ON COMPANION OF OSTROWSKI INEQUALITY FOR MAPPINGS WHOSE FIRST DERIVATIVES ABSOLUTE VALUE ARE CONVEX WITH APPLICATIONS

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1 Miskolc Mthemticl Notes HU ISSN Vol. 3 (), No., pp ON OMPANION OF OSTROWSKI INEQUALITY FOR MAPPINGS WHOSE FIRST DERIVATIVES ABSOLUTE VALUE ARE ONVEX WITH APPLIATIONS MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMA Received Ferury, Astrct. Severl inequlities for compnion of Ostrowski inequlity for solutely continuous mppings whose first derivtives solute vlue re conve (resp. concve) re estlished. Applictions to composite qudrture rule, to p.d.f. s, nd to specil mens re provided. Mthemtics Suject lssifiction: 6D; 6A5; 6A6; 6A5 Keywords: onve mppings, Hermite-Hdmrd inequlity, Ostrowski inequlity. INTRODUTION In 938, Ostrowski estlished very interesting inequlity for differentile mppings with ounded derivtives, s follows [8]: Theorem. Let f W I R! R e differentile mpping on I ı ; interior of the intervl I; such tht f LŒ;, where ; I with <. If jf./j M, then the following inequlity, f./ 6 f.u/du M. /. / holds for ll Œ;. The constnt is the est possile in the sense tht it cnnot e replced y smller constnt. For recent results concerning Ostrowski inequlity see [], [] nd [3]. Also, the reder my e refer to the monogrph [8] where vrious inequlities of Ostrowski type re discussed. In [9], Guess nd Schmeisser hve proved mong others, the following compnion of Ostrowski inequlity: c Miskolc University Press

2 3 MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMA Theorem. Let f W Œ;! R stisfy the Lipschitz condition, i.e., jf.t/ f.s/j M jt sj. Then for ech Œ;, we hve the inequlity, " # f./f. / R 8 3. / M: (.) The constnt =8 is the est possile in the sense tht it cnnot e replced y smller constnt. We my lso note tht the est inequlity in (.) is otined for D 3, giving the trpezoid type inequlity, f 3 f 3 The constnt =8 is shrp in (.) in the sense mentioned ove.. / M (.) 8 ompnions of Ostrowski integrl inequlity for solutely continuous functions ws considered y Drgomir in [6], pp.8, s follows : Theorem 3. Let f W I R! R e n solutely continuous function on Œ;. Then we hve the inequlities, Z f./ f. / (.3) 8 " # 8 3. /kf k ; f L Œ; ˆ< " =q q q q # =q. / =q kf k Œ; ;p ; ˆ: 3 kf k Œ; ; p > ; p q D ; nd f L p Œ; for ll Œ;. In [7], the following theorem which ws otined y Drgomir nd Agrwl contins the Hermite-Hdmrd type integrl inequlity:

3 ON OMPANION OF OSTROWSKI INEQUALITY 35 Theorem. Let f W I ı R! R e differentile mpping on I ı, ; I ı with < : If jf j is conve on Œ;, then the following inequlity holds: Z f./ f./ f.u/du. /.jf./j jf./j/ : (.) 8 In [5], S.S. Drgomir estlished some inequlities for this compnion for mppings of ounded vrition. Also, Z. Liu introduced some compnions of n Ostrowski type integrl inequlity for functions whose derivtives re solutely continuous in []. Recently, N.S. Brnett et l. hve proved some compnions for the Ostrowski inequlity nd the generlized trpezoid inequlity in []. The im of this pper is to study compnion of Ostrowski inequlity Theorem for the clss of functions whose derivtives in solute vlue re conve (concve) functions.. RESULTS In order to prove our results, we need the following lemm (see [6]): Lemm. Let f W I R! R e n solutely continuous mpping on I ı ; where ; I with <, such tht f L Œ;. Then, the following equlity holds where f./ f. / for ll Œ;. D 8 t ; t Œ; ˆ< p.;t/ D t ; t.; ; ˆ: t ; t. ; p.;t/f.t/dt; A simple proof of the equlity cn e done y performing integrtion y prts. The detils re left to the interested reder (see [6]). Let us egin with the following result: Theorem 5. Let f W I R! R e n solutely continuous mpping on I ı, where ; I with <, such tht f L Œ;. If jf j is conve on Œ;, then we hve the following inequlity: f./ f. / (.)

4 36 MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMA. / 6. / for ll Œ;. f./ f./ 8. / 3. /. / Proof. Using Lemm nd the modulus, we hve Z f./ f. / jp.;t/jf.t/dt " Z D Z jp.;t/jf.t/dt Z # jp.;t/jf.t/dt f./ f. / jp.;t/jf.t/dt Since jf j is conve on Œ; D Œ; [.; [. ;, therefore we hve f t.t/ f t./ f./; t Œ; I f t.t/ f t. / f./; t.; I nd f t.t/ f t./ which follows tht, Z f./ f. / Z t jt j f t./ Z jt t t f. /; t. ; I f./ dt f. f./ t t j f./ / dt t f. / dt

5 D ON OMPANION OF OSTROWSKI INEQUALITY 37 "f. /3./ f./. /. / 3. /3 6. / 3 f. / f./. /. / 8 "f. /3./ f #. /3. /. /. / 6 3 D D. / f./ f./ f. 6. /. / f. / f./ 8. /. / 6. / f./ f./ 8. / 3. /. / where R.t / t which completes the proof. f./ f. dt D R. t/ t An Ostrowski type inequlity my e deduced s follows: # / f./ /; dt D. /3 8 ; orollry. Let f s in Theorem 5. Additionlly, if f is symmetric out the -is, i.e., f. / D f./, we hve Z f./. / 6. / for ll Œ;. f./ f./ 8. / 3. /. / f./ f. / Remrk. In Theorem 5, if we choose D, then we get Z f./ f./ f./ f./: 8 which is the inequlity in (.).

6 38 MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMA orollry. In Theorem 5, if we choose () D 3, then we get Z 3 3 f f. / 3 f./ 5f f f./ (.) f () D, then we get R. / h jf./j f jf./ji : (.3) Another result my e considered using the Hölder inequlity, s follows: Theorem 6. Let f W I R! R e n solutely continuous mpping on I ı ; where ; I with <, such tht f L Œ;. If jf j q, q > is conve on Œ;, then we hve the following inequlity: Z f./ f. / (.) n hf. /./q f./qi =q =q. /.p / =p. / hf./q f. /qi =q. / hf. /q f./qi =q o ; for ll Œ;, where p q D. Proof. Using Lemm nd the Hölder inequlity, we hve Z f./ f. / jp.;t/jf.t/dt " Z D Z jt jf.t/dt t Z # jt jf.t/dt f.t/dt (.5)

7 ON OMPANION OF OSTROWSKI INEQUALITY 39 " Z jt Z t jt =p Z j p dt f.t/q dt j p dt p dt! =p Z! =p =q f.t/q dt! 3 =q f.t/q dt 5! =q Since jf j q is conve on Œ; D Œ; [.; [. ;, therefore we hve f.t/q t f./q t f./q ; t Œ; I f.t/q t f. /q t f./q ; t.; I nd f.t/q t f./q t f. /q ; t. ; I which follows tht, Z f./ f. / " Z jt Z Z D jt t =p Z t j p dt p dt t j p dt! =p! =p f./q f. /q t t f./q. /p.p / t f. t =q f./q dt! =q f./q dt! 3 =q /q dt 5! =p =q f./q f./q =q

8 MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMA.p / p! =p. /p.p / D =q. /.p / =p. / =q f./q f. /q =q 3! =p =q f. /q f./q =q 5 f. /./q f./q =q f./q f. /q =q. / f. /q f./q =q ; since p q D, q >, which completes the proof. orollry 3. Let f s in Theorem 6. Additionlly, if f is symmetric out the -is, i.e., f. / D f./, we hve Z f./ =q. /.p / =p n hf. /./q f./qi =q for ll Œ;. orollry. In Theorem 6, if we choose () D, then we get f./f./. / hf./q f. /qi =q hf. /. /q f./qi =q R. / q.p/ =p jf./j q jf./j q =q : (.6) () D 3, then we get 3 3 f f (.7)

9 ON OMPANION OF OSTROWSKI INEQUALITY ( q. / f =q./q 3 q. p/ =p f 3 q q f =q 3 f 3 q f f =q )./q : (3) D, then we get Z f " q. / f =q./q q. p/ =p f q f f =q #./q : The following result holds for concve mppings. Theorem 7. Let f W I R! R e n solutely continuous mpping on I ı ; where ; I with <, such tht f L Œ;. If jf j q, q > is concve on Œ;, then we hve the following inequlity: f./ f. /. /. p/ =p ". / f f for ll Œ;, where p q D. Proof. From Lemm, nd y (.5), we hve Z f./ f. / " Z =p Z jt j p dt f.t/q dt. / =q (.8) # f

10 MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMA Z t jt j p dt p dt! =p Z! =p f.t/q dt! 3 =q f.t/q dt 5:! =q Now, let us write, Z f.t/q dt D. / Z f.. //q d; nd Z f.t/q dt D. / f.t/q dt D. / Z Z f.. /. //q d; f.. /. //q d: Since jf j q, q > is concve on Œ; D Œ; [.; [. ;, we cn use the Jensen integrl inequlity to otin. / Z D. / f.. Z f.. //q d //q d Z. / d f R d D. / q f Z.. //d! nd nlogously Z. / f.. /. //q d. / f Z. / f.. /. //q d. / f omining ll ove inequlities, we get Z f./ f. / q q : q ;

11 ON OMPANION OF OSTROWSKI INEQUALITY 3! 3 =p. /p. / =q.p / f 5.p /. /p.p /! p =p.! =p. / =q f D. /. p/ =p ". / f f / f =q. / # f ; for ll Œ;, where p q D, q >, which is required. Therefore, we my stte the following Ostrowski type inequlity: orollry 5. Let f s in Theorem 7. Additionlly, if f is symmetric out the -is, i.e., f. / D f./, we hve Z f./. /. p/ =p ". / f f for ll Œ;. orollry 6. In Theorem 7, if we choose () D, then we get f./ f./. /. p/ =p. / # f ; f : (.9) () D 3, then we get 3 3 f f

12 MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMA. / 7 f 6. p/ =p 8 f 7 f : (.) 8 (3) D, then we get f R. /.p/ =p hf 3 f 3. A OMPOSITE QUADRATURE FORMULA i 3 : Let I n W D < < < n D e division of the intervl Œ; nd h i D i i, (i D ;;; ;n ). onsider the generl qudrture formul Q n.i n ;f / WD nx 3i i f f id The following result holds. i 3 i h i : (3.) Theorem 8. Let f W I R! R e n solutely continuous mpping on I ı ; where ; I with <, such tht f L Œ;. If jf j is conve on Œ;. Then, we hve D Q n.i n ;f / R n.i n ;f /: (3.) where, Q n.i n ;f / is defined y formul (3.), nd the reminder term R n.i n ;f / stisfies the error estimtes jr n.i n ;f /j nx h i f. i / 5f 3i i 96 id 5 f i 3 i f. i/ : Proof. Applying inequlity (3.) nd (3.) on the intervls Œ i ; i, we my stte tht Z i 3i i i 3 i R i.i i ;f / D f f h i : i Summing the ove inequlity over i from to n, we get nx Z i n X 3i i i 3 i R n.i n ;f / D f f D id i nx id f id 3i i f i 3 i h i ; h i

13 which follows from (.), tht jr n.i n ;f /j D which completes the proof. ON OMPANION OF OSTROWSKI INEQUALITY 5 nx id f 3i i f nx h i f. i / 5f 3i i 96 id 5 f i 3 i f. i/ : i 3 i h i Remrk. One my stte more inequlities, using (.7) nd (.). We shll omit the detils.. APPLIATIONS FOR P.D.F. S Let X e rndom vrile tking vlues in the finite intervl Œ;, with the proility density function f W Œ;! Œ; with the cumultive distriution function F./ D P r.x / D R. Theorem 9. With the ssumptions of Theorem 5, we hve the inequlity ŒF./ F. / E.X/. / 6. / F./ F./ 8. / 3. /. / F./ F. for ll Œ;, where E.X/ is the epecttion of X. / Proof. In the proof of Theorem 5, let f D F, nd tking into ccount tht E.X/ D tdf.t/ D We left the detils to the interested reder. F.t/dt: orollry 7. In Theorem 9, if we choose D 3, then we get 3 3 E.X/ F F. / 3 F./ 5F F F./ :

14 6 MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMA orollry 8. In Theorem 9, if F is symmetric out the -is, i.e., F. F./, we hve F./ E.X/. / 6. / F./ F./ 8. / 3. /. / F./ F. / / D for ll Œ;. Remrk 3. One my stte more inequlities, using Theorem 6 nd Theorem 7. We shll omit the detils. 5. APPLIATIONS FOR SPEIAL MEANS Recll the following mens which could e considered etensions of rithmetic, logrithmic nd generlized logrithmic for positive rel numers. () The rithmetic men: A D A.;/ D I ; R () The logrithmic men: L.;/ D I jj jj; ; R lnjj lnjj (3) The generlized logrithmic men: n n n L n.;/ D I n Znf ;g; ; R ;. /.n / () The identric men: 8 ˆ< e ; I.;/ D ; R : ˆ: ; D Now using our results, we give some pplictions to specil mens for positive rel numers. Proposition. Let ; R ; <. Then, we hve h A 3 ; L. /.;/ i.3/.3/ Proof. The ssertion ws otined y the inequlity in (.) pplied to the conve mpping f W Œ;! R; f./ D : :

15 ON OMPANION OF OSTROWSKI INEQUALITY 7 Proposition. Let ; R ; < ; nd p Z; jpj :Then, Ap.;/ Lp p.;/ p. / p A p.;/ p : Proof. The ssertion ws otined y the inequlity in (.3) pplied to the conve mpping f W Œ;! R; f./ D p : Proposition 3. Let ; R ; <. Then, we hve jlni A.ln;ln/j. / q. p/ p q q : q Proof. The ssertion ws otined y the inequlity in (.6) pplied to the conve mpping f W Œ;! Œ;/; f./ D ln: REFERENES [] M. Alomri nd M. Drus, Some Ostrowski type inequlities for qusi-conve functions with pplictions to specil mens, RGMIA Preprint, vol. 3, no., p. rticle No. 3.,. [] M. Alomri, M. Drus, S. S. Drgomir, nd P. erone, Ostrowski type inequlities for functions whose derivtives re s-conve in the second sense, Appl. Mth. Lett., vol. 3, no. 9, pp. 7 76,. [3] M. W. Alomri, A compnion of Ostrowski s inequlity with pplictions, Trnsylv. J. Mth. Mech., vol. 3, no., pp. 9,. [] N. S. Brnett, S. S. Drgomir, nd I. Gomm, A compnion for the Ostrowski nd the generlised trpezoid inequlities, Mth. omput. Modelling, vol. 5, no. -, pp , 9. [5] S. S. Drgomir, A compnion of Ostrowski s inequlity for functions of ounded vrition nd pplictions, RGMIA Preprint, vol. 5, p. rticle No. 8,. [6] S. S. Drgomir, Some compnions of Ostrowski s inequlity for solutely continuous functions nd pplictions, Bull. Koren Mth. Soc., vol., no., pp. 3 3, 5. [7] S. S. Drgomir nd R. P. Agrwl, Two inequlities for differentile mppings nd pplictions to specil mens of rel numers nd to trpezoidl formul, Appl. Mth. Lett., vol., no. 5, pp. 9 95, 998. [8] S. S. Drgomir nd T. M. Rssis, Eds., Ostrowski type inequlities nd pplictions in numericl integrtion. Dordrecht: Kluwer Acdemic Pulishers,. [9] A. Guess nd G. Schmeisser, Shrp integrl inequlities of the Hermite-Hdmrd type, J. Approimtion Theory, vol. 5, no., pp. 6 88,. [] Z. Liu, Some compnions of n Ostrowski type inequlity nd pplictions, JIPAM, J. Inequl. Pure Appl. Mth., vol., no., p., 9. Authors ddresses Mohmmd W. Alomri Deprtment of Mthemtics, Fculty of Science, Jersh University, Jersh, Jordn E-mil ddress: mwomth@gmil.com

16 8 MOHAMMAD W. ALOMARI, M. EMIN ÖZDEMIR, AND HAVVA KAVURMA M. Emin Özdemir Ağrı İrhim Çeçen University, Fculty of Science nd Letters, Deprtment of Mthemtics,, Ağrı, Turkey E-mil ddress: Hvv Kvurmc Ağrı İrhim Çeçen University, Fculty of Science nd Letters, Deprtment of Mthemtics,, Ağrı, Turkey E-mil ddress: