SOME SHARP OSTROWSKI-GRÜSS TYPE INEQUALITIES

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1 Uiv. Beogrd. Publ. Elektroteh. Fk. Ser. Mt Avilble electroiclly t http: //pefmth.etf.bg.c.yu SOME SHARP OSTROWSKI-GRÜSS TYPE INEQUALITIES Zheg Liu Usig vrit of Grüss iequlity to give ew proof of well kow result o Ostrowski-Grüss type iequlities d shrpess of this iequlity is obtied. Moreover ew geerl shrp Ostrowski-Grüss type iequlity is give.. INTRODUCTION I 00 Cheg i [] hs improved d further geerlized some Ostrowski-Grüss type iequlities ivolvig bouded oce d twice differetible mppigs. I 00 lmost t the sme time Cheg d Su i [4] s well s Mtić i [5] hve estblished the followig vrit of Grüss iequlity. Lemm. Let h g : [ b] R be two itegrble fuctios such tht γ gt Γ for ll t [ b] where γ Γ R re costts. The b htgt ht b gt b Γ γ ht hy dy b. Moreover Mtić hs proved tht there eists fuctio g to tti the equlity i Ceroe d Drgomir hve proved i [] tht / i is shrp costt. I Theorem of [] Ceroe d Drgomir hve treted Theorem.5 of [] i more geerl setup by usig Lemm d obti Theorem. Let f : [ b] R be fuctio which is bsolutely cotiuous o [ b] d there eist costts γ Γ R such tht γ f t Γ for.e.t [ b]. The for ll [ b] we hve f fb f ft b b b 8 b Γ γ where the costt /8 is shrp. 000 Mthemtics Subject Clssifictio: 6D5 Keywords d Phrses: Ostrowski-Grüss type iequlity bsolutely cotiuous shrp boud. 4

2 Some shrp Ostrowski-Grüss type iequlities 5 I this pper we will lso tret Theorem.6 Theorem. d Theorem. of [] by usig Lemm to obti some shrp Ostrowski-Grüss type iequlities s follows: Theorem. Let f : [ b] R be such tht f is bsolutely cotiuous o [ b] d there eist costts γ Γ R such tht γ f t Γ for.e.t [ b]. The for ll [ b] we hve where f b f 4 b b b ft Γ γ G b f b f b G b b b b = b b / b b b b b b / b b. The iequlity with is shrp. Theorem. Let the ssumptios of Theorem hold. The for ll [ b] we hve 4 f bfb f ft b b The costt /8 is shrp. b Γ γ. 8b Theorem 4. Let the ssumptios of Theorem hold. The for ll [ b] we hve 5 f b f b f b f ft 6b b The costt 9 is shrp. 9 b Γ γ. b

3 6 Zheg Liu Here we hve give revised versio for 5 sice the epressio i [] cotied misprit. I Sectio we will use Lemm to provide ew proof of Theorem. Isted of provig Theorem d Theorem 4 i Sectio we will give ew geerl shrp Ostrowsky-Grüss type iequlity.. A NEW PROOF OF THEOREM We choose i ht = K t d gt = f t where K : [ b] R is give by t t < K t := t b t b. The we hve K t = b = 6 4 b b b d so ht b b hy dy = b t b t Deote t = b b /. It is cler tht < t < t < b. b I cse b ht b = t t b t hy dy = b 4 4 b b we see tht t d hece 4 b b. / d t = b b 4 b b 4 b b b t b b b b t /.

4 I cse b ht b = 4 Some shrp Ostrowski-Grüss type iequlities 7 b t t t t hy dy = t b we see tht t t d hece t 4 b b 4 b b 4 b 4 b b b /. b b t b t I cse b ht b = b we see tht t b d hece t t hy dy = t 4 b b 4 b b 4 b b b t b b b b t /. Thus by Lemm we c derive f b = b f 4 b K tf t b K t b f b f b b f t ft

5 8 Zheg Liu Γ γ b b b b b Γ γ b b b / Γ γ b b b b b / / b b b b b i.e. we hve obtied the iequlity with. It is ot difficult to fid tht the iequlity with is shrp. Ideed we c costruct the fuctio ft = t y jz dz dy to tti the equlity i where jt = jt = jt = γ t < Γ t < t γ t t b γ t < t Γ t t < t γ t t b γ t < t Γ t t < γ t b The proof of Theorem is complete. b b b b b.. A NEW GENERAL OSTROWSKY-GRÜSS TYPE INEQUALITY We eed the followig two itegrl idetities: Lemm []. Let f : [ b] R be such tht f is bsolutely cotiuous o [ b] for some. The for ll [ b] we hve the idetity: b ft = k k k f k K tf t k! k=0 where the kerel K : [ b] R is give by t t < K t :=! t b t b.!

6 Some shrp Ostrowski-Grüss type iequlities 9 Lemm. Let f : [ b] R be such tht f is bsolutely cotiuous o [ b] for some. The for ll [ b] we hve the idetity: 6 b ft = k k k f k k! k=0 b f! b f b f! H tf t where the kerel H : [ b] R is give by Proof. It is immedite tht t!! t < H t := t b b!! t b. H tf t = K tf t Cosequetly 6 follows from Lemm. Now let us observe tht H t = t! b! f b f b f!! t b Further deote t = d t = b < t < t < b. If is odd we get t H t =! t b! = t!!! t t! t b t b! t! b b = 0.! b. Clerly!. b!

7 0 Zheg Liu d if is eve we get t t H t = t!! t! t t b b b b!! t! =! b! t b! Thus by Lemm d Lemm we c obti geerl Ostrowsky-Grüss type iequlity s follows: Theorem 5. Let f : [ b] R be such tht f is bsolutely cotiuous o [ b] for some d there eist costts γ Γ R such tht γ f t Γ for.e.t [ b]. The for ll [ b] we hve 7 f b f!b b k k k k!b f k k= b f b f ft!b b! b Γ γ. The equlity i 7 is ttied by choosig ft = t y y jy dy dy dy where jt = if is odd d if is eve. γ t t = Γ t t < γ t < t = b Γ t t b γ t < t Γ jt = t t < Γ t t γ t t b b

8 Some shrp Ostrowski-Grüss type iequlities Remrk. It is esy to fid tht Theorem 5 reduces to Theorem or Theorem 4 if put = or = d by the wy the shrpess of iequlities 4 d 5 re proved. Ackowledgmet. The uthor wishes to thk the referee for his istructios d iformtio bout the referece [] which the uthor hs bee uwre of. REFERENCES. P. Ceroe S. S Drgomir: Midpoit type rules from iequlities poit of view. Hdbook of Alytic-Computtiol Methods i Applied Mthemtics. CRC Press N.Y P. Ceroe S. S. Drgomir: A refiemet of the Grüss iequlity d pplictios. RGMIA Reserch Report Collectio 5 00 Article 4.. X. L. Cheg: Improvemet of some Ostrowsky-Grüss type iequlities. Computers Mth. Applic X. L. Cheg J. Su: A ote o the perturbed trpezoidl iequlity. Jourl of Iequlities i Pure d Applied Mthemtics 00 issue Article M. Mtić: Improvemet of some estimtios relted to the remider i geerlized Tylor s formul. Mthemticl Iequlities & Applictios Istitute of Applied Mthemtics Received Mrch Fculty of Sciece Lioig Uiversity of Sciece d Techology Ash 4044 Lioig Chi E mil: lewzheg@6.et