Some inequalities of Hermite-Hadamard type for n times differentiable (ρ, m) geometrically convex functions
|
|
- Daniella Neal
- 5 years ago
- Views:
Transcription
1 Avilble online t 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 Hussin b Centre for Advnced Studies in Pure nd Applied Mthemtics CASPAM, Bhuddin Zkriy University, Multn 68, Pkistn. b Deprtment of Mthemtics, King Abdulziz University, P.O. Box 83, Jeddh, 589, Sudi Arbi. Communicted by Mnuel De l Sen Abstrct In this pper, some generlized Hermite Hdmrd type ineulities for n times differentible ρ, m geometriclly convex function re estblished. The new ineulities recpture nd give new estimtes of the previous ineulities for first differentible functions s specil cses. The estimtes for trpezoid, midpoint, verged mid-point trpezoid nd Simpson s ineulities cn lso be obtined for higher differentible generlized geometriclly convex functions. c 5 All rights reserved. Keywords: Hermite Hdmrd ineulity, ρ, m geometriclly convex functions, n times differentible function. MSC: 6D5, 6A5.. Introduction Since the estblishment of theory of convex functions in the lst century by Dnish mthemticin, Jensen , the reserch on convex functions hs gined much ttention. However, the geometriclly convex functions only ppered in,, ] but hs now become n ctive domin of definition. Convex nd geometriclly convex functions re used in prllel s tools to prove ineulities. The notion of geometric convexity ws introduced by Montel 6], nlogous to the notion of convex function in n vribles. Now, we restte some bsic convexity domins nd relted results. Corresponding uthor Emil ddresses: fizzfr@gmil.com Fiz Zfr, humir.k86@gmil.com Humir Klsoom, nhusin@ku.edu.s Nwb Hussin Received 4--8
2 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 Definition.. A function f : I R R is convex, if the ineulity holds for ll x, y I, nd t, ]. ftx ty tfx tfy,. The following is well known Hermite Hdmrd ineulity which holds for convex function f : I R R where, b I with < b, f b b fxdx b f f b.. As soon s n ineulity ppers, n ttempt is mde to generlize it 5]. The most erlier ttempts of refining Hermite Hdmrd ineulity cn be found in, 3, 4]. In 4, Zhng 9] presented the following concept of geometriclly convex functions. Definition.. Let fx be positive function on, b]. If fx t y t fx] t fy] t,.3 holds for ll x, y, b] nd t, ], then we sy tht the function fx is geometriclly convex on, b]. In 3, Ozdmir nd Yildiz 7], presented some Ostrowski type ineulities for geometriclly convex functions involving Logrithmic men. Xi et l. 8] in, introduced the concept of m geometriclly convex functions nd presented Hermite Hdmrd type ineulities for the generlized m geometriclly convex functions. Definition.3. Let fx be positive function on, b] nd m, ]. If fx t y m t fx] t fy] m t,.4 holds for ll x, y, b] nd t, ], then we sy tht the function fx is m geometriclly convex on, b]. Theorem.4. 8] Let I, be n open intervl nd f : I, is differentible. If f L, b] nd f x is decresing nd m-geometriclly convex on min{, }, b] for, b,, with < b nd b, nd m, ], then f f b b b fxdx b f b m G α, m, ],.5 is vlid for, where G, m, = f t f b m t dt. Theorem.5. 8] Let I, be n open intervl nd f : I, is differentible. If f L, b], f x is decresing nd m geometriclly convex on min{, }, b] for,, b, nd m, ], then b f b fxdx b b 3 f n b m G, m, ], 4
3 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 3 is vlid for,where G, m, = f t f b m t dt f t f b m Recently, some Ostrowski type ineulities for m-geometriclly convex functions re lso estblished by Ozdmir nd Yildiz 7]. Xi et l. 8] in, introduced the concept of α, m geometriclly convex functions nd estblished the generlized ineulities of this domin. Definition.6. Let fx be positive function on, b] nd α, m, ], ]. If t dt fx t y m t fx] tα fy] m tα,.6 holds for ll x, y, b], nd t, ], then we sy tht the function fx is α, m geometriclly convex on, b]. Remrk.7. If α = m = in.6, then α, m geometriclly convex functions become geometriclly convex functions. Lemm.8. If fx is geometriclly convex, nd then g is convex function. gx = ln fe x,.7 Theorem.9. 8] Let I, be n open intervl nd f : I, is differentible. If f L, b] nd f x is decresing nd α, m-geometriclly convex on min{, }, b] for,, b, nd α, m, ], ], then f f b is vlid for, where b b G α, m, = fxdx b f t f b m f b n m G α, m, ],.8 tα dt. Theorem.. 8] Let I, be n open intervl nd f : I, is differentible. If f L, b] nd f x is decresing nd α, m geometriclly convex on min{, }, b] for,, b nd α, m, ], ], then b f is vlid for, where G α, m, = b b f t f b m fxdx b 4 tα dt f t f b m tα dt. 3 f b n m G α, m, ],.9.
4 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 4 Ostrowski type ineulity for α, m geometriclly convex functions obtined by Ozdmir nd Yildiz in 7] is stted in the form of the following theorem. Theorem.. Let I R be n open intervl nd f : I, is differentible. If f L, b] nd f x is decresing nd α, m-geometriclly convex on,, b] with < b nd α, m, ], ], then we hve f x b fudu b p p { x b f m M t,, α b x b f b } m N t,, α,. where nd M t,, α = N t,, α = f x t f f x t f b tα dt tα dt. Lemm.. 8] For x, y, nd m, t, ], if x < y nd y, then x t y m t tx ty. In this pper, we give some generlized ineulities for ρ, m geometriclly convex n-times differentible function. The specil cse for first differentible ρ, m geometriclly convex functions is three point ineulity of Hermite Hdmrd type which is cpble of recpturing the previous results of this domin s well s some new ineulities cn be obtined s nturl conseuence.. Min Results Lemm.. Let f be rel vlued n-times differentible mpping defined on, b] such tht f n x be bsolutely continuous on, b] with α :, b], b] nd β :, b], b], α x x β x, then for ll x, b], the following identity holds b n k n x, t f n t tbdt = where the kernel k n :, b], ] R is given by k n x, t = b f u du t n k= ] R k x f k x S k x., t, ]. t n, t, ] R k x = β x x k k x α x k, S k x = α x k f k k b β x k f k b..3
5 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 5 Proof. Under the given conditions, the following identity ws proved by Cerone nd Drgomir ] b n K n x, u f n u du = where the kernel K n :, b] R is given by K n x, u = b f u du Moreover, R k x nd S k x re defined by.3. Now, by considering the L. H. S. of the identity.4, b n K n x, u f n u du = Let u = t tb in.5, then x k= { u αx n, u, x] u βx n, u x, b] αx u n ] R k x f k x S k x,.4 b f n u du x βx u n f n u du = I I b.5 I = b n t x n f n t tb dt, nd Thus, I b = b n t x n f n t tb dt. where b n Hence, proved. k n x, t f n t tb dt = k n x, t = b t n f u du k=, t, ] t n, t, ] ] R k x f k x S k x, The generlized Hermite Hdmrd type ineulity for ρ, m geometriclly convex n differentible function is stted s follows. Theorem.. Let I, be n open intervl nd f : I, is n-differentible. Let f n x L, b] is decresing nd ρ, m-geometriclly convex on min{, }, b] for,, b, x, b] nd ρ, m, ], ], then b x] fudu R k x f k x S k k= n b n n! f n b m E n ρ, m,,.6
6 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 6 for, b β n β x n x α n α n], n n for µ = E n ρ, m, = b β n β x n x α n α n ρ µ n γn, ln µ ρ ρ ln u n µ ρ n k β x n k k x α n k n k!ρ ln µ k k= ρ µ n γn, ln µ ρα ρ ln µ n, for < µ <.7 for µ, ρ >, where b β n β x n x α n α n ρ n µ ρ n γn, ln µ ln µ n ρ n µ k ρ k β x ρ n k k ρ k x α n k n k! ln µ k k= ρn µ ρ n γn, ln µ α ln µ n ρ for µ > µ = f n f n b m,, nd γ, x is the lower incomplete gmm function defined s γ, x = x t e t dt, R k x = βx x k k x αx k, S k x = αx k f k k b βx k f k b..8 Proof. Applying the definition of kernel, properties of modulus nd Hölder s ineulity on., we get b x] f u du R k x f k x S k b n b n k= k n x, t dt b β b tn dt k n x, t f n t tb t b β b n dt
7 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 7 b α b tn dt tn t b α b n dt f n t tb dt t n f n t tb dt tn f n t tb dt Applying the definition of α, m geometric convexity on.9, we obtin b x] f u du R k x f k x S k b n k= b β b tn dt t n f n t tb dt t b β b n dt b α b tn dt..9 t b α b n dt t b β b n f n t b m t dt t b α f b n n t b m t dt b β b tn f n t b m t dt Therefore, upon simplifiction b x] f u du R k x f k x S k k= b n f n b m b β b tn dt. b α b tn f n t b m t dt t b β b n dt b α b tn dt
8 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 8 t b α b n dt t b β f n t ρ b n f n b m dt b β b f n tn f n b m t ρ dt b α b f n tn f n b m t ρ dt Let µ = t b α f n b n f n b m f n f n b t ρ dt m in., we hve three cses: Cse : For µ =,. tkes the form b b n f n b b α b tn dt Thus, we hve b f u du k= m f u du k=. x] R k x f k x S k b β b tn dt t b α b n dt. x] R k x f k x S k b n f n b m n n β x n f n b m n! n t b β b n dt x α n n b β n β x n x α n α n]. ] α n, n Cse : For µ <, < t, ρ, we hve µ tρ µ ρt. Thus,. becomes b x] f u du R k x f k x S k k=..
9 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 9 n b n n! α n f n b b α b tn µ ρt dt m b β n β x n x α n b β b tn µ ρt dt t b α b n µ ρt dt t b β b n µ ρt dt. Let, nd Then, upon simplifiction, we hve I = I = I 3 = I 4 = b β b tn µ ρt dt, t b β b n µ ρt dt, b α b tn µ ρt dt, t b α b n µ ρt dt.. nd n ρ I = µ I = µρ γn, ln uρ, ρ ln u n k= n ρ I 3 = µ k β x n k n k! ρ ln µ k n µ k= I 4 = n ρ µ ρ ln µ n k x α n k n k! ρ ln µ k µ Re substituting the vlues of the integrls in., we hve b x] f u du R k x f k x S k k= ρ ρ ln µ n, ρ ρ ln µ n, γn, ln µ ρ α.
10 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 n b n n! f n b m b β n β x n x α n α n ρ µ ρ ln u n n ρ γn, ln µ n ρ µ k= k β x n k k x α n k n k! ρ ln µ k µρ ρ ln µ n n α ρ γn, ln µ, for < µ <. Cse 3: For µ >, < t, ρ, we hve µ tρ b f u du k= x] R k x f k x S k µ t ρ. Thus,. becomes.3 n b n n! f n b m b β n β x n x α n Let, α n b α b tn µ t ρ dt b β b tn µ t ρ dt t b α b n µ t ρ dt I = I = I 3 = t b β b n µ t ρ dt b β b tn µ t ρ dt, t b β b n µ t ρ dt, b α b tn µ t ρ dt,..4
11 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 nd Then, upon simplifiction, we hve I 4 = t b α b n µ t ρ dt. I = ρn µ ρ γn, ln µ ρ, ln u n nd I = µ n ρ k= I 3 = µ n ρ ρ k k β x n k n k! ln µ k ρn n µ k= ρ k k x α n k n k! ln µ k ρn µ I 4 = ρn n µ ρ ln µ n γn, ln µ ρ ρ ln µ n, ρ ρ ln µ n, ρ α. Re substituting the vlues of the integrls in.4, we hve b x] f u du R k x f k x S k k= n b n n! f n b m m b β n β x n x α n α n ρn µ ρ ln µ n n γn, ln µ µ n ρ k= ρn µ ρ ln µ n k ρ k β x n k k x α n k n k! ln µ k n γn, ln µ α ρ ρ, for µ >. Therefore,.,.3 nd.5 re reuired ineulities..5 Remrk.3. If we tke α x =, β x = b, x = b nd n = in., then the ineulities for the m nd α, m geometriclly convex functions by Xi et l.8] re recptured. The generlized ineulity for m geometriclly convex n differentible function is stted s follows.
12 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 Corollry.4. Let f, α, β, µ be defined s in Theorem.. If f n is m geometriclly convex on min{, }, b], for m,, then for ll x, b], b x] fudu R k xf k x S k k= for, n b n n! f n b n n m E n ρ, m,,.6 b β n β x n x α n α n], for µ = E n, m, = b β n β x n x α n α n µ µ ln u n n k= n γn, ln µ k β x n k k x α n k n k! ln µ k µ n γn, ln µ α ln µ n for < µ <,.7 b β n β x n x α n α n µ µ ln µ n n k= n γn, ln µ k β x n k k x α n k n k! ln µ k µ n γn, ln µ α ln µ n, for µ >. where γ, x is the lower incomplete gmm function. Moreover, R k x nd re defined by.8. S k x Proof. Substituting ρ =, in.6.8, we hve the reuired ineulity. The generlized three point Hermite Hdmrd type ineulity for ρ, m geometriclly convex n- differentible function is stted s,
13 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 3 Theorem.5. Let f, ρ, m, µ be defined s in Theorem., then the following ineulity holds for ny r, ] nd for ll x, b], b f u du r k b x k k x k f k x for, nd k= r k x k f k k b x k f k b] n b n n! f n b m E n ρ, m,, r n r n b x n x n.8 E n ρ, m, = n b n r n r n b x n x n, for µ =.9 E n ρ, m, = µ ρ rρ µ ρ ln µ n n k= rx µρ ρ ln µ n n γn, ln µ rρ k r n k k rx n k n k!ρ ln µ k n γn, ln µ rρx for < µ < r n r n b x n x n,. E n ρ, m, = µ ρ ρ n µ ρ ln µ n n γn, ln µ r ρ r n k= ρ k k r n k k rx n k n k! ln µ k. for ρ >. ρn ρ rx µ ln µ n n γn, ln µ rx for µ > ρ, Proof. Let αx nd βx be defined s follows in Theorem., αx = rx r,
14 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 4 nd Then, we hve the reuired result. βx = rx rb. The generlized three point ineulity for ρ, m geometriclly convex first differentible function is stted s follows. Corollry.6. Let f, ρ, m, µ be defined s in Theorem.5, then the following ineulity holds for ny r, ] nd x, b], b r x f b xfb f u du rf x b b b f b m E ρ, m,,. for, where r r b x x E ρ, m, =, for µ =..3 b r r b x x E ρ, m, = E ρ, m, = rρ µ µ rρ ρ ln µ ln µ rρ µ ρ rx µρ ρ ln µ µ rρ r ρ ln µ rx ρ ln µ ρ ln µ µ rρ ln µ rρx µ rρx, for < µ < r r b x x r ρ µ ρ µ r ln µ ρ ln µ r ρ µ r ρ µ ρ ρ ρ rx µ ln µ µ ρ r ln µ r ρ ρ rx ln µ ρ ln µ ln µ rx ρ µ rx ρ, for µ >..4.5 Proof. Substituting n =, in.8-., we hve the reuired ineulity. Remrk.7. For r = in.-.5, we cn get three-point Ostrowski type ineulity for ρ, m geometriclly convex function. Remrk.8. The following re the Hermite Hdmrd type ineulities for ρ, m geometriclly convex first differentible function for different choices of r in..5.
15 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 5. When r = nd x = b in..5, then b b f u du f b b for nd µ is defined s where f b m G ρ, m, ],.6 G ρ, m, = µ = f f b m, µ ρ 4µ ρ ρ ln µ, for µ = ρ µ ρ 4µ ρ ln µ, for < µ <, for µ >.7 for ρ >.. When r = nd x = b in.-.5, then b f fb f u du b b f b m G ρ, m, ],.8 for where µ is defined s where for ρ >. 3. When r = b µ is defined s G ρ, m, = nd x = b in.-.5, then b 4 b f u du 4 µ = f f b m,, for µ = µ ρ ln u ρ ρ µ ρ ln µ ρ ρ ρ ρ µ ln µ µ f f b ln µ fb], for < µ <, for µ >.9 f b m G 3 ρ, m, ],.3 µ = f f b m,
16 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 6 nd, for µ = for ρ >. 4. When r = 3 b G 3 ρ, m, = 4µ ρ 8µ 3ρ 4 8µ ρ 8µ ρ 4 4µ ρ ln µ ρ 4 4 ρ ln µ 3 ρ 4µ ρ 8µ 4ρ 8µ ρ 8µ 4ρ 4 µ ρ ln µ 4ρ 4 nd x = b in.-.5, then b 5b 8 f u du 6 for where µ is defined s where f 4f b ln µ fb], for < µ <, for µ >.3 f b m G 4 ρ, m, ],.3 µ = f f b m,, for µ = for ρ >. G 4 ρ, m, = 9 µ ρ 4µ 5ρ 6 4µ ρ 4µ ρ 6 µ ρ ln µ ρ 6 5ρ ln µ 5 9ρ µ ρ 4µ 6ρ 4µ ρ 4µ 6ρ µ ρ ln µ 6ρ 5 ln µ, for < µ <, for µ > Conclusions Some generlized ineulities for ρ, m geometriclly convex nd n-differentible mppings re given which re cpble of giving bounds of the one point, two point nd three point Hdmrd type ineulities for first nd higher differentible functions. The specil cses recpture ineulities given by Xi et l. 8]. Some new estimtes of the verge mid-point trpezoid nd Simpson s ineulity re given for first differentible generlized geometriclly convex function s specil cses. The estimtes for higher differentible function cn lso be obtined from.8-. for ll these cses. Acknowledgements The uthors re thnkful to the referee for giving vluble comments nd suggestions which helped to improve the finl version of this pper. References ] P. Cerone, S. S. Drgomir, Three point identities nd ineulities for n time differentible functions, SUT. J. 36, ] S. S. Drgomir, Two mppings in connection to Hdmrd s ineulity, J. Mth. Anl. Appl., 67 99, ] S. S. Drgomir, R. P. Agrwl, Two ineulities for differentible mppings nd pplictions to specil mens of rel numbers nd to trpezoidl formul, Appl. Mth. Lett., 998, 9 95.
17 F. Zfr, H. Klsoom, N. Hussin, J. Nonliner Sci. Appl. 8 5, 7 7 4] S. S. Drgomir, J. E. Pečrić, J. Sándor, A note on the Jensen-Hdmrd s ineulity, Anl. Num. Ther. Approx., 9 99, ] M. Msjed-Jmei, N. Hussin, More results on functionl generliztion of the Cuchy-Schwrz ineulity, J. Ineul. Appl.,, 9 pges. 6] P. Montel, Sur les functions convexes et les fonctions soushrmoniues, J. Mth. Ineul., 9 98, ] M. E. Özdemir, C. Yildiz, New Ostrowski type ineulities for geometriclly convex functions, Int. J. Mod. Mth. Sci., 8 3, 7 35.,, 8] Bo-Yn Xi, Rui-Fng Bi, Feng Qi, Hermite Hdmrd type ineulities for the m-nd α, m-geometriclly convex functions, Aeutiones Mth., 84, 6 69.,.4,.5,,.9,.,.,.3, 3 9] X. M. Zhng, Geometriclly Convex Functions, Anhui University Press, Hefei, 4 In Chinese. ] X. M. Zhng, Y. M. Chu, The geometricl convexity nd concvity of integrl for convex nd concve functions, Int. J. Mod. Mth., 3 8, ] X. M. Zhng, Z. H. Yng, Differentil criterion of n-dimensionl geometriclly convex functions, J. Appl. Anl., 3 7, ] X. M. Zhng, An ineulity of the Hdmrd type for the geometriclly convex functions in Chinese, Mth. Prct. Theory, 34 4, 7-76.
New 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 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 informationBulletin 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 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 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 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 informationSome new integral inequalities for n-times differentiable convex and concave functions
Avilble online t wwwisr-ublictionscom/jns J Nonliner Sci Al, 10 017, 6141 6148 Reserch Article Journl Homege: wwwtjnscom - wwwisr-ublictionscom/jns Some new integrl ineulities for n-times differentible
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 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 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 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 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 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 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 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 informationHadamard-Type Inequalities for s Convex Functions I
Punjb University Journl of Mthemtics ISSN 6-56) Vol. ). 5-6 Hdmrd-Tye Ineulities for s Convex Functions I S. Hussin Dertment of Mthemtics Institute Of Sce Technology, Ner Rwt Toll Plz Islmbd Highwy, Islmbd
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 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 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 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 Hermite-Hadamard type integral inequalities for functions whose second derivative are nonconvex
Mly J Mt 34 93 3 On Hermite-Hdmrd tye integrl ineulities for functions whose second derivtive re nonconvex Mehmet Zeki SARIKAYA, Hkn Bozkurt nd Mehmet Eyü KİRİŞ b Dertment of Mthemtics, Fculty of Science
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 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 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 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 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 informationRIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES THROUGH GENERALIZED (m, h 1, h 2 )-PREINVEXITY
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 38 7 345 37 345 RIEMANN-LIOUVILLE FRACTIONAL SIMPSON S INEQUALITIES THROUGH GENERALIZED m h h -PREINVEXITY Cheng Peng Chng Zhou Tingsong Du 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 informationProperties and integral inequalities of Hadamard- Simpson type for the generalized (s, m)-preinvex functions
Avilble online t wwwtjnscom J Nonliner Sci Appl 9 6, 3 36 Reserch Article Properties nd integrl ineulities of Hdmrd- Simpson type for the generlized s, m-preinvex functions Ting-Song Du,b,, Ji-Gen Lio,
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 informationON THE HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATOR
Krgujevc ournl of Mthemtics Volume 44(3) (), Pges 369 37. ON THE HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATOR H. YALDIZ AND M. Z. SARIKAYA Abstrct. In this er, using generl clss
More informationOn new Hermite-Hadamard-Fejer type inequalities for p-convex functions via fractional integrals
CMMA, No., -5 7 Communiction in Mthemticl Modeling nd Applictions http://ntmsci.com/cmm On new Hermite-Hdmrd-Fejer type ineulities or p-convex unctions vi rctionl integrls Mehmet Kunt nd Imdt Iscn Deprtment
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 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 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 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 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 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 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 informationHermite-Hadamard and Simpson-like Type Inequalities for Differentiable p-quasi-convex Functions
Filomt 3:9 7 5945 5953 htts://doi.org/.98/fil79945i Pulished y Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: htt://www.mf.ni.c.rs/filomt Hermite-Hdmrd nd Simson-like Tye Ineulities for
More informationOn some inequalities for s-convex functions and applications
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 htt://wwwjournlofineulitiesndlictionscom/content/3//333 R E S E A R C H Oen Access On some ineulities for s-convex functions nd lictions Muhmet Emin
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 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 informationINEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV FUNCTIONAL. Mohammad Masjed-Jamei
Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: http://www.pmf.ni.c.rs/filomt Filomt 25:4 20) 53 63 DOI: 0.2298/FIL0453M INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV
More informationGENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS. (b a)3 [f(a) + f(b)] f x (a,b)
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
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 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 informationHERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES ARE (α, m)-convex
HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES ARE (α -CONVEX İMDAT İŞCAN Dertent of Mthetics Fculty of Science nd Arts Giresun University 8 Giresun Turkey idtiscn@giresunedutr Abstrct:
More informationGeometrically Convex Function and Estimation of Remainder Terms in Taylor Series Expansion of some Functions
Geometriclly Convex Function nd Estimtion of Reminder Terms in Tylor Series Expnsion of some Functions Xioming Zhng Ningguo Zheng December 21 25 Abstrct In this pper two integrl inequlities of geometriclly
More informationHermite-Hadamard inequality for geometrically quasiconvex functions on co-ordinates
Int. J. Nonliner Anl. Appl. 8 27 No. 47-6 ISSN: 28-6822 eletroni http://dx.doi.org/.2275/ijn.26.483 Hermite-Hdmrd ineulity for geometrilly usionvex funtions on o-ordintes Ali Brni Ftemeh Mlmir Deprtment
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 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 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 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 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 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 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 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 informationAN UPPER BOUND ESTIMATE FOR H. ALZER S INTEGRAL INEQUALITY
SARAJEVO JOURNAL OF MATHEMATICS Vol.4 (7) (2008), 9 96 AN UPPER BOUND ESTIMATE FOR H. ALZER S INTEGRAL INEQUALITY CHU YUMING, ZHANG XIAOMING AND TANG XIAOMIN Abstrct. We get n upper bound estimte for H.
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 informationON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS
PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 9811 015, 43 49 DOI: 10.98/PIM15019019H ON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS
More informationGENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES ARE GENERALIZED s-convex IN THE SECOND SENSE
Journl of Alied Mthemtics nd Comuttionl Mechnics 6, 5(4), - wwwmcmczl -ISSN 99-9965 DOI: 75/jmcm64 e-issn 353-588 GENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES
More informationGeneralized Hermite-Hadamard Type Inequalities for p -Quasi- Convex Functions
Ordu Üniv. Bil. Tek. Derg. Cilt:6 Syı: 683-93/Ordu Univ. J. Sci. Tech. Vol:6 No:683-93 -QUASİ-KONVEKS FONKSİYONLAR İÇİN GENELLEŞTİRİLMİŞ HERMİTE-HADAMARD TİPLİ EŞİTSİZLİKLER Özet İm İŞCAN* Giresun Üniversitesi
More informationHERMITE-HADAMARD TYPE INEQUALITIES OF CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF QUASI-ARITHMETIC MEANS
HERMITE-HADAMARD TYPE INEQUALITIES OF CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF QUASI-ARITHMETIC MEANS FLAVIA CORINA MITROI nd CĂTĂLIN IRINEL SPIRIDON In this pper we estblish some integrl inequlities
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 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 informationOn the Co-Ordinated Convex Functions
Appl. Mth. In. Si. 8, No. 3, 085-0 0 085 Applied Mthemtis & Inormtion Sienes An Interntionl Journl http://.doi.org/0.785/mis/08038 On the Co-Ordinted Convex Funtions M. Emin Özdemir, Çetin Yıldız, nd Ahmet
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 informationResearch Article On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation
Journl of Applied Mthemtics Volume 2011, Article ID 743923, 7 pges doi:10.1155/2011/743923 Reserch Article On Existence nd Uniqueness of Solutions of Nonliner Integrl Eqution M. Eshghi Gordji, 1 H. Bghni,
More informationLecture notes. Fundamental inequalities: techniques and applications
Lecture notes Fundmentl inequlities: techniques nd pplictions Mnh Hong Duong Mthemtics Institute, University of Wrwick Emil: m.h.duong@wrwick.c.uk Februry 8, 207 2 Abstrct Inequlities re ubiquitous in
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 informationSUPERSTABILITY OF DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS
Electronic Journl of Differentil Equtions, Vol. 01 (01), No. 15, pp. 1. ISSN: 107-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu SUPERSTABILITY OF DIFFERENTIAL
More informationSome New Inequalities of Simpson s Type for s-convex Functions via Fractional Integrals
Filomt 3:5 (7), 4989 4997 htts://doi.org/.98/fil75989c Published by Fculty o Sciences nd Mthemtics, University o Niš, Serbi Avilble t: htt://www.m.ni.c.rs/ilomt Some New Ineulities o Simson s Tye or s-convex
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 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 informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics ON LANDAU TYPE INEQUALITIES FOR FUNCTIONS WIT ÖLDER CONTINUOUS DERIVATIVES LJ. MARANGUNIĆ AND J. PEČARIĆ Deprtment of Applied Mthemtics Fculty of Electricl
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 informationSome Improvements of Hölder s Inequality on Time Scales
DOI: 0.55/uom-207-0037 An. Şt. Univ. Ovidius Constnţ Vol. 253,207, 83 96 Some Improvements of Hölder s Inequlity on Time Scles Cristin Dinu, Mihi Stncu nd Dniel Dănciulescu Astrct The theory nd pplictions
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 informationMultiple Positive Solutions for the System of Higher Order Two-Point Boundary Value Problems on Time Scales
Electronic Journl of Qulittive Theory of Differentil Equtions 2009, No. 32, -3; http://www.mth.u-szeged.hu/ejqtde/ Multiple Positive Solutions for the System of Higher Order Two-Point Boundry Vlue Problems
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 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 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 informationSeveral Answers to an Open Problem
Int. J. Contemp. Mth. Sciences, Vol. 5, 2010, no. 37, 1813-1817 Severl Answers to n Open Problem Xinkun Chi, Yonggng Zho nd Hongxi Du College of Mthemtics nd Informtion Science Henn Norml University Henn
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 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 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 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 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 informationMUHAMMAD MUDDASSAR AND AHSAN ALI
NEW INTEGRAL INEQUALITIES THROUGH GENERALIZED CONVEX FUNCTIONS WITH APPLICATION rxiv:138.3954v1 [th.ca] 19 Aug 213 MUHAMMAD MUDDASSAR AND AHSAN ALI Abstrct. In this pper, we estblish vrious inequlities
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 informationON CO-ORDINATED OSTROWSKI AND HADAMARD S TYPE INEQUALITIES FOR CONVEX FUNCTIONS II
TJMM 9 (7), No., 35-4 ON CO-ORDINATED OSTROWSKI AND HADAMARD S TYPE INEQUALITIES FOR CONVEX FUNCTIONS II MUHAMMAD MUDDASSAR, NASIR SIDDIQUI, AND MUHAMMAD IQBAL Abstrt. In this rtile, we estblish vrious
More informationLyapunov-type inequalities for Laplacian systems and applications to boundary value problems
Avilble online t www.isr-publictions.co/jns J. Nonliner Sci. Appl. 11 2018 8 16 Reserch Article Journl Hoepge: www.isr-publictions.co/jns Lypunov-type inequlities for Lplcin systes nd pplictions to boundry
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 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 informationSome integral inequalities of the Hermite Hadamard type for log-convex functions on co-ordinates
Avilble online t www.tjns.om J. Nonliner Si. Appl. 9 06), 5900 5908 Reserh Artile Some integrl inequlities o the Hermite Hdmrd type or log-onvex untions on o-ordintes Yu-Mei Bi, Feng Qi b,, College o Mthemtis,
More informationIntegral inequalities via fractional quantum calculus
Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 DOI.86/s366-6-4- R E S E A R C H Open Access Integrl ineulities vi frctionl untum clculus Weerwt Sudsutd, Sotiris K Ntouys,3 nd Jessd Triboon * *
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 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 information