FURTHER GENERALIZATIONS. QI Feng. The value of the integral of f(x) over [a; b] can be estimated in a variety ofways. b a. 2(M m)

Transcription

1 Univ. Beogrd. Pul. Elekroehn. Fk. Ser. M. 8 (997), 79{83 FUTHE GENEALIZATIONS OF INEQUALITIES FO AN INTEGAL QI Feng Using he Tylor's formul we prove wo inegrl inequliies, h generlize K. S. K. Iyengr's inequliy nd some oher inegrl inequliies. Le f(x) e dierenile funcion sisfying jf 0 (x)j N for x. The vlue of he inegrl of f(x) over [ ] cn e esimed in vriey ofwys. For exmple, he useful inequliy of Iyengr in [{6] ses h () ( )(f()+f()) ( ) N 4 (f() f()) 4N which reduces o [,3,4] () ( ) N 4 when f() = f() = 0:If he ounds for he derivive of f(x) re expressed in he form m f 0 (x) M, hen oher useful inequliies re [, 3] (3) ( ) m ( )f() ( ) M (4) mm( ) +( )(Mf() mf())+(f() f()) (M m) mm( ) +( )(mf() Mf())+(f() f()) : (M m) 0 99 Mhemics Sujec Clssicion: Primry 6D5 Secondry 6D0 79

2 80 QI Feng (5) When f() =f() = 0, he inequliy (4) reduces o mm( ) (M m) : If we se M = m = N, hen () follows from (4) if we se f() =f()=0 nd M = m = N, hen () follows from (5). Suppose h f(x) isn-imes dierenile for x [ ], nd f (n) (x) N, f (r) () =f (r) ()=0r=0:::n. Then we hve[, ] (6) () N (n)!(n + )! ( )n+ : We remrk h he inequliies from () o (5) re no included in (6). There is reled inegrl inequliy []: Le f(x) e dierenile of clss C jf 00 (x)j N x [ ]: Then we hve (7) N( ) 4 f()+f() ( 3Q ) + +Q ( ) f 0 () f 0 () 3 where Q = f() f() +f0 ()+f 0 () N ( ) f 0 () f 0 () : In his ricle, we generlize hese resuls, giving shor proof which is sed on he Tylor's formul. Theorem. Le f(x) e dierenile funcion of C n [ ] sisfying m f (n) (x) M. If we denoe (8) S n (u v w) = k= ( ) k k! u k f (k ) (v)+ w un k S k = S(k) n (u v w) hen, when n is even, we hve for ny [ ] (0) n+ X ( ) i n+ X n+ ( m) S(i) n+ ( m) i ( ) i n+ ( M) S(i) n+ ( M) i

3 Furher generlizions of inequliies for n inegrl 8 wheres when n is odd, we hve () n+ X n+ X ( ) i ( ) i n+ ( m) S(i) n+ ( M) i n+ ( M) S(i) n+ ( m) i [ ]: Proof. Le e prmeer sisfying <<, nd wrie () = The Tylor's formul ses h (3) f(x) = (4) f(x) = f (i) () + : (x ) i + f (n) () (x ) n ( x) (x ) i + f (n) () (x ) n (x ): Inegring on oh sides of (3) over [ ], we oin (5) = Z f (i) () (i + )! ( )i+ + f (n) () (x ) n dx: Since m f (n) (x) M, hen (6) m (n + )! ( )n+ f (i) () (i + )! ( )i+ M (n + )! ( )n+ : Inegring on oh sides of (4) over [ ], we ge (7) = Z (i + )! ( )i+ + f (n) () (x ) n dx: When n is even, from (7), i follows h (8) m (n + )! ( )n+ f + (i) () (i + )! ( )i+ M (n+ )! ( )n+ : When n is odd, he reversed inequliies of (8) hold.

4 8 QI Feng (9) From (), (6) nd (8), when n is even, we hve f (i) () ( )i+ (i + )! (i + )! ( )i+ + m (n + )! ( )n+ ( ) n+ f (i) () ( )i+ (i + )! + M (n + )! ( )n+ ( ) n+ ( )i+ (i + )! when n is odd, we oin (0) f (i) () ( )i+ (i + )! ( )i+ (i + )! + m (n + )! ( M )n+ ( )n+ (n + )! f (i) () ( )i+ (i + )! + M (n + )! ( )n+ m (n + )! ( )n+ : ( )i+ (i + )! Considering (8) nd (9), rewriing (9) nd (0), he desired inequliies (0) nd () follow. Corollry. Le f(x) e wo imes dierenile funcion sisfying m f 00 (x) M. Then (() m( 3 3 ) 6 f() f()+f 0 () f 0 ()+m( )= ( )m+f 0 () f 0 () f()+f()+ f 0 () f 0 () M(3 3 ) 6 Proof. Tking n = in inequliy (0) nd f() f()+f 0 () f 0 ()+M( )= ( )M+f 0 () f 0 () : = f() f()+f 0 () f 0 ()+m( )= ( )m+f 0 () f 0 () he lef-hnd side inequliy of () follows. By he symmery of inequliy (0), he righ-hnd side inequliy of () follows esily.

5 Furher generlizions of inequliies for n inegrl 83 emrk. If we se n = = M m + f() f() =(M m), nd = m M + f() f() =(m M), hen (4) follows from (), so (4) is included in (0). If we se M = m = N, hen (7) follows from (), herefore (7) is included in (0). Since he inequliies (), (), (4), (5) nd (7) re ll included in (0) nd (), inequliies (0) nd () re generl. Moreover, n unied proof nd n unied represenive of hese inequliies re given y he heorem in his ricle. EFEENCES. D. S. Mirinovic: Anlyic Inequliies. Springer-Verlg, Kung Jichng: Applied Inequliies (in Chinese), nd ediion. Hunn Educion Press, Chngsh, Chin, QI Feng: Inequliies for n inegrl. Mh. Gz. 80, No 488 (996), 376{ Xu Lizhi, Wng Xinghu: Mehods of Mhemicl Anlysis nd Seleced Exmples (in Chinese). evised Ediion. Higher Educion Press, Beijing, Chin, D. S. Mirinovic, A. M. Fink: Inequliies Involving Funcions nd Their Inegrls nd Derivives, Chper XV. Kluwer Acdemic Pulishers, G. V. Milovnovic, J. E. Pecric: Some considerions on Iyengr's inequliy nd some reled pplicions. Univ. Beogrd. Pul. Elekroehn. Fk. Ser. M. Fiz. No 544{576 (976), 66{70. Deprmen of Mhemics, (eceived Jnury 7, 997) Universiy of Science nd Technology of Chin, Hefei 3006, Anhui, People's epulic of Chin Curren E-mil ddress o Decemer of 998: qi feng@s.usc.edu.cn