GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS. (b a)3 [f(a) + f(b)] f x (a,b)
|
|
- Susan Parrish
- 5 years ago
- Views:
Transcription
1 GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Abstrct. In this pper, we estblish some generliztions of weighted trpezoid inequlity for monotonic mppings, nd give severl pplictions for r moment, the expecttion of continuous rndom vrible nd the Bet mpping. 1. Introduction The trpezoid inequlity, sttes tht if f exists nd is bounded on (, b), then (1.1) f(x)dx b (b )3 [f() f(b) f 1, where f := sup f <. x (,b) Now if we ssume tht I n : = x 0 < x 1 < < x n = b is prtition of the intervl [, b nd f is s bove, then we cn pproximte the integrl f (x) dx by the trpezoidl qudrture formul A T (f, I n ), hving n error given by R T (f, I n ), where (1.) A T (f, I n ) := 1 [f ( ) f (1 ) l i, nd the reminder stisfies the estimtion (1.3) R T (f, I n ) 1 1 f li 3, with l i := 1 for i = 0, 1,..., n 1. For some recent results which generlize, improve nd extend this clssic inequlity (1.1), see the ppers [ [8. Recently, Cerone-Drgomir [3 proved the following two trpezoid type inequlities: 1991 Mthemtics Subject Clssifiction. Primry: 6D15; Secondry: 41A55. Key words nd phrses. Trpezoid inequlity, Monotonic mppings, r moment nd the expecttion of continuous rndom vrible, the Bet mpping. 1
2 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Theorem A. Let f : [, b R be monotonic non-decresing mpping. Then (1.4) f(t)dt [(x ) f () (b x) f (b) (b x) f (b) (x ) f () sgn (x t) f (t) dt (x ) [f (x) f () (b x) [f (b) f (x) (b ) x b [f (b) f (), for ll x [, b. The bove inequlities re shrp. Let I n, l i (i = 0, 1,..., ) be s bove nd let ξ i [, 1 (i = 0, 1,..., ) be intermedite points. Define the sum T P (f, I n, ξ) := [(ξ i ) f ( ) (1 ξ i ) f (1 ). We hve the following result concerning the pproximtion of the integrl f(x)dx in terms of T P. Theorem B. Let f be defined s in Theorem A, then we hve (1.5) f(x)dx = T P (f, I n, ξ) R P (f, I n, ξ). The reminder term R P (f, I n, ξ) stisfies the estimte (1.6) R P (f, I n, ξ) [(1 ξ i ) f (1 ) (ξ i ) f ( ) xi1 sgn (ξ i t) f (t) dt (ξ i ) [f (ξ i ) f ( ) (1 ξ i ) [f (1 ) f (ξ i ) l i ξ i x i 1 [f (1 ) f ( ) ν (l) mx,1,..., ξ i x i 1 [f (b) f () ν (l) [f (b) f () where ν (l) := mx {l i i = 0, 1,..., n 1}. In this pper, we estblish weighted generliztions of Theorems A-B, nd give severl pplictions for r moments nd the expecttion of continuous rndom vrible, the Bet mpping nd the Gmm mpping.. Some Integrl Inequlities Theorem 1. Let g : [, b R be non-negtive nd continuous with g (t) > 0 on (, b) nd let h : [, b R be differentible such tht h (t) = g (t) on [, b.
3 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS 3 (.1) (.) () Suppose f : [, b R is monotonic non-decresing mpping. Then f(t)g (t) dt [(x h()) f () (h(b) x) f (b) (h(b) x) f (b) (x h()) f () sgn ( h 1 (x) t ) f (t) g (t) dt (x h()) [f ( h 1 (x) ) f () (h(b) x) [f (b) f ( h 1 (x) ) [ 1 b g (t) dt h() h(b) x [f (b) f () for ll x [h(), h(b). (b) Suppose f : [, b R is monotonic non-incresing mpping. Then f(t)g (t) dt [(x h()) f () (h(b) x) f (b) (x h()) f () (h(b) x) f (b) sgn ( t h 1 (x) ) f (t) g (t) dt (x h()) [f () f ( h 1 (x) ) (h(b) x) [f ( h 1 (x) ) f (b) [ 1 b g (t) dt h() h(b) x [f () f (b) for ll x [h(), h(b). The bove inequlities re shrp. Proof. (1) () Let x [h(), h(b). identity Using integrtion by prts, we hve the following (.3) (.4) (x h(t)) df (t) = (x h(t)) f (t) = b f(t)g (t) dt f(t)g (t) dt [(x h()) f () (h(b) x) f (b). It is well known [3, p. 813 tht if µ, ν : [, b R re such tht µ is continuous on [, b nd ν is monotonic non-decresing on [, b, then µ (t) dν (t) µ (t) dν (t).
4 4 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR (.5) Now, using identity (.3) nd inequlity (.4), we hve f(t)g (t) dt [(x h()) f () (h(b) x) f (b) = x h(t) df (t) h 1 (x) (x h(t)) df (t) (h(t) x) df (t) h 1 (x) = (x h(t)) f (t) (h(t) x) h 1 (x) b h 1 (x) h 1 (x) f (t) g (t) dt h 1 (x) = (h(b) x) f (b) (x h()) f () f (t) g (t) dt sgn ( h 1 (x) t ) f (t) g (t) dt nd the first inequlities in (.1) re proved. As f is monotonic non-decresing on [, b, we obtin nd then h 1 (x) h 1 (x) f (t) g (t) dt f ( h 1 (x) ) h 1 (x) g (t) dt = (x h()) f ( h 1 (x) ) f (t) g (t) dt f ( h 1 (x) ) h 1 (x) g (t) dt = (h(b) x) f ( h 1 (x) ), sgn ( h 1 (x) t ) f (t) g (t) dt (x h()) f ( h 1 (x) ) (x h(b)) f ( h 1 (x) ). Therefore, (.6) (h(b) x) f (b) (x h()) f () sgn ( h 1 (x) t ) f (t) g (t) dt (h(b) x) f (b) (x h()) f () (x h()) f ( h 1 (x) ) (x h(b)) f ( h 1 (x) ) = (x h()) [f ( h 1 (x) ) f () (h(b) x) [f (b) f ( h 1 (x) ) which proves tht the second inequlity in (.1). As f is monotonic non-decresing on [, b, we hve f () f ( h 1 (x) ) f (b)
5 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS 5 (.7) nd (x h()) [f ( h 1 (x) ) f () (h(b) x) [f (b) f ( h 1 (x) ) mx{x h(), h(b) x} [f ( h 1 (x) ) f () f (b) f ( h 1 (x) ) [ h(b) h() = h() h(b) x [f (b) f () [ 1 b = g (t) dt h() h(b) x [f (b) f (). Thus, by (.5), (.6) nd (.7), we obtin (.1). Let g (t) 1, t [, b h (t) = t, t [, b { 0, t [, b) f(t) = 1, t = b nd x = b. Then f(t)g (t) dt [(x h()) f () (h(b) x) f (b) = (h(b) x) f (b) (x h()) f () sgn ( h 1 (x) t ) f (t) g (t) dt = (x h()) [f ( h 1 (x) ) f () (h(b) x) [f (b) f ( h 1 (x) ) [ 1 b = g (t) dt h() h(b) x [f (b) f () = b which proves tht the inequlities (.1) re shrp. (b) If f is replced by f in (), then (.) is obtined from (.1). This completes the proof. Remrk 1. If we choose g (t) 1, h (t) = t on [, b, then the inequlities (.1) reduce to (1.4). Corollry 1. If we choose x = h()h(b), then we get f () f (b) b (.8) f(t)g (t) dt g (t) dt 1 1 g (t) dt [f (b) f () ( ( ) ) h() h(b) sgn h 1 t f (t) g (t) dt g (t) dt [f (b) f ()
6 6 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR where f nd g re defined s in () of Theorem 1, nd f () f (b) b (.9) f(t)g (t) dt g (t) dt 1 1 g (t) dt [f () f (b) ( ( )) h() h(b) sgn t h 1 f (t) g (t) dt g (t) dt [f () f (b) where f nd g re defined s in (b) of Theorem 1. The inequlities (.8) nd (.9) re the weighted trpezoid inequlities. Note tht the trpezoid inequlity (.8) nd (.9) re, in sense, the best possible inequlities we cn obtin from (.1) nd (.). Moreover, the constnt 1 is the best possible for both inequlities in (.8) nd (.9), respectively. Remrk. The following inequlity is well-known in the literture s the Fejér inequlity (see for exmple [9): ( ) b b f () f (b) b (.10) f g (t) dt f(t)g (t) dt g (t) dt, where f : [, b R is convex nd g : [, b R is positive integrble nd symmetric to b. Using the bove results nd (.8) (.9), we obtin the following error bound of the second inequlity in (.10): (.11) f () f (b) g (t) dt g (t) dt [f (b) f () f(t)g (t) dt ( ) sgn h 1 h() h(b) ( ) t f (t) g (t) dt g (t) dt [f (b) f () provided tht f is monotonic non-decresing on [, b. (.1) f () f (b) g (t) dt g (t) dt [f () f (b) f(t)g (t) dt ( ) sgn t h 1 h() h(b) ( ) f (t) g (t) dt g (t) dt [f () f (b) provided tht f is monotonic non-incresing on [, b.
7 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS 7 3. Applictions for Qudrture Formul Throughout this section, let g nd h be defined s in Theorem 1. Let f : [, b R, nd let I n : = x 0 < x 1 < < x n = b be prtition of [, b nd ξ i [h( ), h(1 ) (i = 0, 1,..., n 1) be intermedite points. Put l i := h(1 ) h( ) = 1 g (t) dt nd define the sum T P (f, g, h, I n, ξ) := [(ξ i h( )) f ( ) (h(1 ) ξ i ) f (1 ). We hve the following result concerning the pproximtion of the integrl f(t)g (t) dt in terms of T P. Theorem. Let ν (l) := mx {l i i = 0, 1,..., n 1}, f be defined s in Theorem 1 nd let (3.1) f(t)g (t) dt = T P (f, g, h, I n, ξ) R P (f, g, h, I n, ξ). Then, the reminder term R P (f, g, h, I n, ξ) stisfies the following estimtes: () Suppose f is monotonic non-decresing on [, b, then (3.) R P (f, g, h, I n, ξ) [(h (1 ) ξ i ) f (1 ) (ξ i h ( )) f ( ) xi1 sgn ( h 1 (ξ i ) t ) f (t) g (t) dt (ξ i h ( )) [ f ( h 1 (ξ i ) ) f ( ) (h (1 ) ξ i ) [ f (1 ) f ( h 1 (ξ i ) ) l i ξ i h (x i) h (1 ) [f (1 ) f ( ) ν (l) mx,1,..., ξ i h (x i) h (1 ) [f (b) f () ν (l) [f (b) f (). (b) Suppose f is monotonic non-incresing on [, b, then (3.3) R P (f, g, h, I n, ξ) [(ξ i h ( )) f ( ) (h (1 ) ξ i ) f (1 ) xi1 sgn ( t h 1 (ξ i ) ) f (t) g (t) dt
8 8 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR (ξ i h ( )) [ f ( ) f ( h 1 (ξ i ) ) (h (1 ) ξ i ) [ f ( h 1 (ξ i ) ) f (1 ) l i ξ i h (x i) h (1 ) [f ( ) f (1 ) ν (l) mx,1,..., ξ i h (x i) h (1 ) [f () f (b) ν (l) [f () f (b). Proof. (1) () Apply Theorem 1 on the intervls [, 1 (i = 0, 1,..., n 1) to get xi1 f(t)g (t) dt [(ξ i h( )) f ( ) (h(1 ) ξ i ) f (1 ) (h(1 ) ξ i ) f (1 ) (ξ i h( )) f ( ) xi1 sgn ( h 1 (ξ i ) t ) f (t) g (t) dt (ξ i h( )) [f ( h 1 (ξ i ) ) f ( ) (h(1 ) ξ i ) [f (1 ) f ( h 1 (ξ i ) ) l i ξ i h(x i) h(1 ) [f (1 ) f ( ) for ll i {0, 1,..., n 1}.
9 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS 9 (3.4) (3.5) Using this nd the generlized tringle inequlity, we hve R P (f, g, h, I n, ξ) xi1 f(t)g (t) dt [(ξ i h( )) f ( ) (h(1 ) ξ i ) f (1 ) [(h (1 ) ξ i ) f (1 ) (ξ i h ( )) f ( ) xi1 sgn ( h 1 (ξ i ) t ) f (t) g (t) dt (ξ i h ( )) [ f ( h 1 (ξ i ) ) f ( ) (h (1 ) ξ i ) [ f (1 ) f ( h 1 (ξ i ) ) l i ξ i h (x i) h (1 ) [f (1 ) f ( ) ν (l) mx,1,..., ξ i h (x i) h (1 ) [f (b) f (). Next, we observe tht ξ i h() h(1 ) 1 l i (i = 0, 1,..., n 1); nd then (3.6) mx,1,..., ξ i h() h(1 ) 1 ν (l). Thus, by (3.4), (3.5) nd (3.6), we obtin (3.). (b) The proof is similr s () nd we omit the detils. This completes the proof. Remrk 3. If we choose g (t) 1, h (t) = t on [, b, then the inequlities (3.) reduce to (1.6). Now, let ξ i = h(xi)h(xi1) (i = 0, 1,..., n 1) nd let T P W (f, g, h, I n ) nd R P (f, g, h, I n ) be defined s T P W (f, g, h, I n ) = T P (f, g, h, I n, ξ) = 1 xi1 [f ( ) f (1 ) g (t) dt nd R P W (f, g, h, I n ) = R P (f, g, h, I n, ξ) = f (t) g (t) dt 1 xi1 [f ( ) f (1 ) g (t) dt. If we consider the weighted trpezoidl formul T P W (f, g, h, I n ), then we hve the following corollry:
10 10 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Corollry. Let f, g, h be defined s in Theorem nd let ξ i = h(xi)h(xi1) (i = 0, 1,..., n 1). Then f (t) g (t) dt = T P W (f, g, h, I n ) R P W (f, g, h, I n ) where the reminder stisfies the following estimtes: (3.7) (3.8) () Suppose f is monotonic non-decresing on [, b, then R P W (f, g, h, I n ) 1 ( xi1 ) g (t) dt [f (1 ) f ( ) 1 xi1 ( xi1 ν (l) g (t) dt [f (b) f (). ( ( ) ) h sgn h 1 (xi ) h (1 ) t f (t) g (t) dt ) [f (1 ) f ( ) (b) Suppose f is monotonic non-incresing on [, b, then R P W (f, g, h, I n ) 1 ( xi1 ) g (t) dt [f ( ) f (1 ) 1 xi1 ( xi1 ν (l) g (t) dt [f () f (b). ( ( )) h sgn t h 1 (xi ) h (1 ) f (t) g (t) dt ) [f ( ) f (1 ) Remrk 4. In Corollry, suppose f is monotonic on [, b, [ = h 1 i (h(b) h()) h () (i = 0, 1,..., n), n nd l i := h(1 ) h( ) = h(b) h() n = 1 n g (t) dt. (i = 0, 1,..., n 1). If we wnt to pproximte the integrl f (t) g (t) dt by T P W (f, g, h, I n ) with n ccurcy less tht ε > 0, we need t lest n ε N points for the prttion I n, where n ε := [ 1 ε g (t) dt f (b) f () nd [r denotes the Gussin integer of r (r R). 1
11 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS Some Inequlities for Rndom Vribles Throughout this section, let 0 < < b, r R, nd let X be continuous rndom vrible hving the continuous probbility density mpping g : [, b R with g (t) > 0 on (, b), h : [, b R with h (t) = g (t) for t (, b) nd the r moment which is ssumed to be finite. E r (X) := Theorem 3. The inequlities E r (X) r b r (4.1) 1 (br r ) t r g (t) dt, 1 (br r ) s r 0 ( ( ) ) 1 sgn h 1 t t r g (t) dt nd (4.) E r (X) r b r 1 (r b r ) 1 (r b r ) s r < 0, ( ( )) 1 sgn t h 1 t r g (t) dt hold. Proof. If we put f (t) = t r (t [, b), h (t) = t g (x) dx (t [, b)nd x = h()h(b) = 1 in Corollry 1, then we obtin (4.1) nd (4.). This completes the proof. The following corollry which is specil cse of Theorem 3. Corollry 3. The inequlities (4.3) E (X) b b ( ( ) ) 1 sgn h 1 t tg (t) dt b hold where E (X) is the expecttion of the rndom vrible X. 5. Inequlities for Bet Mpping nd Gmm Mpping The following two mppings re well-known in the literture s the Bet mpping nd the Gmm mpping, respectively: β (x, y) := Γ(x) = t x 1 (1 t) y 1 dt, x > 0, y > 0. e t t x 1 dt, x > 0. The following inequlity which is n pplicton of Theorem 1 for the Bet mpping holds:
12 1 KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Theorem 4. Let p, q > 0. Then we hve the inequlity (5.1) for ll x [ 1 0, p1. β (p 1, q 1) x x sgn [t ((p 1) x) 1 p1 ( 1 x p 1 x 1 (p 1) x 1 (p 1) t p (1 t) q dt ) [ 1 ((p 1) x) 1 q p1 Proof. If we put = 0, b = 1, f(t) = (1 t) q, g(t) = t p nd h (t) = tp1 p1 (t [0, 1) [ 1 in Theorem 1, we obtin the inequlity (5.1) for ll x 0, p1. This completes the proof. The following remrk which is n pplicton of Theorem 4 for the Gmm mpping holds: Remrk 5. Tking into ccount tht β (p 1, q 1) = Γ(p1)Γ(q1) Γ(pq), the inequlity (5.1) is equivlent to i.e., Γ(p 1)Γ(q 1) Γ(p q ) x x ( 1 x p 1 x 1 (p 1) x 1 (p 1) sgn [t ((p 1) x) 1 p1 t p (1 t) q dt ) [ 1 ((p 1) x) 1 q p1 (p 1)Γ(p 1)Γ(q 1) x(p 1)Γ(p q ) [ b x sgn [t ((p 1) x) 1 p1 t p (1 t) q dt (p 1)Γ(p q ) [ ( ) 1 [ x p 1 x 1 ((p 1) x) 1 q p1 (p 1)Γ(p q ) x(p 1) 1 Γ(p q ) nd s (p 1)Γ(p 1) = Γ(p ), we get (5.) Γ(p )Γ(q 1) x(p 1)Γ(p q ) [ b x sgn [t ((p 1) x) 1 p1 t p (1 t) q dt (p 1)Γ(p q ) [ ( ) 1 [ x p 1 x 1 ((p 1) x) 1 q p1 (p 1)Γ(p q ) x(p 1) 1 Γ(p q ).
13 WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS 13 References [1 T.M. Apostol, Mthemticl Anlysis, Second Edition, Addision-Wesley Publishing Compny, [ N.S. Brnett, S.S. Drgomir nd C.E.M. Perce, A qusi-trpezoid inequlity for double integrls, submitted. [3 P. Cerone nd S.S. Drgomir, Trpezoidl Type Rules from An Inequlities Point of View, Hndbook of Anlytic-Computtionl Methods in Applied Mthemtics, Editor: G. Anstssiou, CRC Press, N.Y., (000), [4 S.S. Drgomir, P. Cerone nd A. Sofo, Some remrks on the trpezoid rule in numericl integrtion, Indin J. of Pure nd Appl. Mth., 31(5) (000), [5 S.S. Drgomir nd T.C. Pechey, New estimtion of the reminder in the trpezoidl formul with pplictions, Studi Universittis Bbes-Bolyi Mthemtic, XLV(4) (000), [6 S.S. Drgomir, On the trpezoid qudrture formul nd pplictions, Krgujevc J. Mth., 3 (001), [7 S.S. Drgomir, On the trpezoid qudrture formul for Lipschitzin mppings nd pplictions, Tmkng J. of Mth., 30 () (1999), [8 P. Cerone, S.S. Drgomir nd C.E.M. Perce, A generlized trpezoid inequlity for functions of bounded vrition, Turkish J. Mth., 4() (000), [9 L. Fejér, Uberdie Fourierreihen, II, Mth. Ntur. Ungr. Akd Wiss., 4 (1906), [In Hungrin Deprtment of Mthemtics, Alethei University, Tmsui, Tiwn E-mil ddress: kltseng@emil.u.edu.tw Deprtment of Mthemtics, Tmkng University, Tmsui, Tiwn School of Computer Science nd Mthemtics, Victori University of Technology, PO Box 1448, MCMC 8001, Victori, Austrli. E-mil ddress: sever@mtild.vu.edu.u URL:
Bulletin of the. Iranian Mathematical Society
ISSN: 07-060X Print ISSN: 735-855 Online Bulletin of the Irnin Mthemticl Society Vol 3 07, No, pp 09 5 Title: Some extended Simpson-type ineulities nd pplictions Authors: K-C Hsu, S-R Hwng nd K-L Tseng
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.
More informationS. S. Dragomir. 2, we have the inequality. b a
Bull Koren Mth Soc 005 No pp 3 30 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Abstrct Compnions of Ostrowski s integrl ineulity for bsolutely
More informationNEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS. := f (4) (x) <. The following inequality. 2 b a
NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS MOHAMMAD ALOMARI A MASLINA DARUS A AND SEVER S DRAGOMIR B Abstrct In terms of the first derivtive some ineulities of Simpson
More informationS. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:
FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy
More informationAN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir
RGMIA Reserch Report Collection, Vol., No., 999 http://sci.vu.edu.u/ rgmi AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS I. Fedotov nd S. S. Drgomir Astrct. An
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARI- ABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NEIL S. BARNETT, PIETRO CERONE, SEVER S. DRAGOMIR
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics http://jipmvueduu/ Volume, Issue, Article, 00 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NS BARNETT,
More informationSOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL
SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS Abstrct Some ineulities for the dispersion of rndom
More informationTRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS
TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS S.S. DRAGOMIR AND A. SOFO Abstrct. In this pper by utilising result given by Fink we obtin some new results relting to the trpezoidl inequlity
More informationON PERTURBED TRAPEZOIDAL AND MIDPOINT RULES. f (t) dt
ON PERTURBED TRAPEZOIDAL AND MIDPOINT RULES P. CERONE Abstrct. Explicit bounds re obtined for the perturbed or corrected trpezoidl nd midpoint rules in terms of the Lebesque norms of the second derivtive
More informationNew Integral Inequalities for n-time Differentiable Functions with Applications for pdfs
Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353-362 New Integrl Inequlities for n-time Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute
More information0 N. S. BARNETT AND S. S. DRAGOMIR Using Gruss' integrl inequlity, the following pertured trpezoid inequlity in terms of the upper nd lower ounds of t
TAMKANG JOURNAL OF MATHEMATICS Volume 33, Numer, Summer 00 ON THE PERTURBED TRAPEZOID FORMULA N. S. BARNETT AND S. S. DRAGOMIR Astrct. Some inequlities relted to the pertured trpezoid formul re given.
More informationA unified generalization of perturbed mid-point and trapezoid inequalities and asymptotic expressions for its error term
An. Ştiinţ. Univ. Al. I. Cuz Işi. Mt. (N.S. Tomul LXIII, 07, f. A unified generliztion of perturbed mid-point nd trpezoid inequlities nd symptotic expressions for its error term Wenjun Liu Received: 7.XI.0
More informationAN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
Applied Mthemtics E-Notes, 5(005), 53-60 c ISSN 1607-510 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
More informationf (a) + f (b) f (λx + (1 λ)y) max {f (x),f (y)}, x, y [a, b]. (1.1)
TAMKANG JOURNAL OF MATHEMATICS Volume 41, Number 4, 353-359, Winter 1 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI, M. DARUS
More informationSome estimates on the Hermite-Hadamard inequality through quasi-convex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper
More informationKeywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.
Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. Lmi-Athens Lmi 3500 Greece Abstrct Using
More informationBounds for the Riemann Stieltjes integral via s-convex integrand or integrator
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 6, Number, 0 Avilble online t www.mth.ut.ee/ct/ Bounds for the Riemnn Stieltjes integrl vi s-convex integrnd or integrtor Mohmmd Wjeeh
More informationWEIGHTED INTEGRAL INEQUALITIES OF OSTROWSKI, 1 (b a) 2. f(t)g(t)dt. provided that there exists the real numbers m; M; n; N such that
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted 6 June 8 doi.944/preprints86.4.v WEIGHTED INTEGRAL INEQUALITIES OF OSTROWSKI, µcebyšev AND LUPAŞ TYPE WITH APPLICATIONS SILVESTRU SEVER DRAGOMIR Abstrct.
More informationNew general integral inequalities for quasiconvex functions
NTMSCI 6, No 1, 1-7 18 1 New Trends in Mthemticl Sciences http://dxdoiorg/185/ntmsci1739 New generl integrl ineulities for usiconvex functions Cetin Yildiz Atturk University, K K Eduction Fculty, Deprtment
More informationA Generalized Inequality of Ostrowski Type for Twice Differentiable Bounded Mappings and Applications
Applied Mthemticl Sciences, Vol. 8, 04, no. 38, 889-90 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.988/ms.04.4 A Generlized Inequlity of Ostrowski Type for Twice Differentile Bounded Mppings nd Applictions
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics http://jipm.vu.edu.u/ Volume 3, Issue, Article 4, 00 ON AN IDENTITY FOR THE CHEBYCHEV FUNCTIONAL AND SOME RAMIFICATIONS P. CERONE SCHOOL OF COMMUNICATIONS
More informationImprovement of Ostrowski Integral Type Inequalities with Application
Filomt 30:6 06), 56 DOI 098/FIL606Q Published by Fculty of Sciences nd Mthemtics, University of Niš, Serbi Avilble t: http://wwwpmfnicrs/filomt Improvement of Ostrowski Integrl Type Ineulities with Appliction
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics MOMENTS INEQUALITIES OF A RANDOM VARIABLE DEFINED OVER A FINITE INTERVAL PRANESH KUMAR Deprtment of Mthemtics & Computer Science University of Northern
More informationA Companion of Ostrowski Type Integral Inequality Using a 5-Step Kernel with Some Applications
Filomt 30:3 06, 360 36 DOI 0.9/FIL6360Q Pulished y Fculty of Sciences nd Mthemtics, University of Niš, Seri Aville t: http://www.pmf.ni.c.rs/filomt A Compnion of Ostrowski Type Integrl Inequlity Using
More informationON AN INTEGRATION-BY-PARTS FORMULA FOR MEASURES
Volume 8 (2007), Issue 4, Article 93, 13 pp. ON AN INTEGRATION-BY-PARTS FORMULA FOR MEASURES A. ČIVLJAK, LJ. DEDIĆ, AND M. MATIĆ AMERICAN COLLEGE OF MANAGEMENT AND TECHNOLOGY ROCHESTER INSTITUTE OF TECHNOLOGY
More informationRGMIA Research Report Collection, Vol. 1, No. 1, SOME OSTROWSKI TYPE INEQUALITIES FOR N-TIME DIFFERENTIA
ttp//sci.vut.edu.u/rgmi/reports.tml SOME OSTROWSKI TYPE INEQUALITIES FOR N-TIME DIFFERENTIABLE MAPPINGS AND APPLICATIONS P. CERONE, S.S. DRAGOMIR AND J. ROUMELIOTIS Astrct. Some generliztions of te Ostrowski
More informationON THE WEIGHTED OSTROWSKI INEQUALITY
ON THE WEIGHTED OSTROWSKI INEQUALITY N.S. BARNETT AND S.S. DRAGOMIR School of Computer Science nd Mthemtics Victori University, PO Bo 14428 Melbourne City, VIC 8001, Austrli. EMil: {neil.brnett, sever.drgomir}@vu.edu.u
More informationSOME INTEGRAL INEQUALITIES OF GRÜSS TYPE
RGMIA Reserch Report Collection, Vol., No., 998 http://sci.vut.edu.u/ rgmi SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE S.S. DRAGOMIR Astrct. Some clssicl nd new integrl inequlities of Grüss type re presented.
More informationAn inequality related to η-convex functions (II)
Int. J. Nonliner Anl. Appl. 6 (15) No., 7-33 ISSN: 8-68 (electronic) http://d.doi.org/1.75/ijn.15.51 An inequlity relted to η-conve functions (II) M. Eshghi Gordji, S. S. Drgomir b, M. Rostmin Delvr, Deprtment
More informationResearch Article On New Inequalities via Riemann-Liouville Fractional Integration
Abstrct nd Applied Anlysis Volume 202, Article ID 428983, 0 pges doi:0.55/202/428983 Reserch Article On New Inequlities vi Riemnn-Liouville Frctionl Integrtion Mehmet Zeki Sriky nd Hsn Ogunmez 2 Deprtment
More informationResearch Article On The Hadamard s Inequality for Log-Convex Functions on the Coordinates
Hindwi Publishing Corportion Journl of Inequlities nd Applictions Volume 29, Article ID 28347, 3 pges doi:.55/29/28347 Reserch Article On The Hdmrd s Inequlity for Log-Convex Functions on the Coordintes
More informationRIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROXIMATION OF CSISZAR S f DIVERGENCE
SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17 (2009, 3 12 RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROIMATION OF CSISZAR S f DIVERGENCE GEORGE A. ANASTASSIOU Abstrct. Here re estblished vrious tight probbilistic
More informationThe Hadamard s inequality for quasi-convex functions via fractional integrals
Annls of the University of Criov, Mthemtics nd Computer Science Series Volume (), 3, Pges 67 73 ISSN: 5-563 The Hdmrd s ineulity for usi-convex functions vi frctionl integrls M E Özdemir nd Çetin Yildiz
More informationAn optimal 3-point quadrature formula of closed type and error bounds
Revist Colombin de Mtemátics Volumen 8), págins 9- An optiml 3-point qudrture formul of closed type nd error bounds Un fórmul de cudrtur óptim de 3 puntos de tipo cerrdo y error de fronter Nend Ujević,
More informationINEQUALITIES FOR BETA AND GAMMA FUNCTIONS VIA SOME CLASSICAL AND NEW INTEGRAL INEQUALITIES
INEQUALITIES FOR BETA AND GAMMA FUNCTIONS VIA SOME CLASSICAL AND NEW INTEGRAL INEQUALITIES S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT Abstrct. In this survey pper we present the nturl ppliction of
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued
More informationQUADRATURE is an old-fashioned word that refers to
World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 A New Qudrture Rule Derived from Spline Interpoltion with Error Anlysis Hdi Tghvfrd
More informationINNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS AND APPLICATIONS
INNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS AND APPLICATIONS S. S. DRAGOMIR Abstrct. Some inequlities for two inner products h i nd h i which generte the equivlent norms kk nd kk with pplictions
More informationGENERALIZED ABSTRACTED MEAN VALUES
GENERALIZED ABSTRACTED MEAN VALUES FENG QI Abstrct. In this rticle, the uthor introduces the generlized bstrcted men vlues which etend the concepts of most mens with two vribles, nd reserches their bsic
More informationThe Hadamard s Inequality for s-convex Function
Int. Journl o Mth. Anlysis, Vol., 008, no. 3, 639-646 The Hdmrd s Inequlity or s-conve Function M. Alomri nd M. Drus School o Mthemticl Sciences Fculty o Science nd Technology Universiti Kebngsn Mlysi
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationRevista Colombiana de Matemáticas Volumen 41 (2007), páginas 1 13
Revist Colombin de Mtemátics Volumen 4 7, págins 3 Ostrowski, Grüss, Čebyšev type inequlities for functions whose second derivtives belong to Lp,b nd whose modulus of second derivtives re convex Arif Rfiq
More informationON COMPANION OF OSTROWSKI INEQUALITY FOR MAPPINGS WHOSE FIRST DERIVATIVES ABSOLUTE VALUE ARE CONVEX WITH APPLICATIONS
Miskolc Mthemticl Notes HU ISSN 787-5 Vol. 3 (), No., pp. 33 8 ON OMPANION OF OSTROWSKI INEQUALITY FOR MAPPINGS WHOSE FIRST DERIVATIVES ABSOLUTE VALUE ARE ONVEX WITH APPLIATIONS MOHAMMAD W. ALOMARI, M.
More informationHermite-Hadamard type inequalities for harmonically convex functions
Hcettepe Journl o Mthemtics nd Sttistics Volume 43 6 4 935 94 Hermite-Hdmrd type ineulities or hrmoniclly convex unctions İmdt İşcn Abstrct The uthor introduces the concept o hrmoniclly convex unctions
More informationOn New Inequalities of Hermite-Hadamard-Fejer Type for Harmonically Quasi-Convex Functions Via Fractional Integrals
X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey On New Ineulities of Hermite-Hdmrd-Fejer Type for Hrmoniclly Qusi-Convex Functions Vi Frctionl Integrls Mehmet Kunt * nd İmdt İşcn Deprtment
More informationON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs
ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin,
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationOn some refinements of companions of Fejér s inequality via superquadratic functions
Proyecciones Journl o Mthemtics Vol. 3, N o, pp. 39-33, December. Universidd Ctólic del Norte Antogst - Chile On some reinements o compnions o Fejér s inequlity vi superqudrtic unctions Muhmmd Amer Lti
More informationASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 3, September 2010 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II TIBERIU TRIF Dedicted to Professor Grigore Ştefn
More informationNew Integral Inequalities of the Type of Hermite-Hadamard Through Quasi Convexity
Punjb University Journl of Mthemtics (ISSN 116-56) Vol. 45 (13) pp. 33-38 New Integrl Inequlities of the Type of Hermite-Hdmrd Through Qusi Convexity S. Hussin Deprtment of Mthemtics, College of Science,
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationINEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION
INEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION BAI-NI GUO AND FENG QI Abstrct. In the rticle, using the Tchebycheff s integrl inequlity, the suitble properties of double integrl nd
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationLOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS VIA TAYLOR S EXPANSION WITH INTEGRAL REMAINDER
LOGARITHMIC INEQUALITIES FOR TWO POSITIVE NUMBERS VIA TAYLOR S EXPANSION WITH INTEGRAL REMAINDER S. S. DRAGOMIR ;2 Astrct. In this pper we otin severl new logrithmic inequlities for two numers ; minly
More informationFUNCTIONS OF α-slow INCREASE
Bulletin of Mthemticl Anlysis nd Applictions ISSN: 1821-1291, URL: http://www.bmth.org Volume 4 Issue 1 (2012), Pges 226-230. FUNCTIONS OF α-slow INCREASE (COMMUNICATED BY HÜSEYIN BOR) YILUN SHANG Abstrct.
More informationSome Hermite-Hadamard type inequalities for functions whose exponentials are convex
Stud. Univ. Beş-Bolyi Mth. 6005, No. 4, 57 534 Some Hermite-Hdmrd type inequlities for functions whose exponentils re convex Silvestru Sever Drgomir nd In Gomm Astrct. Some inequlities of Hermite-Hdmrd
More informationImprovements of some Integral Inequalities of H. Gauchman involving Taylor s Remainder
Divulgciones Mtemátics Vol. 11 No. 2(2003), pp. 115 120 Improvements of some Integrl Inequlities of H. Guchmn involving Tylor s Reminder Mejor de lguns Desigulddes Integrles de H. Guchmn que involucrn
More informationOstrowski Grüss Čebyšev type inequalities for functions whose modulus of second derivatives are convex 1
Generl Mthemtics Vol. 6, No. (28), 7 97 Ostrowski Grüss Čebyšev type inequlities for functions whose modulus of second derivtives re convex Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn Abstrct In this pper,
More informationDIFFERENCE BETWEEN TWO RIEMANN-STIELTJES INTEGRAL MEANS
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-
More informationINEQUALITIES OF HERMITE-HADAMARD TYPE FOR
Preprints (www.preprints.org) NOT PEER-REVIEWED Posted: 7 June 8 doi:.944/preprints86.44.v INEQUALITIES OF HERMITE-HADAMARD TYPE FOR COMPOSITE h-convex FUNCTIONS SILVESTRU SEVER DRAGOMIR ; Abstrct. In
More informationAPPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A TRAPEZOIDAL QUADRATURE RULE WITH APPLICATIONS
APPROXIMATING THE RIEMANN-STIELTJES INTEGRAL BY A TRAPEZOIDAL QUADRATURE RULE WITH APPLICATIONS S.S. DRAGOMIR Astrct. In this pper we provide shrp ounds for the error in pproximting the Riemnn-Stieltjes
More informationGeneralized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral
DOI 763/s4956-6-4- Moroccn J Pure nd Appl AnlMJPAA) Volume ), 6, Pges 34 46 ISSN: 35-87 RESEARCH ARTICLE Generlized Hermite-Hdmrd-Fejer type inequlities for GA-conve functions vi Frctionl integrl I mdt
More informationWENJUN LIU AND QUÔ C ANH NGÔ
AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of Ostrowski-Grüss type on time scles nd thus unify corresponding continuous
More informationDEFINITE INTEGRALS. f(x)dx exists. Note that, together with the definition of definite integrals, definitions (2) and (3) define b
DEFINITE INTEGRALS JOHN D. MCCARTHY Astrct. These re lecture notes for Sections 5.3 nd 5.4. 1. Section 5.3 Definition 1. f is integrle on [, ] if f(x)dx exists. Definition 2. If f() is defined, then f(x)dx.
More informationHomework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.
Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationResearch Article Fejér and Hermite-Hadamard Type Inequalities for Harmonically Convex Functions
Hindwi Pulishing Corportion Journl of Applied Mthemtics Volume 4, Article ID 38686, 6 pges http://dx.doi.org/.55/4/38686 Reserch Article Fejér nd Hermite-Hdmrd Type Inequlities for Hrmoniclly Convex Functions
More informationLYAPUNOV-TYPE INEQUALITIES FOR NONLINEAR SYSTEMS INVOLVING THE (p 1, p 2,..., p n )-LAPLACIAN
Electronic Journl of Differentil Equtions, Vol. 203 (203), No. 28, pp. 0. ISSN: 072-669. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu LYAPUNOV-TYPE INEQUALITIES FOR
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationMAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL
MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA-302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition
More informationINEQUALITIES OF HERMITE-HADAMARD S TYPE FOR FUNCTIONS WHOSE DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX
INEQUALITIES OF HERMITE-HADAMARD S TYPE FOR FUNCTIONS WHOSE DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI A, M. DARUS A, AND S.S. DRAGOMIR B Astrct. In this er, some ineulities of Hermite-Hdmrd
More informationON A CONVEXITY PROPERTY. 1. Introduction Most general class of convex functions is defined by the inequality
Krgujevc Journl of Mthemtics Volume 40( (016, Pges 166 171. ON A CONVEXITY PROPERTY SLAVKO SIMIĆ Abstrct. In this rticle we proved n interesting property of the clss of continuous convex functions. This
More informationHermite-Hadamard Type Inequalities for the Functions whose Second Derivatives in Absolute Value are Convex and Concave
Applied Mthemticl Sciences Vol. 9 05 no. 5-36 HIKARI Ltd www.m-hikri.com http://d.doi.org/0.988/ms.05.9 Hermite-Hdmrd Type Ineulities for the Functions whose Second Derivtives in Absolute Vlue re Conve
More informationSet Integral Equations in Metric Spaces
Mthemtic Morvic Vol. 13-1 2009, 95 102 Set Integrl Equtions in Metric Spces Ion Tişe Abstrct. Let P cp,cvr n be the fmily of ll nonempty compct, convex subsets of R n. We consider the following set integrl
More informationParametrized inequality of Hermite Hadamard type for functions whose third derivative absolute values are quasi convex
Wu et l. SpringerPlus (5) 4:83 DOI.8/s44-5-33-z RESEARCH Prmetrized inequlity of Hermite Hdmrd type for functions whose third derivtive bsolute vlues re qusi convex Shn He Wu, Bnyt Sroysng, Jin Shn Xie
More informationImprovement of Grüss and Ostrowski Type Inequalities
Filomt 9:9 (05), 07 035 DOI 098/FIL50907A Pulished y Fculty of Sciences nd Mthemtics, University of Niš, Seri Aville t: http://wwwpmfnicrs/filomt Improvement of Grüss nd Ostrowski Type Inequlities An Mri
More informationCommunications inmathematicalanalysis Volume 6, Number 2, pp (2009) ISSN
Communictions inmthemticlanlysis Volume 6, Number, pp. 33 41 009) ISSN 1938-9787 www.commun-mth-nl.org A SHARP GRÜSS TYPE INEQUALITY ON TIME SCALES AND APPLICATION TO THE SHARP OSTROWSKI-GRÜSS INEQUALITY
More informationarxiv: v1 [math.ca] 28 Jan 2013
ON NEW APPROACH HADAMARD-TYPE INEQUALITIES FOR s-geometrically CONVEX FUNCTIONS rxiv:3.9v [mth.ca 8 Jn 3 MEVLÜT TUNÇ AND İBRAHİM KARABAYIR Astrct. In this pper we chieve some new Hdmrd type ineulities
More informationON SOME NEW INEQUALITIES OF HADAMARD TYPE INVOLVING h-convex FUNCTIONS. 1. Introduction. f(a) + f(b) f(x)dx b a. 2 a
Act Mth. Univ. Comenine Vol. LXXIX, (00, pp. 65 7 65 ON SOME NEW INEQUALITIES OF HADAMARD TYPE INVOLVING h-convex FUNCTIONS M. Z. SARIKAYA, E. SET nd M. E. ÖZDEMIR Abstrct. In this pper, we estblish some
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function
More informationON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES
Volume 1 29, Issue 3, Article 86, 5 pp. ON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES SOUMIA BELARBI AND ZOUBIR DAHMANI DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MOSTAGANEM soumi-mth@hotmil.fr zzdhmni@yhoo.fr
More informationON CLOSED CONVEX HULLS AND THEIR EXTREME POINTS. S. K. Lee and S. M. Khairnar
Kngweon-Kyungki Mth. Jour. 12 (2004), No. 2, pp. 107 115 ON CLOSED CONVE HULLS AND THEIR ETREME POINTS S. K. Lee nd S. M. Khirnr Abstrct. In this pper, the new subclss denoted by S p (α, β, ξ, γ) of p-vlent
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, Bbeş-Bolyi University Str. Koglnicenu
More informationChapter 6. Riemann Integral
Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl
More informationOn the Generalized Weighted Quasi-Arithmetic Integral Mean 1
Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, 2039-2048 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2013.3499 On the Generlized Weighted Qusi-Arithmetic Integrl Men 1 Hui Sun School
More informationSome inequalities of Hermite-Hadamard type for n times differentiable (ρ, m) geometrically convex functions
Avilble online t www.tjns.com J. Nonliner Sci. Appl. 8 5, 7 Reserch Article Some ineulities of Hermite-Hdmrd type for n times differentible ρ, m geometriclly convex functions Fiz Zfr,, Humir Klsoom, Nwb
More informationResearch Article On Hermite-Hadamard Type Inequalities for Functions Whose Second Derivatives Absolute Values Are s-convex
ISRN Applied Mthemtics, Article ID 8958, 4 pges http://dx.doi.org/.55/4/8958 Reserch Article On Hermite-Hdmrd Type Inequlities for Functions Whose Second Derivtives Absolute Vlues Are s-convex Feixing
More informationUNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationWHEN IS A FUNCTION NOT FLAT? 1. Introduction. {e 1 0, x = 0. f(x) =
WHEN IS A FUNCTION NOT FLAT? YIFEI PAN AND MEI WANG Abstrct. In this pper we prove unique continution property for vector vlued functions of one vrible stisfying certin differentil inequlity. Key words:
More informationNEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX
Journl of Mthemticl Ineulities Volume 1, Number 3 18, 655 664 doi:1.7153/jmi-18-1-5 NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX SHAHID
More informationIntegral inequalities for n times differentiable mappings
JACM 3, No, 36-45 8 36 Journl of Abstrct nd Computtionl Mthemtics http://wwwntmscicom/jcm Integrl ineulities for n times differentible mppings Cetin Yildiz, Sever S Drgomir Attur University, K K Eduction
More informationA HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES. 1. Introduction
Ttr Mt. Mth. Publ. 44 (29), 159 168 DOI: 1.2478/v1127-9-56-z t m Mthemticl Publictions A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES Miloslv Duchoň Peter Mličký ABSTRACT. We present Helly
More informationGrüss Type Inequalities for Complex Functions Defined on Unit Circle with Applications for Unitary Operators in Hilbert Spaces
v49n4-drgomir 5/6/3 : pge 77 # Revist Colombin de Mtemátics Volumen 49(5), págins 77-94 Grüss Type Inequlities for Complex Functions Defined on Unit Circle with Applictions for Unitry Opertors in Hilbert
More informationLecture 14 Numerical integration: advanced topics
Lecture 14 Numericl integrtion: dvnced topics Weinn E 1,2 nd Tiejun Li 2 1 Deprtment of Mthemtics, Princeton University, weinn@princeton.edu 2 School of Mthemticl Sciences, Peking University, tieli@pku.edu.cn
More informationKRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION
Fixed Point Theory, 13(2012), No. 1, 285-291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd
More informationA Note on Feng Qi Type Integral Inequalities
Int Journl of Mth Anlysis, Vol 1, 2007, no 25, 1243-1247 A Note on Feng Qi Type Integrl Inequlities Hong Yong Deprtment of Mthemtics Gungdong Business College Gungzhou City, Gungdong 510320, P R Chin hongyong59@sohucom
More information