Krgujev Journl of Mthemtis Volume 38() (204), Pges 35 49. DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS MOHAMMAD W. ALOMARI Abstrt. In this pper, severl bouns for the ifferene between two Riemn- Stieltjes integrl mens uner vrious ssumptions re prove.. Introution In 938, Ostrowski estblishe very interesting inequlity for ifferentible mppings with boune erivtives, s follows. Theorem.. Let f : I R R be ifferentible mpping on I, the interior of the intervl I, suh tht f L, b, where, b I with < b. If f (x) M, then the following inequlity, f (x) b f (u) u M (b ) 4 ( x b 2 (b ) 2 hols for ll x, b. The onstnt is the best possible in the sense tht it nnot 4 be reple by smller onstnt. In 200, Mtić n Pečrić 7 hve prove the following estimtes of the ifferene of two integrl mens. Theorem.2. Given funtion f :, b R stisfying the Lipshitz onition with onstnt M > 0, n < b, then the following two-point Ostrowski inequlity b f (t) t f (s) s b M ( )2 (b ) 2 2 ( b ), hols. Key wors n phrses. Ostrowski s inequlity, Funtion of boune vrition, Riemn-Stieltjes integrl. 200 Mthemtis Subjet Clssifition. Primry: 26D5. Seonry: 26A42, 26A6, 26A45. Reeive: November, 202 Revise: Deember 22, 203. ) 2 35
36 M. W. ALOMARI Another result ws prove by Brnett et l. 4, s follows. Theorem.3. Let f :, b R be n bsolutely ontinuous mpping with the property tht f L, b, i.e., f : ess sup f (t). t,b Then for < b, we hve the inequlity b f (t) t f (s) s b ( ) 2 ( b) /2 ( ) /2 (.) 4 (b ) ( ) f (b ) ( ) 2 (b ) ( ) f. The onstnt /4 in the first inequlity n /2 in the seon inequlity re the best possible. After tht, severl uthors obtine interesting bouns for the ifferene of two integrl mens uner vrious ssumptions. For other results, the reer my refer to 0 n the referenes therein. The im of this pper n for the first time severl bouns for the ifferene between two Stieltjes integrl mens re obtine. Nmely, bouns for (.2) where (.3) n (.4) D (f, u;, b;, ) : S (f, u;, b) S (f, u;, ), S (f, u;, b) : S (f, u;, ) : u (b) u () u () u () f (x) u (x) f (t) u (t) < b suh tht the integrn f is ssume to be of r H Höler type mpping on, b n the integrtor u is to be of boune vrition, Lipshitzin n monotoni mppings; respetively on, b re given. The following result hols. 2. The Result Theorem 2.. Let f :, b R be of r-h Höler type mpping on, b, where, H > 0 n r (0, re given, n u :, b R is mpping of boune vrition
DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS 37 on, b. Then we hve the inequlity D (f, u;, b;, ) H (b ) r u (b) u () u () u () for ll < b suh tht x, b n t,. b (u), Proof. First we observe tht, using the integrtion by prts formul for Riemnn Stieltjes integrl, we hve f (x) f (t) u (t) u (x) ( ) f (x) f (t) u (t) u (x) ( f (x) u () u () u () u () ) f (t) u (t) u (x) f (x) u (x) u (b) u () f (t) u (t) for ll < b suh tht x, b n t,. It is well-known tht for ontinuous funtion p :, b R n funtion ν :, b R of boune vrition, one hs the inequlity (2.) p (t) ν (t) sup p (t) t,b b (ν). Therefore, s u is of boune vrition on, b (n then on, ), we hve f (x) f (t) u (t) u (x) D (f, u;, b;, ) u (b) u () u () u () ( b ) f (x) f (t) u (t) u (x) (2.2) u (b) u () u () u () sup f (x) f (t) u (t) x,b b (u) u (b) u () u () u (). Now, pply (2.) gin on the right hn sie of the bove inequlity, we get f (x) f (t) u (t) sup f (x) f (t) t, (u).
38 M. W. ALOMARI Sine f is of r-holer type on,, then sup f (x) f (t) H sup x t r t, t, mx {( x) r, ( x) r }, if x < H mx {(x ) r, ( x) r }, if < x < therefore by (2.2), we hve mx {(x ) r, (x ) r }, if < x mx {( x), ( x)} r, if x < H mx {(x ), ( x)} r, if < x < mx {(x ), (x )} r, if < x ( x) r, if x < H 2 x r 2, if < x < (x ) r, if < x D (f, u;, b;, ) H b sup u (b) u () u () u () f (x) f (t) u (t) (u) xb ( x) r, if x < H sup 2 x r b 2, if < x < x,b (u) u (b) u () u () u () (x ) r, if < x H ( ) r, if x < b (u) u (b) u () u () u () ( ) r, if < x < (b ) r, if < x H b (u) u (b) u () u () u () mx {( )r, ( ) r, (b ) r } H b (u) u (b) u () u () u () mx {( )r, (b ) r }
DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS 39 H b (u) mx {( ), (b )}r u (b) u () u () u () H b (u) (b ) ( ) r u (b) u () u () u () 2 2 ( ) ( b) H (b ) r u (b) u () u () u () b (u) sine < b, then b > n therefore ( ) ( b) b it follows tht (b ) ( ) ( ) ( b) b, 2 2 whih ompletes the proof. Corollry 2.. Let u be s in Theorem 2. n f :, b R be K Lipshitzin mpping on, b. Then we hve the inequlity K (b ) b D (f, u;, b;, ) u (b) u () u () u () (u), for ll < b suh tht x, b n t,. Corollry 2.2. Let f be s in Theorem 2.. Let u C (), b. Then we hve the inequlity H (b ) r D (f, u;, b;, ) u (b) u () u () u () u,,b u,, where is the L norm, nmely u,α,β : β α u (t) t. Corollry 2.3. Let f be s in Theorem 2.. Let u :, b R be L-Lipshitzin mpping with the onstnt L > 0. Then we hve the inequlity D (f, u;, b;, ) HL2 (b ) ( ) (b ) r u (b) u () u () u (). Corollry 2.4. Let f be s in Theorem 2.. mpping. Then we hve the inequlity D (f, u;, b;, ) H u (b) u () u () u () u (b) u () u () u () Let u :, b R be monotoni (b ) r. Remrk 2.. Let us ssume tht g :, b R is Lebesgue integrble on, b, then u (z) z g (s) s is ifferentible lmost everywhere. Using the properties of the Stieltjes integrl, we hve f (x) u (x) f (x) g (x) x, f (t) u (t) f (t) g (t) t
40 M. W. ALOMARI n b (u) u (s) s g (s) s, (u) Therefore, the weighte version of D (f, u;, b;, ) is WD (f, u;, b;, ) : g (t) t WD (f, u;, b;, ) : f (x) u (x) u (s) s g (x) x g (s) s. f (t) u (t). A generl weighte version of the Stieltjes-integrl men, my be eue s follows f (x) g (x) x g (x) x f (t) g (t) t g (t) t, for ll < b, provie tht g(s) 0, for lmost every s, b n g (x) x 0 n g (t) t 0. Therefore, we n stte the following result. Corollry 2.5. Let f :, b R be of r-h Höler type mpping on, b, where, H > 0 n r (0, re given, n g :, b R is ontinuous mpping on, b. Then we hve the inequlity WD (f, u;, b;, ) H (b ) r g (t) t g (x) x, for ll < b suh tht x, b n t,. Moreover, we hve WD (f, u;, b;, ) H (b ) r, We n eue the following result. Corollry 2.6. If g (s), for ll s, b then we get the following two-point Ostrowski inequlity for mpping f efine on, b whih is of r-höler type b f (x) x f (t) t b H (b )r. Moreover, if n b, then we hve b f (x) x f (t) t b H (b )r. Also, if b n, then we hve f (x) x f (t) t H ( )r. Boun for the ifferene between two Stieltjes integrl mens for L-Lipshitz integrtor is inorporte in the following result.
DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS 4 Theorem 2.2. Let f :, b R be of r-h Höler type mpping on, b, where, H > 0 n r (0, re given, n u :, b R be L Lipshitzin mpping of on, b. Then we hve the inequlity D (f, u;, b;, ) HL 2 ( ) r2 ( ) r2 (b ) r2 (b ) r2, (r ) (r 2) u (b) u () u () u () for ll < b suh tht x, b n t,. Proof. It is well-known tht for Riemnn integrble funtion p :, b R n L Lipshitzin funtion ν :, b R, one hs the inequlity p (t) ν (t) L p (t) t. Therefore, s u is L-Lipshitzin on, b, we hve (2.3) f (x) f (t) u (t) u (x) D (f, u;, b;, ) u (b) u () u () u () ( b ) f (x) f (t) u (t) u (x) u (b) u () u () u () f (x) f (t) u (t) x L u (b) u () u () u () Now, pply (2.3) gin on the right hn sie of the bove inequlity, we get f (x) f (t) u (t) L f (x) f (t) t n sine f is of r-holer type on,, then f (x) f (t) u (t) L f (x) f (t) t L x t r t
42 M. W. ALOMARI L L whih gives by (2.3) tht, x)r t, if x < x t)r t x x)r t, if < x < (x t)r t, if < x ( x) r ( x) r r, if x < (x ) r ( x) r r, if < x < (x ) r (x ) r r, if < x D (f, u;, b;, ) f (x) f (t) u (t) x L u (b) u () u () u () L 2 u (b) u () u () u () ( x) r ( x) r r, if x < L 2 (x ) r ( x) r, if < x < u (b) u () u () u () r (x ) r (x ) r, if, < x r ( L 2 ( x) r ( x) r x u (b) u () u () u () r (x ) r ( x) r ) b (x ) r (x ) r x x r r ( ) r2 ( ) r2 ( ) r2 (r ) (r 2) 2 ( )r2 (r ) (r 2) (b )r2 ( ) r2 (b ) r2 (r ) (r 2) L 2 ( ) r2 ( ) r2 (b ) r2 (b ) r2 (r ) (r 2) u (b) u () u () u () whih is require. x Remrk 2.2. Let us ssume tht g :, b R is Lebesgue integrble on, b, then u (z) z g (s) s is ifferentible lmost everywhere. Therefore, g is L-Lipshitzin
DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS 43 with the onstnt L g. Using the properties of the Stieltjes integrl, we hve f (x) u (x) f (x) g (x) x, Therefore, we n stte the following result. f (t) u (t) f (t) g (t) t Corollry 2.7. Let f :, b R be of r-h Höler type mpping on, b, where, H > 0 n r (0, re given, n g :, b R is ontinuous mpping on, b. Then we hve the inequlity WD (f, u;, b;, ) H g 2 ( ) r2 ( ) r2 (b ) r2 (b ) r2 (r ) (r 2) for ll < b suh tht x, b n t,. Moreover, we hve WD (f, u;, b;, ) H g 2 ( )r2 ( ) r2 (b ) r2 (b ) r2 ( ) ( b ) (r ) (r 2) g (x) x, g (t) t We my eue the following result. Corollry 2.8. If g (s), for ll s, b then we get the following two-point Ostrowski inequlity for mpping f efine on, b whih is of r-höler type b f (x) x f (t) t b H g 2 ( ) r2 ( ) r2 (b ) r2 (b ) r2. (r ) (r 2) (b ) ( ) Moreover, if n b, then we hve b f (x) x f (t) t b H g 2 ( ) r2 (b ) r2 (b ) r2. (r ) (r 2) (b ) ( ) Also, if b n, then we hve f (x) x f (t) t H g 2 ( ) r2 ( ) r2 ( ) r2. (r ) (r 2) ( ) ( ) Finlly, we point out the ifferene between two Stieltjes integrls men for monotoni integrtor.
44 M. W. ALOMARI Theorem 2.3. Let f :, b R be of r-h Höler type mpping on, b, where, H > 0 n r (0, re given, n u :, b R be monotoni non-eresing mpping of on, b. Then we hve the inequlity D (f, u;, b;, ) H ( ) r u () u () u (b) u () ( u () u () )r u (b) u () (b u (b) u () )r, u (b) u () for ll < b suh tht x, b n t,. Proof. It is well-known tht for monotoni non-eresing funtion ν :, b R n ontinuous funtion p :, b R, one hs the inequlity (2.4) p (t) ν (t) p (t) ν (t). Therefore, s u is monotoni non-eresing on, b, we hve (2.5) D (f, u;, b;, ) ( ) f (x) f (t) u (t) u (x) u (b) u () u () u () f (x) f (t) u (t) u (x) u (b) u () u () u () Now, pply (2.4) gin on the right hn sie of the bove inequlity, we get f (x) f (t) u (t) f (x) f (t) u (t) n sine f is of r-holer type on,, then f (x) f (t) u (t) f (x) f (t) u (t) x t r u (t) x)r u (t), if x < x t)r u (t) x x)r u (t), if < x < (x t)r u (t), if < x
DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS 45 ( x) r u () ( x) r u () r (t x)r u (t) t, if x < ( x) r u () (x ) r u () x r (x t)r u (t) t (t x x)r u (t) t, if < x < (x ) r u () (x ) r u () r (x t)r u (t) t, if < x Sine u is monotoni noneresing on, b n then on,, hene n x x (t x) r u (t) t u () (x t) r u (t) t u (x) (t x) r u (t) t u (x) (x t) r u (t) t u () x x (t x) r t r u () ( x)r ( x) r, (x t) r t r (x )r u (x), (t x) r t r ( x)r u (x), (x t) r t r u () (x )r (x ) r from whih it follows tht ( x) r u () ( x) r u () r x)r u (t) t, if x < ( x) r u () (x ) r u () x r (x t)r u (t) t (t x x)r u (t) t, if < x < (x ) r u () (x ) r u () r (x t)r u (t) t, if < x ( x) r u () ( x) r u () u () ( x) r ( x) r, if x < ( x) r u () (x ) r u () (x ) r u (x) ( x) r u (x), if < x < (x ) r u () (x ) r u () u () (x ) r (x ) r, if < x ( x) r u () u (), if x < ( x) r u () u (x) (x ) r u (x) u (), if < x < (x ) r u () u (), if < x whih, by (2.5), gives tht,
46 M. W. ALOMARI f (x) f (t) u (t) u (x) u (b) u () u () u () ( x) r u() u(), if x < u(b) u() u() u() b ( x) H r u() u(x)(x ) r u(x) u(), if < x < u(b) u() u() u() u (x) (x ) r u() u(), if < x u(b) u() u() u() { H ( x)r u (x) ( x)r u () u (x) u (x) u (b) u () u (b) u () u () u () (2.6) { H (x )r u (x) u () u (x) u (b) u () u () u () u (b) u () ( )r u () u (b) u () ( )r u () u (b) u () r u (b) u () u () u () r u (b) u () u () u () u (b) u () (x )r u () u () u (x) u (b) u () ( ) r u () ( ) r u () r (b ) r u (b) ( ) r u () r Sine u is monotoni noneresing on, b, hene (2.7) Now, for the term (2.8) n ( x) r u (t) t u () ( x) r u () u (x) u (x) x ( x) r u () u (x) x ( x) r u () u (x) x u 2 () ( x) r u () u (x) u (x) x (x ) r u (x) u () u (x) x } ( x) r u (x) x } (x ) r u (x) x. ( x) r t r u () ( )r ( ) r. ( x) r u 2 (x) x ( x) r x r u2 () ( ) r,
DIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS 47 sine u () u (), then we hve (2.9) ( x) r u 2 (x) x u 2 () ( x) r x r u2 () ( ) r, whih implies by (2.8) (2.9), tht (2.0) ( x) r u () u (x) u (x) x r ( )r u 2 () u 2 (). Similrly, for the term (2.) (x ) r u () u (x) u (x) x (x ) r u () u (x) x (x ) r u 2 (x) x sine u () u (), then we hve (x ) r u () u (x) x u () u () (x ) r x (2.2) r u2 () ( ) n (2.3) (x ) r u 2 (x) x r u2 () ( ) r whih implies by (2.) (2.3), tht (2.4) (x ) r u (x) u () u (x) x r ( )r u 2 () u 2 () n finlly, we hve (2.5) (x ) r u (x) x r (b )r ( ) r u ()
48 M. W. ALOMARI n if we substitute (2.7), (2.0), (2.4) n (2.5) in (2.6), we get f (x) f (t) u (t) u (x) { H u (b) u () ( ) r u () ( ) r u () r ( x) r u (x) x ( )r u () u (b) u () ( )r u () u (b) u () r u (b) u () u () u () r u (b) u () u () u () u (b) u () { H (b ) r u (b) ( ) r u () r ( x) r u () u (x) u (x) x (x ) r u (x) u () u (x) x } (x ) r u (x) x. u (b) u () ( )r u () ( ) r u () u () ( ) r ( ) r ( )r u () u (b) u () ( )r u () u (b) u () H } u (b) u () (b )r u (b) ( ) r u () (b ) r ( ) r u () ( ) r u () u () u (b) u () ( u () u () )r u (b) u () (b u (b) u () )r u (b) u () whih proves this theorem. Corollry 2.9. If in Theorem 2.3, we hoose u (s) s, s, b, then we hve b f (x) x f (t) t b H ( ) r ( ) ( ) r (b ) r (b ). (b ) Moreover, if n b, then we hve b f (x) x f (t) t b H ( ) r (b ) r (b ). (b ) Also, if b n, then we hve f (x) x f (t) t. H ( ) r ( ) ( ) r. ( )
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