AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir

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1 RGMIA Reserch Report Collection, Vol., No., rgmi AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS I. Fedotov nd S. S. Drgomir Astrct. An integrl inequlity of Ostrowski type nd its pplictions for specil mens nd error evlution in Simpson s qudrture rule re given. Introduction The following integrl inequlity which estlishes connection etween the integrl of the product nd the product of the integrls is well known in literture s Grüss inequlity [, p. 96]. Theorem.. Let f, g : [, ] R e two integrle functions such tht ϕ f (x) Φ nd γ g (x) Γ for ll x [, ] ; ϕ, Φ, γ nd Γ re constnts. Then we hve the inequlity (.) f (x) g (x) dx f (x) dx g (x) dx (φ ϕ) (Γ γ) 4 nd the inequlity is shrp in the sense tht the constnt 4 one. cn not e replced y smller In 938, Ostrowski (cf., for exmple [3, p. 468]), proved the following inequlity which gives n pproximtion of the integrl s follows: Theorem.. Let f : [, ] R e continuous on [, ] nd differentile on (, ) whose derivtive f : (, ) R is ounded on (, ), i.e., f := f (t) dt <. Then: (.) f (x) [ 4 + ( x + ( ) ) ] sup t (,) ( ) f for ll x [, ]. The constnt 4 is est. In the recent pper [], S.S. Drgomir nd S. Wng hve proved the following version of Ostrowski s inequlity y using Grüss inequlity (.). Dte. Jnury, Mthemtics Suject Clssifiction. Primry 6 D 5; Secondry 4 A 55. Key words nd phrses. Ostrowski Inequlity, Simpson s Rule.

2 4 Fedotov nd Drgomir Theorem.3. Let f : I R R e differentile mpping in the interior of I nd let, int(i) with <. If f L [, ] nd then we hve the following inequlity: f (x) (.3) for ll x [, ]. γ f (x) Γ for ll x [, ], f () f () ( ) (Γ γ) 4 ( x + ) They lso pplied this result for specil mens nd in Numericl Integrtion otining some qudrture formule generlizing the mid-point qudrture rule nd the trpezoid rule. Note tht the error ounds they otined re in terms of the first derivtive which re prticulrly useful in the cse when f does not exists or is very lrge t some points in [, ]. In this pper, we give generliztion of the ove inequlity which contins in prticulr cse the clssicl Simpson formul. Appliction for specil mens nd in Numericl Integrtion re lso given. An Integrl Inequlity of Grüss Type For ny rel numers <, let us consider the function t + A, if t x p (t) p x (t) = t + B if x < t. It is cler tht p x hs the following properties. () It hs the jump [p] x = (B A) ( ) t the point t = x nd dp x (t) dt = + [p] x δ (t x). () Let M x := sup p x (t) nd m x := inf p x (t). Then the difference M x m x cn e t (,) t (,) evluted s follows :. For B A 0 we hve M x m x = [p] x.. For B A > 0, the following three cses re possile (i) If 0 B A ( ), then x + for x + (B A) ; M x m x = [p] x for + (B A) < x (B A) ; x for (B A) < x.

3 An Inequlity of Ostrowski Type 5 (ii) If ( ) < B A ( ), then x + for x < (B A) ; M x m x = B A for (B A) x < + (B A) ; x for q + (B A) x. (iii) If B A >, then M x m x = B A. The following inequlity of Ostrowski type holds. Theorem.. Let f : [, ] R e continuous on [, ] nd differentile on (, ) whose derivtive stisfies the ssumption (.) γ f (t) Γ for ll t (, ), where γ, Γ re given rel numers. Then we hve the inequlity: (.) (C A) f () + ( B + A) f (x) + (B C) f () 4 (Γ γ) (M x m x ) ( ) where C = C (x) := [(x ) (x + A) (x ) (x + B)], ( ) nd A, B, M x nd m x re s ove, x [, ]. Proof. Using Grüss inequlity (.) we cn stte tht p x (t) f f () f () (.3) (t) dt p x (t) dt for ll x [, ]. Integrting first term, y prts, we otin: Also, s 4 (Γ γ) (M x m x ), p x (t) f (t) dt = Bf () Af () [p] x f (x). p x (t) dt = [(x ) (x + A) (x ) (x + B)]

4 6 Fedotov nd Drgomir then (.3) gives the inequlity: Bf () Af () [p] x f (x) C f () f () 4 (Γ γ) (M x m x ) which is clerly equivlent with the desired result (.). Remrk.. Setting in (.) A = B = 0 nd tking into ccount, y property (), tht M x m x =, we otin the inequlity (.3) y Drgomir nd Wng. The following corollry is interesting: Corollry.. Let A, B rel numers so tht 0 B A ( ). If f is s ove, then we hve the inequlity (.4) ( ) B A + f () + [ (B A)] f + B A f () Proof. Consider x = +. Then By property () we hve (Γ γ) ( B + A) ( ). 4 x =, x = C = A + B, x [ + (B A), (B A)]. M x m x = ( ) (B A). Applying Theorem. for x = +, we get esily (.4). Remrk.. (.5). If we choose in the ove corollry B A =, then we get [ ( )] f () + f () + + f ( ) (Γ γ) ( ) 8 which is comintion etween mid-point nd trpezoid formul.. If we choose in (.4), B = A, then we get the mid-point inequlity ( ) + (.6) ( ) f (Γ γ) ( ) 4 proved y S.S. Drgomir nd S. Wng in [] (Corollry.3).

5 An Inequlity of Ostrowski Type 7 3. If we choose in (.4), B A = 3, then we otin Simpson s formul (.7) 6 [ f () + 4f ( + ) ] + f () (Γ γ) ( ) 6 for which we hve n estimtion in terms of the first derivtive not s in the clssicl cse in which the forth derivtive is required, i.e., [ ( ) ] + (.8) f () + 4f + f () 6 s f (4) 880 ( ) 5. Remrk.3. The method of evlution of the error for the Simpson rule considered ove cn e pplied for ny qudrture formul of Newton-Cotes type. For exmple, to get the similr evlution of the error for the Newton-Cotes rule of order 3, it is sufficient to replce the function p x (t) in (.3) y the function: t A if t + h p x (t) := t + + A + B if + h < t h t B if h < t where B A = 4, h = 3. 3 Appliction For Specil Mens Let us recll some importnt mens of positive rel numers. () The rithmetic men : () The geometric men: (c) The hrmonic men: A = A (, ) := +,, > 0; G = G (, ) :=,, 0; H = H (, ) := +,, > 0; (d) The logrithmic men: L = L (, ) := if =, ln ln if,, > 0;

6 8 Fedotov nd Drgomir (e) The identric men : I = I (, ) := ( ) e if, if =,, > 0; (f) The p-logrithmic men [ p+ p+ ] p if, L p = L p (, ) := (p ) ( ) if =,, > 0, p R\ {, 0}. (3.) In wht follows we shll pply the inequlity (.7) written in the following form for the previous mens. ( ) + 3 f f () + () + 6 (Γ γ) ( ). 6 Consider the mpping f (x) = x p (p > ), x > 0.Then ( ) + f = A p f () + f () (, ), = A ( p, p ), = L p p (, ), Γ γ = ( ) (p ) L p p for, R with 0 < <. Consequently, we hve the inequlity 3 Ap (, ) + 3 A (3.) (p, p ) L p p (, ) 6 ( ) (p ) L p p Consider the mpping f (x) = x, x > 0. Then ( ) + f = A (, ), f () + f () = H (, ), = L (, ), Γ γ = ( ) A (, ) = G 4 (, )

7 An Inequlity of Ostrowski Type 9 for 0 < <. Consequently, we hve the inequlity: 3 A (, ) + 3 H (, ) L (, ) 3 ( A (, ) ) G 4 (, ) which is equivlent to (3.3) 3 HL + AL AH 3 3 ( A HL ) G 4. Consider the mpping f (x) = ln x, x > 0. Then we hve ( ) + f () + f () f = ln A, = ln G, = ln I, Γ γ = G for, R with 0 < <. Consequently, we hve the inequlity 3 ln A + ln G ln I 3 ( ) 6 G which is equivlent to (3.4) ( ) ln A 3 G 3 I 6 ( ) G. 4 A New Estimtion of The Error Bound In Simpson s Rule The following theorem holds. Theorem 4.. Let f : [, ] R e continuous on [, ] nd differentile on (, ) whose derivtive stisfies the condition (.), i.e., γ f (t) Γ for ll t (, ) ; where γ, Γ re given rel numers. Then we hve (4.) = S n (I n, f) + R n (I n, f) where S n (I n, f) = n h i [f (x i ) + 4f (x i + h i ) + f (x i+ )], 3 i=o I n is the prtition given y I n : = x 0 < x <... < x n < x n =, h i := (x i+ x i ), i = 0,..., n ; nd the reminder term R n (I n, f) stisfies the estimtion: (4.) R n (I n, f) n 3 (Γ r) h i. i=0

8 0 Fedotov nd Drgomir Proof. Let us set in (.7) = x i, = x i+, h i = x i+ x i, where i = 0,..., n. Then we hve the estimtion: x i+ h i 3 [f (x i) + 4f (x i + h i ) + f (x i+ )] 3 (Γ r) h i, for ll i = 0,..., n. After summing nd using the tringle inequlity, we otin n h i 3 [f (x i) + 4f (x i + h i ) + f (x i+ )] i=0 which proves the required estimtion (4.). x i n 3 (Γ γ) Corollry 4.. Under the ove ssumptions nd if we put f := i=o h i sup f (t) <, t (,) then we hve the following estimtion of the reminder term in Simpson s formul R n (I n, f) 4 n 3 f (4.3) h i. Remrk 4.. The clssicl error estimtes sed on the Tylor expnsion for the Simpson s rule involve the forth derivtive f (4). In the cse when f (4) does not exist, or is very lrge t some points in [, ], the clssicl estimtes cn not e pplied, nd thus (4.) nd (4.3) provide lterntive error estimtes for the Simpson s rule. References [] S.S. DRAGOMIR nd S. WANG, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtions of error ounds for some specil mens nd for some numericl qudrture rules, Computing Mth. Appl. 33() (997), 5-0. [] D.S. MITRINOVIĆ, J.E. PE CARIĆ nd A.M. FINK, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Pulishers, 993. [3] D.S. MITRINOVIĆ, J.E. PE CARIĆ nd A.M. FINK, Inequlities for Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Pulishers, 994. i=0 (I. Fedotov) Deprtment of Applied Mthemtics, University of Trnskei, Privte Bg X, UNITRA Umtt, 57, South Afric (S. S. Drgomir) School of Communictions nd Informtics, Victori University of Technology, PO Box 448, Melourne City MC, Victori 800, Austrli. E-mil ddress: I. Fedotovfedotov@getfix.utr.c.z S. S. Drgomir sever@mtild.vu.edu.u