Improvement of Grüss and Ostrowski Type Inequalities
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1 Filomt 9:9 (05), DOI 098/FIL50907A Pulished y Fculty of Sciences nd Mthemtics, University of Niš, Seri Aville t: Improvement of Grüss nd Ostrowski Type Inequlities An Mri Acu University Lucin Blg of Siiu, Deprtment of Mthemtics nd Informtics, Str Dr I Rtiu, No5-7, RO-5500 Siiu, Romni Astrct Severl inequlities of Ostrowski-Grüss-type ville in the literture re generlized considering the weighted cse of them The inequlity of Grüss type proved y P Cerone nd SS Drgomir [3] is extended for the weighted cse Introduction In 935 Grüss [8] proved integrl inequlity tht estlishes connection etween the integrl of the product of two functions nd the product of the integrls In 938, Ostrowski [3] estlished n interesting integrl inequlity which gives n upper ound for the pproximtion of the integrl verge y the vlue of mpping in certin point of the intervl In 997, Drgomir nd Wng [6] comined the Ostrowski inequlity with Grüss inequlity nd otined new result for ounded differentile mppings which is well known in the literture s Ostrowski-Grüss inequlity In 000, B Gvre nd IGvre [7] otined some generliztions of these inequlities using the lest concve mjornt of the modulus of continuity nd the second order modulus of smoothness During the lst few yers, mny reserchers focused their ttention on the study nd generliztions of these inequlities ([3], [], [9], [0], [], [], [5]) In this pper we generlize these type of inequlities considering the weighted cse of them nd we improve some Ostrowski-Grüss type inequlities ville in the literture The functionl given y T( f, ) := f (t) (t)dt f (t)dt (t)dt, () where f, : [, ] R re integrle functions, is well known in the literture s the Cheyshev functionl (see []) In 935, G Grüss [8] otined the following result Theorem Let f nd e two functions defined nd integrle on [, ] If m f (x) M nd p (x) P for ll x [, ], then we hve T( f, ) (M m)(p p) The constnt / is the est possile () 00 Mthemtics Suject Clssifiction Primry 6D0; Secondry 65D30, 6A5 Keywords Ostrowki type inequlity, Grüss type inequlity, modulus of continuity Received: 05 Ferury 0; Accepted: Ferury 0 Communicted y Drgn Djordjević Emil ddress: cun77@yhoocom (An Mri Acu)
2 AM Acu / Filomt 9:9 (05), Another celerted clssicl inequlity ws proved y A Ostrowski [3] in 938, which we cite elow in the form given y GA Anstssiou in 995 (see []) Theorem Let f e in C [, ], x [, ] Then f (x) f (t)dt (x ) + ( x) f (3) () t, t [, x] In 997, SS Drgomir nd S Wng [6] pplied Theorem to the mppings f (t) nd p(x, t)= t, t (x, ] otining new result for ounded differentile mppings, s shown in the reltion (6), which is known s the Ostrowski-Grüss-type inequlity This inequlity hs een improved y M Mtić et l ([]), nd we recll their result in (5) An improvement of this result is given y XL Cheng in [5], s shown in reltion () He lso proved tht the constnt /8 is shrp Theorem 3 Let f : [, ] R e differentile mpping If γ f (x) Γ, x [, ] for some constnts γ, Γ R, then L( f )(x) ()(Γ γ) 8 () ()(Γ γ) 3 (5) ()(Γ γ), (6) where L( f )(x) := f (x) f (t)dt f () f () ( x + In [7], N Ujević proved the following result, involving the second derivtive of the mpping f Theorem (see Theorem in [7]) Let f : I R, where I R n intervl, e twice continuously differentile mpping in the interior Int(I) of I with f L (, ), nd let, Int(I), < Then we hve, for ll x [, ], L( f )(x) () 3/ π 3 f In this pper we will consider the weighted vrint of the functionl L nd some new inequlities which involve second derivtive of mpping re proved The weight function is ssumed to e non-negtive nd integrle over its entire domin In order to formulte the min results we will define the following: Definition 5 Let w : (, ) (0, ) e integrle, ie, M(α, β) := β α tw(t)dt nd M (α, β) := β ) of the intervl [α, β] with respect to the weight function w s σ(α, β) := α w(t)dt <, then m(α, β) := β α (7) w(t)dt, t w(t)dt re the first moments, for [α, β] [, ] Define the men M(α, β) m(α, β) The weighted vrint of the functionl L cn e written in the following wy L w ( f )(x) := f (x) f (t)w(t)dt m(, ) f () f () (x σ(, )) The structure of this pper is s follows: in Section we give new ounds for the functionl L w nd improve some inequlities ville in literture In Section 3 involving the lest concve mjornt of the modulus of continuity we provide new estimtions of the functionl L w Finlly, in Section we extend for the weighted cse Grüss-type inequlity proved y P Cerone nd SS Drgomir [3]
3 Ostrowski-Grüss-Type Inequlities AM Acu / Filomt 9:9 (05), The im of this section is to give new inequlities for the functionl L w involving the second derivtive of mpping Denote m(, ) K(x, t) := m(, ) t t (u t)w(u)du t (x σ(, )), t [, x), (u t)w(u)du t (x σ(, )), t [x, ] Theorem Let f : [, ] R e twice differentile on the intervl (, ), with the second derivtive ounded on (,), ie, f := sup f (t) < Then, for ll x [, ], we get t (,) L w ( f )(x) u(x;, ) f, where u(x;, ) := K(x, t) dt (9) Proof Integrting y prts, we hve K(x, t) f (t)dt = L w ( f )(x) Therefore, we get L w ( f )(x) = K(x, t) f (t)dt f K(x, t) dt = f u(x;, ) (8) Theorem Let f : [, ] R e twice differentile on the intervl (, ), with f L [, ] Then, for ll x [, ], we hve [ L w ( f )(x) µ(x;, ) f, where µ(x;, ) = (K(x, t)) dt Proof If we consider the function K(x, t) defined in (8), it follows [ L w ( f )(x) = K(x, t) f ]/ (t)dt f (K(x, t)) dt = µ(x;, ) f ]/ (0) Theorem 3 Let f : [, ] R e twice differentile on the intervl (, ), with f L [, ] Then, for ll x [, ], we hve L w ( f )(x) ν(x;, ) f, where ν(x;, ) := sup K(x, t) () t [,] Proof If we consider the function K(x, t) defined y (8), it follows L x [ f ] = K(x, t) f (t)dt f sup K(x, t) = ν(x;, ) f t [,] Remrk For = 0, = nd w(x) = the inequlities (9)-() nd (7) ecome L( f )(x) C (x) f f, where ()
4 AM Acu / Filomt 9:9 (05), [ 6x 3 + 8x 6x + ], x [0, /] C (x) = [ 6x 3 30x + 8x 3 ], x (/, ], L( f )(x) C (x) f f 0093 f, where (3) x C (x) = x3 + x 3 x + 0, L( f )(x) C 3 (x) f 6 f where () C 3 (x) = ( x [, x ) x ( x), x [0, ] \, + ] [, + ], L( f )(x) π 3 f 0099 f (5) In this prticulr cse, our estimtes (3) is etter thn N Ujević s result (5) 3 Ostrowski-Grüss-Type Inequlities in Terms of the Lest Concve Mjornt In order to formulte the next result we need the following Definition 3 Let f C[, ] If, for t [0, ), the quntity ω( f ; t) = sup f (x) f (y), x y t is the usul modulus of continuity, its lest concve mjornt is given y (t x)ω( f ; y) + (y t)ω( f ; x) ω( f ; t) = sup ; 0 x t y, x y y x Let I = [, ] e compct intervl of the rel xis nd f C(I) In [6], the following result for the lest concve mjornt is proved: ( ) ( t K, f ; C[, ], C [, ] := inf f + t ) = ω( f ; t), t 0 C (I) Before we proceed to the result of this section we recll the following lemm estlished y J Roumeliotis in [5] Lemm 3 [5] There exists t [, ] uniquely stisfying m(, ) x σ(, ) = m(t, ), x σ(, ), m(, t ), σ(, ) < x (6) Denote P(x, t) = m(, ) m(, ) t t w(u)du (x σ(, )), t [, x), w(u)du (x σ(, )), t [x, ]
5 Lemm 33 If ũ(x;, ) := ũ(x;, ) = AM Acu / Filomt 9:9 (05), P(x, t) dt, then t (t x)w(t)dt, for ll x [, ] nd t s defined in Lemm 3 (7) m(, ) x Proof If x σ(, ), then P(x, t) 0, for t [, x] [t, ] nd P(x, t) 0, for t [x, t ] Also, if σ(, ) x we hve P(x, t) 0, for t [, t ] [x, ] nd P(x, t) 0, for t [t, x] From proof of Theorem 3 ([5]) it follows ũ(x;, ) = x t t P(x, t)dt + P(x, t)dt P(x, t)dt = (t x)w(t)dt, for ll x [, σ(, )] t x m(, ) x In similr wy the identity (7) cn e proved for x (σ(, ), ] Lemm 3 There exists t [, ] uniquely stisfying t (u t )w(u)du = t (x σ(, )), for x σ(, ), nd (8) m(, ) t (u t )w(u)du = t (x σ(, )), for σ(, ) < x (9) m(, ) Proof Let us consider x σ(, ) nd f (t) = (u t)w(u)du t (x σ(, )), t [x, ] m(, ) t We hve f (t) = w(u)du (x σ(, )) m(, ) t Since f (t) 0 on t [x, t ] nd f (t) 0 on t [t, ], where t is defined in Lemm 3, then to show tht t [x, t ] exists such the identity (8) holds it will suffice to estlish tht f (x) 0 We hve f (x) = m(, ) x (u x)w(u)du x (x σ(, )) m(, ) = (σ(, ) x) x x (x σ(, )) = (σ(, ) x) 0 Similrly, we cn show (9) to e true for σ(, ) < x (u x)w(u)du x (x σ(, )) Lemm 35 If u(x;, ) := K(x, t) dt, then for ll x [, ] nd t s defined in Lemm 3, it follows x σ(, ) u(x;, ) = t ()x + xσ(, ) + x M (t, ) m(, ) M (, ) t m(t, ), for x σ(, ) (0) nd x σ(, ) u(x;, ) = t ()x + + xσ(, ) + x M (, t ) m(, ) M (, ) t m(, t ), for σ(, ) < x ()
6 AM Acu / Filomt 9:9 (05), Proof If x σ(, ), then K(x, t) 0, for t [, x] (x, t ] nd K(x, t) < 0, for t (t, ] () Also, if σ(, ) x we hve K(x, t) 0, for t [t, x] (x, ] nd K(x, t) < 0, for t [, t ) (3) From (), (3) nd Lemm 3 the reltions (0) nd () re proved Theorem 36 If f C [, ], then ( ) ũ(x;, ) L w ( f )(x) ω f u(x;, ) ;, () ũ(x;, ) where ũ nd u re defined in Lemm 33 nd Lemm 35 Proof Let A : C[, ] R e defined y A( f )(x) = A( f )(x) f ũ(x;, ) P(x, t) f (t)dt We hve (5) Let C [, ] nd K(x, t) e the mpping defined y (8) We otin K(x, t) (t)dt = A( )(x), nmely A( )(x) = K(x, t) (t)dt K(x, t) dt = u(x;, ) (6) From reltions (5) nd (6), we hve A( f )(x) = A( f + )(x) A( f )(x) + A( )(x) f ũ(x;, ) + u(x;, ) u(x;, ) ũ(x;, ) inf f + C [,] ũ(x;, ) Therefore, A( f )(x) ũ(x;, ) ω ( f ; ) u(x;, ) (7) ũ(x;, ) If we write (7) for the function f, we otin inequlity () Theorem 37 If f C[, ], then for ll x [, ], we hve ( ) u(x;, ) L( f )(x) K ; f ; C[, ], C [, ], where (8) K ( t; f ; C[, ], C [, ] ) := inf f + t nd u is defined in Lemm 35 C [,] Proof For ny f C[, ], L w ( f )(x) f For C [, ], from Theorem we get L w ( )(x) u(x;, ) So, for f C[, ] fixed nd C [, ] ritrry, we hve L w ( f )(x) L w ( f )(x) + L w ( )(x) f + u(x;, ) u(x;, ) = f + Pssing to the infimum over C [, ] gives reltion (8)
7 The Weighted Grüss Type Inequlities AM Acu / Filomt 9:9 (05), In [3], P Cerone nd SS Drgomir proved the following inequlity for functionl T( f, ) Theorem ([3])Assume tht f : [, ] R is mesurle function on [, ] nd such tht f := f f (t)dt, e f L [, ], where e(t) = t, t [, ] If : [, ] R is solutely continuous nd L [, ], then we hve the inequlity T( f, ) f (t) dt t f ( (t) dt t f (t) dt f (t) dt ) / e f In this section we will prove Grüss type inequlity for the weighted cse Let w : [, ] [0, ) weight function nd we will consider tht w is ounded function, nmely exists K R such tht w(x) K for ll x [, ] Denote T w ( f, ) := f (x) (x)w(x)dx w(x)dx the Čeyšev functionl for the weighted cse f (x)w(x)dx (x)w(x)dx ( ), w(x)dx Consider in Hilert spce L,w [, ] the inner product f, w = ( )/ f,w = f (x) w(x)dx Lemm Assume tht function f : [, ] R is Leesque integrle nd Define F(x) = x f (x) (x)w(x)dx with the norm f (t)w(t)dt, e(x) = x, x [, ] nd ssume tht F, f, e f L,w [, ] Then we hve f (x)w(x)dx = 0 F,w K f,w f e,w ( f, f e w) K f e f,w (9),w Proof We hve F,w = F(x) w(x)dx K F(x) dx = K F(x) (x λ) dx = K F(x) (x λ) F(x) f (x)w(x)(x λ)dx = K (λ x)f(x) f (x)w(x)dx ( )/ ( )/ ( K (λ x) f (x) w(x)dx F(x) w(x)dx = K F,w (λ x) f (x) w(x)dx From the ove reltion we hve ( F,w K (λ x) f (x) w(x)dx nmely )/ F,w K (λ x) f (x) w(x)dx, )/
8 AM Acu / Filomt 9:9 (05), Denote (λ) := (λ x) f (x) w(x)dx = λ f,w λ f, e f w + f e,w Since inf (λ) = f,w f e,w ( f, f e w), we otin the reltion (9) λ R f,w Corollry 3 Assume tht f stisfies the ssumption in Lemm nd 0 < < Then we hve the inequlity F,w K () ( f, e f w) f,w () K e f,w Proof We use the following integrl version of Cssels inequlity p(t)l(t) dt p(t)h(t) dt ( ) p(t)l(t)h(t)dt (M + m) mm, (30) provided 0 < m h(t) M < et t [, ] nd p 0 et on [, ] l(t) Applying (30) for p(t) = f (t) w(t), l(t) =, h(t) = t, t [, ] we get f,w f e,w ( f, f e w ) ( + ), nmely Using reltion (9) we hve f,w f e,w ( f, f e w ) () ( f, f e w ) F,w K () ( f, f e w ) f,w () K e f,w Theorem Assume tht f : [, ] R is mesurle function on [, ] nd such tht f := f w(t)dt e f L,w [, ] If : [, ] R is solutely continuous nd L,w [, ], then we hve the inequlity T w ( f, ) Proof Denote We hve But F() = K,w w(t)dt w(t)dt f (t) w(u)du [ f,w e f,w f, e f ] / w f,w F(x) (x)dx = F(x) = x f (t)w(t)dt, x [, ] F(x) (x) w(t)dt f (u)w(u)du w(t)dt= f (t)w(t)dt K,w e w(t)dt w(t)dt w(u)du f (x)w(x) (x)dx f (t)w(t)dt, f,w (3) w(t)dt f (u)w(u)du=0
9 Therefore F(x) (x)dx = w(t)dt w(t)dt = f (x) (x)w(x)dx w(t)dt From the ove reltion we cn write From Lemm we otin the inequlity (3) AM Acu / Filomt 9:9 (05), w(t)dt f (x) w(t)dt f (t)w(t)dt T w ( f, ) F,w,w w(t)dt f (t)w(t)dt w(x) (x)dx Remrk 5 For w(x) = we hve the results otined y Cerone nd Drgomir in [3] (x)w(x)dx = T w( f, ) References [] GA Anstssiou, Ostrowski type inequlities, Proc Amer Mth Soc, 3 (995), [] PL Cheyshev, Sur les expressions pproximtives des intégrles définies pr les utres prises entre les mêmes limites, Proc Mth Soc Khrkov, (88), (Russin), trnslted in Oeuvres, (907), [3] P Cerone, SS Drgomir, New ounds for the Čeyšev functionl, Applied Mthemtics Letters 8(005), [] P Cerone, SS Drgomir, Mthemticl Inequlities:A Perspective, CRC Press, Tylor & Frncis Group, Boc Rton, London, New York, 0 [5] XL Cheng, Improvement of some Ostrowski-Grüss type inequlities, Comput Mth Appl, (00), 09- [6] SS Drgomir, S Wng, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtion of error ounds for some specil mens nd for some numericl qudrture rules, Computers Mth Applic, 33(997), 6-0 [7] B Gvre, I Gvre, Ostrowski type inequlities from liner functionl point of view, J Inequl Pure Appl Mth, (000), rticle [8] GGrüss, Üer ds Mximum des soluten Betrges von f (x) (x)dx ( ) f (x)dx (x)dx, Mth Z, 39 (935), 5 6 [9] Z Liu, Some inequlities of Ostrowski type nd pplictions, Applied Mthemtics E-Notes, 7(007), 93-0 [0] Z Liu, Some Ostrowski-Grüss type inequlities nd pplictions, Computers nd Mthemtics with Applictions, 53 (007), [] W-J Liu, Q-L Xue, S-F Wng, New generliztion of pertured Ostrowski type inequlities nd pplictions, J Appl Mth Comput 3(00), [] M Mtić, J Pečrić, N Ujević, Improvement nd further generliztion of some inequlities of Ostrowski Grüss type, Computers Mth Applic, 39(000) (3/), 6-75 [3] A Ostrowski, Üer die Asolutweichung einer differentiierren Funktion von ihrem Integrlmittelwert, Comment Mth Helv, 0(938), 6-7 [] J Roumeliotis, P Cerone, SS Drgomir, An Ostrowski type inequlity for weighted mppings with ounded second derivtives, RGMIA Reserch Report Collection, Vol, No, 998, 0- [5] J Roumeliotis, Improved weighted Ostrowski-Grüss type inequlities, Inequlity Theory nd Applictions, Vol3, Nov Sci Pul, Huppuge, NY, 003, [6] EM Semenov, BS Mitjgin, Lck of interpoltion of liner opertors in spces of smooth functions, Mth USSR-Izv, (977), 9-66 [7] N Ujević, New ounds for the first inequlity of Ostrowski-Grüss type nd pplictions, Comput Mth Appl, 6(003), No -3, -7
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