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 Fculty, Deprtment of Mthemtics, 54, Cmpus, Erzurum, Turey School of Engineering nd Science, Victori University, P O Box448, Melbourne City, MC8, Austrli Received: 4 Jn 8, Accepted: My 8 Published online: Nov 8 Abstrct: In this pper, using integrl representtions for n times differentible mppings, we estblish new generliztions of certin Hermite-Hdmrd type ineulity for convex functions by using firly elementry nlysis Also prllel development is mde bse on concvity Keywords: Hermite-Hdmrd Integrl Ineulity, Hölder Ineulity, Jensen Ineulity, Convex Functions Introduction Let f : I R R be convex mpping nd,b I with < b Then b f f xdx b f f b Both the ineulities hold in reversed direction if f is concve A letter ws sent by Hermite 8-9 to the Journl Mthesis on November, 88 nd it ws published in Mthesis 3 883, p: 8 This letter involved n ineulity which is well-nown in the literture s Hermite-Hdmrd integrl ineulity Since its discovery in 883, Hermite- Hdmrd ineulity hs been considered s very useful ineulity in mthemticl nlysisthese ineulities hve been used in numerous settings In ddition, gret number of ineulities of specil mens cn be obtined for prticulr choice of the function f The rich geometricl significnce of Hermite-Hdmrd s ineulity enbles literture to grow by providing its new proofs, extensions, refinements nd generliztions, see for exmple [], [6], [7], []-[4], [6], [8], []-[5] Definition The function f : [,b] R R is sid to be convex if the following ineulity holds: f tx ty t f x t f y for ll x,y [,b] nd t [,] It is sid tht f is concve if f is convex This definition is originted from Jensen s results in [9] nd hs led to the most extended, useful nd multi-disciplinry domin of mthemtics, nmely, convex nlysis Convex curves nd convex bodies hve been seen in mthemticl literture since ntiuity nd nturlly mny significnt results relted to them were obtined Cerone et l see [4] proved the following generliztion for n time differentible functions c 8 BISKA Bilisim Technology Corresponding uthor e-mil: cetin@tuniedutr
37 Cetin Yildiz, Sever S Drgomir: Integrl ineulities for n times differentible mppings Lemm Let f : [,b] R be mpping such tht f is bsolutely continuous on [,b] Then for ll x [,b] we hve the identity: f t = where the ernel K n : [,b] R is given by x [,b] nd n is nturl number, n [ b x x ] f x n K n x,t f n t, K n x,t = t n n! i f t [,x] t b n n! i f t x,b], For other recent results regrding the n time differentible functions see [3]-[5], [8], [], [3], [5], [7], [4] where further references re given In [9], Özdemir et l proved the following Hdmrd type ineulities Theorem Let f : I [, R be differentible mpping on I such tht f L[,b], where,b I with < b If f, is s-convex on [,b], for some fixed s,], then the following ineulity holds: b f f xdx b b 6 Corollry In Theorem, if we choose s = we hve b f f xdx b { p 3 s s s 3 s 3 b 48 b f In [], Alomri nd Drus obtined the following theorem nd corollry f b s 3 f f b } s s s 3 { 3 f b 4 3 f b f f b } 3 Theorem Let f : I [, R be differentible mpping on I such tht f L[,b], where,b I with < b If f p p is concve on [,b], then the following ineulity holds: f x b for ech x [,b], where p > f udu b x b x b p /p f x x b p /p f Corollry In Theorem, choose x = b, then b f [ ] f udu b b 3b 4p /p f 3 b 4 f 4 3 c 8 BISKA Bilisim Technology
JACM 3, No, 36-45 8 / wwwntmscicom/jcm 38 for ech x [,b], where p > The present pper minly ims to estblish severl new ineulities for n time differntible mppings relted to the celebrted Hermite-Hdmrd integrl ineulity Min Results In order to rech our im, the following lemm is necessry: Lemm For n, let f : [,b] R R be n-time differentible functions If f n L[,b], then f t = { n b n n n! b b f t n f n t b t n f n tb t b t 4 } Proof The proof is by mthemticl induction For n =, we hve the eulity in pper [] Assume tht 4 holds for n = m nd let us prove it for n = m Tht is, we hve to obtin the eulity, f xdx = m m b m m m! b b f { t m f m t b t m f m tb t b } t 5 To hndle5, if we choose { b m I = m t m f m t b m! t m f m tb t b t } nd integrting by prts gives I = { b m m t m m! t m b f m = b m m m! f m m b m m m! b f m t b tb t b b b m m m! b f m b m m m! t m t m f m t b t b m t m f m tb t b b t m f m t b t t m f m tb t b } c 8 BISKA Bilisim Technology
39 Cetin Yildiz, Sever S Drgomir: Integrl ineulities for n times differentible mppings Hence b m { m m! m = m m! t m f m t b t b m f m b I t m f m tb t b } Now, using the mthemticl induction hypothesis, upon rerrngement we obtin the following eulity: b m f xdx = m m m m m! b b f b m f m b I 6 Multiplying the both sides of 6 by n nd substituting I in the right prt of 6, we obtin f xdx = m b b f { t m f m t b m b m m m! Thus, the identity 5 nd the Lemm is proved t m f m tb t b } t Theorem 3 Let f : [,b] R be n time differentible function nd < b If f n L[,b] nd on [,b], then we obtin: f n n is convex f xdx b n n n! np p b b f f n f n b f n b f n b 7 where p = Proof Using Lemm nd Hölder integrl ineulity, it follows tht f xdx b n n n! { t np b b f t np p f n t b p f n tb t b t } c 8 BISKA Bilisim Technology
JACM 3, No, 36-45 8 / wwwntmscicom/jcm 4 Since f n is convex on [,b], then we cn write b f xdx b f { b n p [ n t b n! np f n t f n ] p [ t f n b t b ] } np f n b n p f n f n b f n b f n b = n n! np which completes the proof Corollry 3 Under the ssumptions of Theorem 3, we hve b f xdx b f b n f n f n b f n b f n b n n! 8 Proof For p >, since p p lim = nd lim = p np p np n, we hve Hence we obtin the ineulity 8 p n < lim <, p, p np Theorem 4 Let f : [,b] R be n time differentible function nd < b If f n L[,b] nd on [,b], then we get where p > f n n is convex b f xdx b f b n n f n b ] n! n p p p p f n 9 [ b p f n f n b ] } p p c 8 BISKA Bilisim Technology
4 Cetin Yildiz, Sever S Drgomir: Integrl ineulities for n times differentible mppings Proof From Lemm nd using the properties of modulus, we get = f xdx b n n n! b n n n! { t n f n t b { t n t p f n t b t p b b f t t n f n tb t b t t n t p t p f n } tb t b } Using the Hölder integrl ineulity, we cn write f xdx [ ] t n b n n n! [ t n t p ] t p b b f t p f n t b t t p f n tb t b Since f n is convex on [,b], we hve b f xdx b f b n n f n b ] n! n p p p p f n [ b p f n f n b ] } p p which completes the proof of the theorem 4 Corollry 4 In Theorem 4, if we choose n =, we hve b f b 4 [ p b p b f f xdx f b ] p p p f f b ] } p p c 8 BISKA Bilisim Technology
JACM 3, No, 36-45 8 / wwwntmscicom/jcm 4 Corollry 5 In Theorem 4, if we choose n =, then we obtin b f b 6 [ p b 3 p b f f xdx f b ] p p p f f b ] } p p Theorem 5 Let f : [,b] R R be n time differentible function If f n L[,b] nd for, then the following ineulity obtins: f xdx b n n n! b b f f n n n n [ n b f n n n f n is convex on [,b] R, b ] f n f n b ] } Proof Suppose tht = From Lemm, we hve f xdx b n n n! b b f [ f n n b ] f n f n b Suppose now tht > Using the well nown Power-men integrl ineulity nd Lemm, we obtin f xdx b n n n! { t n b b f t n t n f n t b t t n f n tb t b } Since f n is convex on [,b], for, then we obtin f xdx b n n n! b b f { t [t n b f n t f n ] n c 8 BISKA Bilisim Technology
43 Cetin Yildiz, Sever S Drgomir: Integrl ineulities for n times differentible mppings Hence, the proof of the theorem is completed t [t n f n b t b ] } f n b n = n f n n b ] f n n! n n [ n b f n f n b ] } n n Corollry 6 In Theorem 5, if we choose n =, we obtin b f f udu b b 8 f f b 3 f b f b 3 Remr In Theorem 5, if we choose n =, we get the ineulity Now, we give the following Hdmrd type ineulity for concve mppings Theorem 6 Let f : [,b] R be n time differentible function nd < b If f n L[,b] nd f n is concve on [,b], then we hve: b f xdx b f { } b n p 3 b n f n 3b n! np 4 f n 4 where p = Proof From Lemm nd Hölder integrl ineulity, we cn write f xdx b n n n! { t np b b f t np p f n t b p f n tb t b Since f n is concve on [,b], we cn use the Jensen s integrl ineulity to get t } f n t b t = t f n t b t f n = 3 b f n 4 t 3 t b t t c 8 BISKA Bilisim Technology
JACM 3, No, 36-45 8 / wwwntmscicom/jcm 44 nd similrly f n tb t b 3b f n 4 4 Therefore, if we use 3 nd 4 in the ineulity, we obtin the ineulity of Remr In Theorem 6, if we choose n =, we hve the ineulity 3 Corollry 7 In the ineulity if we choose n =, then we cn b f f t b b 6 { p 3 b f p 4 3b f 4 } 3 Applictions to Specil Mens We now consider the mens for rbitrry rel numbers α,β α β We te Arithmetic men : Logrithmic men: 3Generlized log men: Lα,β = Aα,β = α β, α,β R α β ln α ln β, α β, α,β, α,β R [ β m α m ] m L m α,β =, m Z\{,}, α,β R m β α Now using the results of Section, we give some pplictions for specil mens of rel numbers Proposition Let < < b nd m N,m > Then, we hve: A m,b L m m,b b m 4p [ b p [ ] n bn p n p ] b n Proof The ssertion follows from Corollry 4 pplied for f x = x m, x R Proposition Let,b R, < b Then, we hve the following ineulity; A m,b L m m,b b 8 3 [ b ] [ b ] b Proof The ssertion follows from Corollry 6 pplied for f x = x, x [,b] c 8 BISKA Bilisim Technology
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