AN INEQUALITY OF GRÜSS TYPE FOR RIEMANN-STIELTJES INTEGRAL AND APPLICATIONS FOR SPECIAL MEANS

Transcription

1 RGMIA Reserch Report Collectio, Vol., No., rgmi/reports.html AN INEQUALITY OF GRÜSS TYPE FOR RIEMANN-STIELTJES INTEGRAL AND APPLICATIONS FOR SPECIAL MEANS S.S. DRAGOMIR AND I. FEDOTOV Astrct. I this pper we derive ew iequlity of Grüss type for Riem- Stieltjes itegrl d pply it for specil mes (logrithmic me, idetric me, etc...). Itroductio I 935, G. Grüss proved the followig iequlity which estlishes coectio etwee the itegrl of the product of two fuctios d the product of the itegrls: f(x)g(x)dx f(x)dx g(x)dx (Φ φ)(γ γ) 4 provided tht f d g re two itegrle fuctios o [, ] d stisfyig the coditio φ f(x) Φ d γ g(x) Γ for ll x [, ]. The costt 4 is the est possile d is chieved for ( f(x) = g(x) = sg x + ). For other similr results, geerliztios for positive lier fuctiols, discrete versios, determitl versios etc. see the Chpter X of the ook [] due to Mitriović, Pečrić d Fik where further refereces re give. I this pper we poit out Grüss type iequlity for Riem-Stieltjes itegrl d pply it for specil mes, i.e., logrithmic me, idetric me, etc... The Results The followig result of Grüss type holds: Dte. Novemer, Mthemtics Suject Clssifictio. Primry 6D5; Secodry 6Dxx. Key words d phrses. Grüss iequlity, Riem-Stieltjes itegrl. 89

2 90 Drgomir d Fedotov Theorem.. Let f, u : [, ] R e so tht u is L-lipschitzi o [, ], i.e., (.) u(x) u(y) L x y for ll x, y [, ], f is Riem itegrle o [, ] d there exists the rel umers m, M so tht (.) m f(x) M for ll x [, ]. The we hve the iequlity (.3) u() u() f(x)du(x) L (M m) ( ), d the costt is shrp, i the sese tht it c ot e replced y smller oe. Proof. First of ll let oserve tht if v is L-lipschitzi mppig d q is Riem itegrle o [, ], the (.4) q(x)dv(x) L q(t) dt. Ideed, if := x 0 < x <... < ( x < x ) = is sequece of prtitios of [, ] with ν ( ) := mx i=0, x i+ x i 0 (for ) d ξ [ ] i x i, x i+ the q(x)dv(x) = lim ν( ) 0 i=0 L lim lim ν( ) 0 i=0 q (ξ i ) ( v ( x ) i+ v (x i ) ) ( q(ξ v(x i ) i+ ) v(x i )) x i+ x i (x i+ x i ) ν( ) 0 i=0 Now, let oserve tht (.5) f(x)du(x) = q (ξ i ) ( x i+ x i u() u() (f(x) ) = L )du(x) q(t) dt. RGMIA Reserch Report Collectio, Vol., No., 998

3 Grüss Iequlity for Riem-Stieltjes Itegrl 9 L f(x) dx. Now, defie I := f(x) dx. The we hve I = f (x) f(x) = f (x)dx + dx d I = M m (M f(t)) (f(t) m) dt. As m f(x) M for ll x [, ], the which implies I M (M f(t)) (f(t) m) dt 0 Usig the elemetry iequlity m. (M k)(k m) 4 [(M k) + (k m)] = (M m) 4 which holds for k, m, M R, we get (.6) I 4 (M m). RGMIA Reserch Report Collectio, Vol., No., 998

4 9 Drgomir d Fedotov Usig Cuchy-Buikowski-Schwrz s itegrl iequlity we hve I Now, y (.6), we get f(x) f(x) dx. dx (M m)( ) d the y (.5) we oti the desired iequlity (.3). To prove the shrpess of the costt, let choose u(x) := x + (, f(x) := sg x + ), x [, ]. The u(x) u(y) = x + y + x y for ll x, y [, ], which shows tht u is L-lipschitzi with the costt L =. Also, ecuse f(x), for ll x [, ], the M m = d O the other hd, f(x)du(x) L (M m)( ) =. u() u() = ( sg x + ) du(x) = + = du(x) + + ( ) ( ) + + du(x) = u + u() + u() u which shows tht the equlity is relized i (.3). Corollry.. Let f : [, ] R e s ove d u : [, ] R differetile mppig whose derivtive u : (, ) R is ouded o (, ). Deote u := sup t (,) u (t) <. The we hve the iequlity f(t)u (t)dt u() u() u (M m)( ). RGMIA Reserch Report Collectio, Vol., No., 998

5 Grüss Iequlity for Riem-Stieltjes Itegrl 93 Corollry.3. Let f : [, ] R e s ove d g : [, ] R cotiuous mppig. Deote g := sup t [,] g(t) <. The we hve the iequlity f(t)g(t)dt g(t)dt g (M m)( ). Corollry.4. Let f : [, ] R e differetile mppig whose derivtive f : (, ) R is ouded o (, ) d f() f(). Deote f := sup t (,) f (t) <. The we hve the iequlity f() + f() f (M m)( ). (f() f()) The proof is ovious from the ove corollry choosig u = f. Remrk.. If i Corollry. we put u = f p, u = l f, u = si f etc..., we c oti some other iterestig iequlities. We shll omit the detils. 3 Applictios for Specil Mes We first discuss the pplictio of the results i the previous sectio to lower d upper ouds estimtio of some importt reltioships etwee the followig mes: The rithmetic me: The geometric me: The hrmoic me: The logrithmic me: A = A(, ) := ( + )/,, 0. G = G(, ) :=,, 0. H = H(, ) := +,, > 0. L = L(, ) := l l if if =,, > 0. Note tht for the covex mppig f : (0, ) R, f(t) = t we hve = L (, ) for. RGMIA Reserch Report Collectio, Vol., No., 998

6 94 Drgomir d Fedotov The p-logrithmic me [ p+ p+ ] /p if L p = L p (, ) := (p + )( ) if =, p R \ {, 0},, > 0. For the covex (or cocve) mppig f(t) = t p, p (, 0) [, ) \ { } (or p (0, )) we hve = L p p(, ) for. The idetric me: I = I(, ) := ( ) e if if =,, > 0. For the covex mppig f(t) = l t, t > 0 we hve = l I(, ) if. These mes re ofte used i umericl pproximtio d i other res. However, the followig simple reltioships re kow i the literture H G L I A. It is lso kow tht L p is mootoiclly icresig i p R with L 0 = I d L = L. We ow derive vrious sophisticted ouds for some differeces d products of the ove specil mes usig the results otied i the previous sectio. These ouds re useful i pplictios sice the specil mes re ofte used i umericl pproximtios.. If i Corollry. we choose f(x) = x q, u(x) = x p+ (p, q > 0), the we get (3.) L p+q p+q L p pl q q p q ( )Lq.. If i Corollry. we choose f(x) = x q (q > 0), u(x) = x, the we get (3.) L q q G L q q q ( )G L q q. RGMIA Reserch Report Collectio, Vol., No., 998

7 Grüss Iequlity for Riem-Stieltjes Itegrl If i the sme corollry we choose f(x) = x q (q > 0), u(x) = l x, the we get (3.3) L q q LL q q q ( )LLq. We remrk tht if i the Corollry. we choose f d u i other pproprite wys, we get some other iterestig iequlities for specil mes. We omit the detils. Refereces [] D.S. Mitriović, J.E. Pečrić d A.M. Fik, Clssicl d New Iequlities i Alysis, Kluwer Acdemic Pulishers, Dordrecht, 993. (S.S. Drgomir) School of Commuictios d Iformtics, Victori Uiversity of Techology, PO Box 448, MCMC Meloure, Victori 800, Austrli (I. Fedotov) Deprtmet of Applied Mthemtics, Uiversity of Trskei, Privte Bg X, UNITRA, Umtt, 500, South Afric E-mil ddress: S.S. Drgomir sever@mtild.vut.edu.u I. Fedotov fedotov@getfix.utr.c.z RGMIA Reserch Report Collectio, Vol., No., 998