Weighted Inequalities for Riemann-Stieltjes Integrals
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1 Aville hp://pvm.e/m Appl. Appl. Mh. ISSN: ol. Ie Decemer 06 pp Applicion n Applie Mhemic: An Inernionl Jornl AAM Weighe Ineqliie or Riemnn-Sielje Inegrl Hüeyin Bk n Mehme Zeki Sriky Deprmen o Mhemic Fcly o Science n Ar Düzce Univeriy Düzce-Trkey hyn.k@gmil.com; rikymz@gmil.com Receive: My 5 06; Accepe: Ocoer 3 06 Arc In hi pper ir e eine ne ncionl hich i eighe verion o he ncionl eine y Drgomir n Feoov. Then ome ineqliie involving hi ncionl re oine. Finlly e pply hi rel o elih ne on or eighe Cheyev ncionl. Keyor: Fncion o one vriion; Oroki ype ineqliie; Riemnn- Sielje inegrl MSC 00 No.: 6D5 6A45 6D0. Inrocion The olloing einiion ill e reqenly e o prove or rel. Deiniion.. Le P : x0 x... xn e ny priion o n le xi xi xi hen i i o e o one vriion i he m 856
2 AAM: Inern. J. ol. Ie Decemer i one or ll ch priion. m xi i Deiniion.. Le e o one vriion on o he priion P o. The nmer i clle he ol vriion o on.. n P enoe he m x : p P : P P Here P Drgomir n Feoov 998 hve elihe he olloing ncionl D. In he me pper he hor prove he olloing ineqliy. Theorem. Le : R e ch h i o one vriion on he conn L 0. Then e hve D L n i i correponing enoe he mily o priion o. The conn i hrp in he ene h i cnno e replce y mller one. Alomri 0 gve he olloing ineqliy. Theorem. Le x. Le : R e conino mpping on monoonic non-ecreing mpping on n R oh inervl x n x. Then e hve he ineqliy n i Lipchizin ih. Ame h i : i monoonic nonecreing on
3 858 H. Bk e l.. D Drgomir 04 gve ome ne on or he ncion D. One o hem i olloing ineqliy. Theorem 3. Ame h R : re o one vriion n ch h he Riemnn-Sielje inegrl exi. Then. D Lemm. Le. C : I i conino on n i o one vriion on hen he Riemnn-Sielje inegrl exi n. mx A gre mny hor orke on ineqliie or Riemnn-Sielje inegrl vi ncion o one vriion or erivive o one vriion. For ome o hem plee ee in Alomri 0-Li 004. The min prpoe o hi pper i o oin ome eighe ineqliie or Riemnn-Sielje inegrl. Fir o ll e eine eighe verion o he ncionl D. Then e elih ome on or hi ncionl ccoring o ce o he ncion n. Finlly ome ne on or he eighe Cheyev ncionl re lo given.
4 AAM: Inern. J. ol. Ie Decemer Thi pper i ivie ino he olloing ix ecion. In Secion e elih ome ieniie h ill e e o prove or rel. In Secion 3 n Secion 4 ome eighe inegrl ineqliie or he ce hen he ncion i one vriion n hen he ncion i one vriion re given repecively. In he nex ecion e give n ineqliy or he ce hen i l L Lipchizn. Finlly in Secion 6 e preen ome pplicion or eighe Cheyev ncionl ing he rel given in previo ecion.. Some Ieniie Le : R e nonnegive n conino on. We eine m n m o h m 0 or. No e give ome repreenion. Weighe verion o he ncionl eine y Drgomir n Feoov: Weighe Cheyev ncionl: D m. T g m g m m g. Weighe Oroki rnorm: Weighe generlize rpezoi rnorm: m. m g m g g m g. Beore e r or min rel e e n prove olloing lemm: Lemm. re one ncion ch h he Riemnn-Sielje inegrl I : R
5 860 H. Bk e l. n he Riemnn inegrl exi hen e hve D Q here x Q x. Proo: Uing he inegrion y pr in Riemnn-Sielje inegrl e hve Q. Thi complee he proo o he econ eqliy. The ir ieniy i ovio.
6 AAM: Inern. J. ol. Ie Decemer Corollry. Le R g : ncion ch h g i Riemnn inegrle on. I e chooe g in Lemm hen e hve. g Q m g m g T Lemm 3. Wih he mpion in Lemm e hve Q D here he mpping Q i eine y in Lemm. Proo: By he Fini ype heorem or he Riemnn-Sielje inegrl e ge. Q Q Thi complee he proo o he ir n he l erm in. Inegring y pr e oin. m m m Q Thi complee he proo. Corollry.
7 86 H. Bk e l. Ame h g R : Riemnn inegrle on hen e hve g. g m g m g. Remrk. I e chooe in Lemm n Lemm hen or rel rece Lemm n Lemm prove y Drgomir 04 repecively. 3. Ineqliie in he Ce hen i o one vriion No ing he ove ieniie e e n prove he olloing ineqliie in he ce hen i o one vriion. Theorem 4. Le R : e nonnegive n conino on. hen e hve he ineqliie vriion on D m m I : R re o one m m m m m. Proo: Tking he mol in Lemm n ing he Lemm e hve D Q
8 AAM: Inern. J. ol. Ie Decemer Q 4 Since i o one vriion ing Lemm l gin e oin m 5 n. m 6 I e ie he ineqliie 5 n 6 in 4 e elih D m m
9 864 H. Bk e l. m m In l line o 7 e hve m m I e p he eqliy 8 in 7 e oin he ir ineqliy in 3. The oher ineqliie re ovio rom he c h m. Remrk. I e chooe in Theorem 4 hen e oin Theorem in Drgomir 04. Theorem 5.
10 AAM: Inern. J. ol. Ie Decemer Le : R e nonnegive n conino on. I : R i o one vriion on n : R i monoonic nonecreing hen e hve he ineqliy Proo: D m m m m. 9 I i ell knon h i he Sielje inegrl p v n p v exi n v i monoonic non-ecreing on hen 0 p v p v. Uing he ineqliy 0 e hve n m
11 866 H. Bk e l.. m I e ie he ineqliie n in 4 e oin D m m 3 m m. Uing he inegrion y pr in Riemnn-Sielje inegrl e hve 4 n. 5 Ping he eqliie 4 n 5 in 3 e complee he proo he ir ineqliy in 9. The econ ineqliy i ovio. Remrk 3.
12 AAM: Inern. J. ol. Ie Decemer I e chooe in Theorem 5 hen he ir ineqliy in 9 rece o he ineqliy 3.7 in Drgomir Ineqliie in he Ce hen i o one vriion In hi ecion e give me ineqliy in he ce hen i o one vriion ing he ieniie preene in Secion. Theorem 6. Le : R e nonnegive n conino on n R one vriion on. I R 0 n L 0 ih n L or ll hen e hve Proo: D : e ncion o : i conino ch h here exi conn L 6 L 7 L m L m. Tking he mol in Lemm 3 n ing Lemm e hve D 8 Q
13 868 H. Bk e l. m m. 9 Uing properie 6 n 7 in 9 e oin D Lm Lm. 0 L m L m Inegring y pr e hve n m m m m m m m m Thee eqliie complee he proo. Remrk 4. I e chooe in Theorem 6 hen e oin Theorem 4 in Drgomir 04.
14 AAM: Inern. J. ol. Ie Decemer Corollry 3. Le n e in Theorem 6. I i o r H Höler ype i.e. here 0 H n 0 r H or ny r re given hen D r r H r r r m m. Corollry 4. I i Lipchizin ih he conn L 0 hen e hve D m m L. 5. Ineqliie or l L -Lipchizn Fcion The olloing lemm given y Drgomir 04. Lemm 4. : n l LR ih L l. The olloing emen re eqivlen: Le R l L i The ncion. here e ii We hve he ineqliie e l L or ech i L l -Lipchizn; ; iii We hve he ineqliie l L or ech.
15 870 H. Bk e l. Deiniion 3. The ncion : R hich iie one o he eqivlen coniion i - iii rom Lemm l3 i i o e l L -Lipchizn on. I L 0 n l L hen L L - Lipchizn men L -Lipchizn in he clicl ene. Theorem 7. Le : R e nonnegive n conino on n R one vriion on. I : R i n l L ineqliy l L D : e ncion o -Lipchizn ncion hen e hve he m m L l. Proo: From Lemm e hve l l D. e l l l l l L D. Applying Corollry 4 or he ncion l l. e hich i L l -Lipchizin e hve
16 AAM: Inern. J. ol. Ie Decemer l l D. e hich complee he proo. Remrk 5. m m L l I e chooe in Theorem 7 hen e oin Theorem 5 in Drgomir Bon For Weighe Cheyev Fncionl In hi ecion e pply he or rel or he eighe Cheyev ncionl. From Secion e kno h y chooing he g in Lemm. T g D m Moreover i o one vriion on ny inervl n g i conino on hen e hve g. 3 Propoiion. I i o one vriion on hen e hve he ineqliy T g m m g g m
17 87 H. Bk e l. g m m g m g m m g m m g. Proo: I chooe g in Theorem 4 n e he ieniy n e cn prove he reqire rel eily. Propoiion. I i monoonic non-ecreing on hen e hve he ineqliy T g m m m g g g m m m g g. Proo:
18 AAM: Inern. J. ol. Ie Decemer The proo i ovio rom Theorem Conclion Some explici error on re knon or Cheyev ncionl. In hi pper y ing he ie o Drgomir 04 e elih ome eighe verion o inegrl ineqliie oine in Drgomir 04 The meho e in hi pper migh in ome poenil pplicion in he generlizion o ome oher inegrl ineqliie. To o o one hol eine ome ne ncionl e eine in Secion. REFERENCES Alomri M.W. 0. Some Grü ype ineqliie or Riemnn-Sielje inegrl n pplicion. Ac Mh. Univ. Comenine LXXXI -0. Alomri M.W. n Drgomir S.S. 03. Mercer--Trpezoi rle or he Riemnn--Sielje inegrl ih pplicion. Jornl o Avnce in Mhemic Alomri M.W. 04. Dierence eeen o Sielje inegrl men. Krgjevc Jornl o Mhemic Alomri M.W. n Drgomir S.S. 04. Ne Grü ype ineqliie or Riemnn-Sielje inegrl ih monoonic inegror n pplicion. Ann. Fnc. Anl Alomri M.W. n Drgomir S.S. 04. Some Grü ype ineqliie or he Riemnn- Sielje inegrl ih Lipchizin inegror. Konrlp J. Mh Brne N.S. Cheng W.-S. Drgomir S.S. n Soo A Oroki n rpezoi ype ineqliie or he Sielje inegrl ih Lipchizin inegrn or inegror. Comp. Mh. Appl Bk H. n Srky M.Z. 06. On generlizion o Drgomir' ineqliie. Erin Mhemicl Jornl in pre. Bk H. n Srky M.Z. 06. Ne eighe Oroki ype ineqliie or mpping ih ir erivive o one vriion. Trnylvnin Jornl o Mhemic n Mechnic 8-7. Bk H. n Srky M.Z. 05. A ne generlizion o Oroki ype ineqliy or mpping o one vriion. RGMIA Reerch Repor Collecion 8 Aricle 47 9 pp. Bk H. n Srky M.Z. 06. A ne Oroki ype ineqliy or ncion hoe ir erivive re o one vriion. Moroccn Jornl o Pre n Applie Anlyi -. Bk H. n Srky M.Z. 06. A compnion o Oroki ype ineqliie or mpping o one vriion n ome pplicion. RGMIA Reerch Repor Collecion 9 Aricle 4 0 pp.
19 874 H. Bk e l. Bk H. Srky M.Z. n Qyym A. 06. Improvemen in compnion o Oroki ype ineqliie or mpping hoe ir erivive re o one vriion n pplicion. RGMIA Reerch Repor Collecion 9 Aricle 5 pp. Cerone P. n Drgomir S.S Dierence eeen men ih on rom Riemnn- Sielje inegrl. Comp. Mh. Appl Cerone P. Cheng W.S. n Drgomir S.S On Oroki ype ineqliie or Sielje inegrl ih olely conino inegrn n inegror o one vriion. Comp. Mh. Appl Cerone P. n Drgomir S.S. 00. Ne on or he hree-poin rle involving he Riemnn-Sielje inegrl in: C. Gli e l. E. Avnce in Siic Cominoric n Rele Are Worl Science Plihing pp Cerone P. n Drgomir S.S Approximing he Riemnn--Sielje inegrl vi ome momen o he inegrn. Mhemicl n Comper Moelling Drgomir S. S. n Feoov I 998. An ineqliy o Gr ype or RiemnnSielje inegrl n pplicion or pecil men. Tmkng J. Mh Drgomir S. S. n Feoov I 00. A Grü ype ineqliy or mpping o one vriion n pplicion o nmericl nlyi. Nonliner Fnc. Anl. Appl Drgomir S. S The Oroki inegrl ineqliy or mpping o one vriion. Bllein o he Arlin Mhemicl Sociey Drgomir S. S. 00. On he Oroki' inegrl ineqliy or mpping ih one vriion n pplicion. Mhemicl Ineqliie & Applicion Drgomir S. S Ineqliie o Grü ype or he Sielje inegrl n pplicion. Krgjevc J. Mh Drgomir S. S. 04. Approximing he Riemnn-Sielje inegrl vi Cheyhev ype ncionl. Ac Commen. Univ. Tr. Mh Drgomir S.S. Be C. Bole M.. n Brec L. 00. A generliion o he rpezoi rle or he Riemnn-Sielje inegrl n pplicion. Nonliner Anl. Form Drgomir S. S Some perre Oroki ype ineqliie or ncion o one vriion Preprin RGMIA Reerch Repor Collecion 6 03 Ar. 93. Drgomir S. S Approximing rel ncion hich poe nh erivive o one vriion n pplicion. Comper n Mhemic ih Applicion Li Z Reinemen o n ineqliy o Grü ype or Riemnn-Sielje inegrl. Soocho J. Mh
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