Bounds for the Riemann Stieltjes integral via s-convex integrand or integrator
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1 ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 6, Number, 0 Avilble online t Bounds for the Riemnn Stieltjes integrl vi s-convex integrnd or integrtor Mohmmd Wjeeh Alomri Abstrct. Severl bounds in pproximting the Riemnn Stieltjes integrl in terms of s-convex integrnds or integrtor re given.. Introduction A function f : R + R, where R + = 0, ), is sid to be s-convex in the second sense if f (αx + βy) α s f (x) + β s f (y) for ll x, y 0, ), α, β 0 with α + β = nd for some fixed s (0,. This clss of functions is denoted by K s. It cn be esily seen tht for s =, s-convexity reduces to the ordinry convexity of functions defined on 0, ) (see 6). In, Cerone nd Drgomir hve proved some error bounds in pproximting the Riemnn Stieltjes integrl in terms of some moments of the integrnd. Among others, they proved the following result. Theorem. Let u be p-convex with p > 0, f be monotoniclly incresing on, b nd such tht the Riemnn Stieltjes integrl f (t) du (t) nd the Riemnn integrls (t )p f (t) dt, (b t)p f (t) dt exist. Then f (t) du (t) p (b ) p u () u (b) (t ) p f (t) dt (b t) p f (t) dt. Received Februry 7, Mthemtics Subject Clssifiction. 6A4, 6A5, 6D5. Key words nd phrses. Riemnn Stieltjes integrl, s-convex function. 8
2 8 MOHAMMAD WAJEEH ALOMARI For other results concerning different bounds for the Riemnn Stieltjes integrl under vrious ssumptions on f nd u, see the recent ppers 5 nd the references therein. In this pper, severl inequlities for the Riemnn Stieltjes integrl f (x) dg (x) re proved. Nmely, the integrnd f is ssumed to be s-convex (s-concve) nd the integrtor g is monotoniclly incresing, bounded nd s-convex (s-concve).. Inequlities for s-convex integrnds or integrtors We my strt with the following result. Theorem. Let f, g :, b R + R be such tht f is s-convex on, b, g is monotoniclly incresing on, b nd the Riemnn Stieltjes integrl f (x) dg (x) nd the Riemnn integrls (x )s g (x) dx, (b x)s g (x) dx exist. Then we hve the inequlities (b ) s (b ) s g (b) s + f () (b ) s (b ) s g () + s g (x) (x ) s dx g (x) (b x) s dx g (b) g () f () +. (.) Proof. Since f is s-convex on, b nd by using the integrtion by prts formul for Riemnn Stieltjes integrl, we hve ( ) x s s + f () dg (x) b b ( ) x s s = dg (x) + f () dg (x) b b = (b ) s = (b ) s + f () (b ) s (x ) s dg (x) + f () (b ) s (b ) s g (b) s (b ) s g () + s (b x) s dg (x) g (x) (x ) s dx (.) g (x) (b x) s dx, which proves the first inequlity in (.). To prove the second inequlity in (.), using the monotonicity of g on, b, we get (x ) s g (x) dx g () (x ) s dx = g () (b )s s
3 nd BOUNDS FOR THE RIEMANN STIELTJES INTEGRAL 83 (b x) s g (x) dx g (b) Therefore by (.), we get (b ) s (b ) s g (b) s (b x) s dx = s g (b) (b )s. (b ) s g () + s + f () (b ) s (b ) s (b ) s g (b) g () (b ) s g (x) (x ) s dx + f () (b ) s (b ) s g () + g (b) (b ) s = g (b) g () f () +, g (x) (b x) s dx which proves the second inequlity in (.). The following result holds. Theorem 3. Let g :, b R + R be monotoniclly incresing function on, b. () If f :, b R + is convex on, b, then we hve the inequlity { f (x) dg (x) min f ()++ f () g (b) g (), g (b) g () + g () + g (b) } g (x) dx b f () +. () If f is concve, then we hve the inequlity { f (x) dg (x) mx f ()+ f () g (b) g (), g (b) g () g () + g (b) } g (x) dx b f () + provided tht the Riemnn Stieltjes integrl f (x) dg (x) exists. (.3) (.4)
4 84 MOHAMMAD WAJEEH ALOMARI Proof. () In (.), set s =, then we get x b + b x b f () dg (x) = (b ) g (b) g (x) dx + f () (b ) g () + g (x) dx b b = g (b) b g (x) dx + f () g (x) dx g (). b b Thus nd lso mx {f (), } g (b) g () mx g (b) g () = + = f () + + f () g (b) g () { g (b) b g () + g (b) g (x) dx, b } g (x) dx g () f () + b g (x) dx f () + which proves (.3). () If f is concve, then we hve x f (x) dg (x) b + b x b f () dg (x). So, similrly to the proof of (), nd f (x) dg (x) min {f (), } g (b) g () { f (x) dg (x) min g (b) which proves (.4). b g (x) dx, b } g (x) dx g () f () +
5 BOUNDS FOR THE RIEMANN STIELTJES INTEGRAL 85 Theorem 4. Let f, g :, b R + R be, respectively, s -, s -convex functions on, b, s, s (0,. Then we hve the inequlity s s + s g (b) f () g () + s β (s +, s ) f () g (b) g () (.5) provided tht the Riemnn Stieltjes integrl f (x) dg (x) exists. If f, g re s -, s -concve, then the inequlity (.5) is reversed. Proof. Since f is s -convex on, b, by using the integrtion by prts formul for Riemnn Stieltjes integrl, we hve ( ) x s s + f () dg (x) b b ( ) x s s = dg (x) + f () dg (x) b b = (b ) s = (b ) s (x ) s dg (x) + f () (b ) s (b x) s dg (x) (b ) s g (b) s g (x) (x ) s dx (.6) + f () (b ) s (b ) s g ()+s g (x) (b x) s dx. Since g(x) is s -convex on, b, we hve ( ) x s g (x) g (b) + b which, by (.6), gives f (x) dg (x) (b ) s (b )s g (b) (( ) x s s g (b) + b + f () (b ) s (b )s g () (( ) x s +s g (b) + b s g (), b s g ()) (x ) s dx b s g ()) (b x) s dx b
6 86 MOHAMMAD WAJEEH ALOMARI = (b ) s (b ) s g (b) b g (b) s (b ) s (x ) s +s dx g () b s (b ) s (b x) s (x ) s dx + f () (b ) s (b ) s g (b) b g () + s (b ) s (x ) s (b x) s dx g () b +s (b ) s (b x) s +s dx. Simple clcultions yield tht nd where, (b x) s (x ) s dx = (b ) s +s β (s, s + ), (x ) s (b x) s dx = (b ) s +s β (s +, s ), (x ) p (b x) q dx = (b ) p+q+ ( t) p t q dt nd β (, ) is the Euler Bet function. It follows tht f (x) dg (x) (b ) s (b ) s g (b) g (b) s (b ) s (b ) s +s g () s (b ) s β (s, s + ) + f () (b ) s +s g () (b ) s = (b ) p+q+ β (p +, q + ) 0 (b ) s +s s + s (b ) s g () + s (b ) s +s g (b) (b ) s +s s + s (b ) s β (s +, s ) = g (b) s s + s g (b) s β (s, s + ) g () f () g () + s β (s +, s ) f () g (b) + s s + s f () g () = s s + s g (b) f () g ()+s β (s +, s ) f () g (b) g (),
7 BOUNDS FOR THE RIEMANN STIELTJES INTEGRAL 87 since β (s, s + ) = β (s +, s ), which proves (.5). Corollry. In Theorem 4, if s = s =, i.e., f, g re two convex functions on, b, then we hve f () + g (b) g (). Corollry. In Theorem 4, if f is convex nd g is s-convex, then we hve sg (b) g () g (b) sg () + f () (.7) s + s + provided tht the Riemnn Stieltjes integrl f (x) dg (x) exists. concve nd g is s-concve, then the inequlity (.7) is reversed. If f is Theorem 5. Let f, g :, b R + R be such tht g stisfies φ g (t) Φ for ll t, b. () If f is s-convex on, b, then we hve the inequlity g (b) φ + f () Φ g (). (.8) () If f is s-concve on, b, then we hve the inequlity f (x) dg (x) g (b) Φ + f () φ g () (.9) provided tht the Riemnn Stieltjes integrl f (x) dg (x) exists. Proof. () From (.), we get which proves (.8). (b ) s + f () (b ) s (b ) s g (b) sφ (b ) s g () + sφ (b ) s (b ) s g (b) φ (b ) s (x ) s dx + f () (b ) s (b ) s g () + Φ (b ) s = g (b) φ + f () Φ g () (b x) s dx
8 88 MOHAMMAD WAJEEH ALOMARI () If f is s-concve, then, similrly, f (x) dg (x) (b ) s (b ) s g (b) sφ + f () (b ) s (b ) s g () + sφ (b ) s (b ) s g (b) Φ (b ) s (x ) s dx + f () (b ) s (b ) s g () + φ (b ) s = g (b) Φ + f () φ g () (b x) s dx which proves (.9). Remrk. Define the function g :, b R +, g(t) = t u(s)ds. Then g is differentible on (, b) nd g (t) = u(t). And we hve f (x) dg (x) = f (x) u (x) dx. Therefore, we cn point out some results for the Riemnn integrl of product. () Under the ssumptions of Theorem, we hve f (x) u (x) dx f () + u(x)dx. () Under the ssumptions of Theorem 3, we hve the following. () If f :, b R + is convex on, b, then { b f (x) u(x)dx min f () + + f () u(x)dx, b u(x)dx + u(x)dx x } u(t)dtdx b f () +.
9 BOUNDS FOR THE RIEMANN STIELTJES INTEGRAL 89 (b) If f is concve, then { b f (x) u(x)dx mx f () + f () u(x)dx, b u(x)dx u(x)dx x } u(t)dtdx b f () +. (3) Under the ssumptions of Theorem 4, we hve s b f (x) u (x) dx + s β (s +, s ) f () u (x) dx. s + s (4) Under the ssumptions of Theorem 5, we hve the following. () If f is s-convex on, b, then f (x) u (x) dx u (x) dx φ + Φf (). (b) If f is s-concve, then f (x) u (x) dx References u (x) dx Φ + φf (). N. S. Brnett nd S. S. Drgomir, The Beesck Drst Pollrd inequlities nd pproximtions of the Riemnn Stieltjes integrl, Appl. Mth. Lett. (009), P. Cerone nd S. S. Drgomir, Approximting the Riemnn Stieltjes integrl vi some moments of the integrnd, Mth. Comput. Modelling 49 (009), P. Cerone nd S. S. Drgomir, Approximtion of the Stieltjes integrl nd pplictions in numericl integrtion, Appl. Mth. 5() (006), S. S. Drgomir, Inequlities for Stieltjes integrls with convex integrtors nd pplictions, Appl. Mth. Lett. 0() (007), S. S. Drgomir, Inequlities of Grüss type for the Stieltjes integrl nd pplictions, Krgujevc J. Mth. 6 (004), H. Hudzik nd M. Mligrnd, Some remrks on s-convex functions, Aequtiones Mth. 48() (994), 00. Deprtment of Mthemtics, Fculty of Science, Jersh University, 650 Jersh, Jordn E-mil ddress: mwomth@gmil.com
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