct nd Computtionl Mthemtics http://wwwntmscicom/jcm Integrl ineulities for n times differentible mppings Cetin Yildiz, Sever S Drgomir Attur University, K K Eduction
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationHadamard-Type Inequalities for s Convex Functions I
Punjb University Journl of Mthemtics ISSN 6-56) Vol. ). 5-6 Hdmrd-Tye Ineulities for s Convex Functions I S. Hussin Dertment of Mthemtics Institute Of Sce Technology, Ner Rwt Toll Plz Islmbd Highwy, Islmbd
More informationRIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROXIMATION OF CSISZAR S f DIVERGENCE
SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17 (2009, 3 12 RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROIMATION OF CSISZAR S f DIVERGENCE GEORGE A. ANASTASSIOU Abstrct. Here re estblished vrious tight probbilistic
More information1 The Lagrange interpolation formula
Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x
More informationBounds for the Riemann Stieltjes integral via s-convex integrand or integrator
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 6, Number, 0 Avilble online t www.mth.ut.ee/ct/ Bounds for the Riemnn Stieltjes integrl vi s-convex integrnd or integrtor Mohmmd Wjeeh
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationOn the Generalized Weighted Quasi-Arithmetic Integral Mean 1
Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, 2039-2048 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2013.3499 On the Generlized Weighted Qusi-Arithmetic Integrl Men 1 Hui Sun School
More informationHermite-Hadamard type inequalities for harmonically convex functions
Hcettepe Journl o Mthemtics nd Sttistics Volume 43 6 4 935 94 Hermite-Hdmrd type ineulities or hrmoniclly convex unctions İmdt İşcn Abstrct The uthor introduces the concept o hrmoniclly convex unctions
More informationHermite-Hadamard Type Inequalities for the Functions whose Second Derivatives in Absolute Value are Convex and Concave
Applied Mthemticl Sciences Vol. 9 05 no. 5-36 HIKARI Ltd www.m-hikri.com http://d.doi.org/0.988/ms.05.9 Hermite-Hdmrd Type Ineulities for the Functions whose Second Derivtives in Absolute Vlue re Conve
More informationQUADRATURE is an old-fashioned word that refers to
World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 A New Qudrture Rule Derived from Spline Interpoltion with Error Anlysis Hdi Tghvfrd
More informationINNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS AND APPLICATIONS
INNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS AND APPLICATIONS S. S. DRAGOMIR Abstrct. Some inequlities for two inner products h i nd h i which generte the equivlent norms kk nd kk with pplictions
More informationA Note on Feng Qi Type Integral Inequalities
Int Journl of Mth Anlysis, Vol 1, 2007, no 25, 1243-1247 A Note on Feng Qi Type Integrl Inequlities Hong Yong Deprtment of Mthemtics Gungdong Business College Gungzhou City, Gungdong 510320, P R Chin hongyong59@sohucom
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, Bbeş-Bolyi University Str. Koglnicenu
More informationarxiv: v1 [math.ca] 11 Jul 2011
rxiv:1107.1996v1 [mth.ca] 11 Jul 2011 Existence nd computtion of Riemnn Stieltjes integrls through Riemnn integrls July, 2011 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde
More informationOn some refinements of companions of Fejér s inequality via superquadratic functions
Proyecciones Journl o Mthemtics Vol. 3, N o, pp. 39-33, December. Universidd Ctólic del Norte Antogst - Chile On some reinements o compnions o Fejér s inequlity vi superqudrtic unctions Muhmmd Amer Lti
More informationLOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS VIA TAYLOR S EXPANSION WITH INTEGRAL REMAINDER
LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS VIA TAYLOR S EXPANSION WITH INTEGRAL REMAINDER S. S. DRAGOMIR ;2 Astrct. In this pper we otin severl new logrithmic inequlities for two numers ; minly
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationSome new integral inequalities for n-times differentiable convex and concave functions
Avilble online t wwwisr-ublictionscom/jns J Nonliner Sci Al, 10 017, 6141 6148 Reserch Article Journl Homege: wwwtjnscom - wwwisr-ublictionscom/jns Some new integrl ineulities for n-times differentible
More informationINEQUALITIES OF HERMITE-HADAMARD S TYPE FOR FUNCTIONS WHOSE DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX
INEQUALITIES OF HERMITE-HADAMARD S TYPE FOR FUNCTIONS WHOSE DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI A, M. DARUS A, AND S.S. DRAGOMIR B Astrct. In this er, some ineulities of Hermite-Hdmrd
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationKRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION
Fixed Point Theory, 13(2012), No. 1, 285-291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd
More informationAPPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A TRAPEZOIDAL QUADRATURE RULE WITH APPLICATIONS
APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A TRAPEZOIDAL QUADRATURE RULE WITH APPLICATIONS S.S. DRAGOMIR Astrct. In this pper we provide shrp ounds for the error in pproximting the Riemnn-Stieltjes
More informationSome inequalities of Hermite-Hadamard type for n times differentiable (ρ, m) geometrically convex functions
Avilble online t www.tjns.com J. Nonliner Sci. Appl. 8 5, 7 Reserch Article Some ineulities of Hermite-Hdmrd type for n times differentible ρ, m geometriclly convex functions Fiz Zfr,, Humir Klsoom, Nwb
More informationResearch Article On Hermite-Hadamard Type Inequalities for Functions Whose Second Derivatives Absolute Values Are s-convex
ISRN Applied Mthemtics, Article ID 8958, 4 pges http://dx.doi.org/.55/4/8958 Reserch Article On Hermite-Hdmrd Type Inequlities for Functions Whose Second Derivtives Absolute Vlues Are s-convex Feixing
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationNEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX
Journl of Mthemticl Ineulities Volume 1, Number 3 18, 655 664 doi:1.7153/jmi-18-1-5 NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX SHAHID
More informationSPECIAL FUNCTIONS: APPROXIMATIONS AND BOUNDS
Applicble Anlysis nd Discrete Mthemtics, 1 7), 7 91. Avilble electroniclly t http://pefmth.etf.bg.c.yu Presented t the conference: Topics in Mthemticl Anlysis nd Grph Theory, Belgrde, September 1 4, 6.
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationApproximation of functions belonging to the class L p (ω) β by linear operators
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 3, 9, Approximtion of functions belonging to the clss L p ω) β by liner opertors W lodzimierz Lenski nd Bogdn Szl Abstrct. We prove
More informationOn Hermite-Hadamard type integral inequalities for functions whose second derivative are nonconvex
Mly J Mt 34 93 3 On Hermite-Hdmrd tye integrl ineulities for functions whose second derivtive re nonconvex Mehmet Zeki SARIKAYA, Hkn Bozkurt nd Mehmet Eyü KİRİŞ b Dertment of Mthemtics, Fculty of Science
More informationGENERALIZED ABSTRACTED MEAN VALUES
GENERALIZED ABSTRACTED MEAN VALUES FENG QI Abstrct. In this rticle, the uthor introduces the generlized bstrcted men vlues which etend the concepts of most mens with two vribles, nd reserches their bsic
More informationSeveral Answers to an Open Problem
Int. J. Contemp. Mth. Sciences, Vol. 5, 2010, no. 37, 1813-1817 Severl Answers to n Open Problem Xinkun Chi, Yonggng Zho nd Hongxi Du College of Mthemtics nd Informtion Science Henn Norml University Henn
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationn-points Inequalities of Hermite-Hadamard Type for h-convex Functions on Linear Spaces
Armenin Journl o Mthemtics Volume 8, Number, 6, 38 57 n-points Inequlities o Hermite-Hdmrd Tpe or h-convex Functions on Liner Spces S. S. Drgomir Victori Universit, Universit o the Witwtersrnd Abstrct.
More informationarxiv: v1 [math.ca] 7 Mar 2012
rxiv:1203.1462v1 [mth.ca] 7 Mr 2012 A simple proof of the Fundmentl Theorem of Clculus for the Lebesgue integrl Mrch, 2012 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationTHIELE CENTRE. Linear stochastic differential equations with anticipating initial conditions
THIELE CENTRE for pplied mthemtics in nturl science Liner stochstic differentil equtions with nticipting initil conditions Nrjess Khlif, Hui-Hsiung Kuo, Hbib Ouerdine nd Benedykt Szozd Reserch Report No.
More informationON THE C-INTEGRAL BENEDETTO BONGIORNO
ON THE C-INTEGRAL BENEDETTO BONGIORNO Let F : [, b] R be differentible function nd let f be its derivtive. The problem of recovering F from f is clled problem of primitives. In 1912, the problem of primitives
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationINEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION
INEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION BAI-NI GUO AND FENG QI Abstrct. In the rticle, using the Tchebycheff s integrl inequlity, the suitble properties of double integrl nd
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More information3.4 Numerical integration
3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,
More information