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 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 859-95, 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
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
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.
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
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
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
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
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=..
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.
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
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.
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,
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,
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.
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,
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 >.33 3. 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, 35 384. ] S. S. Drgomir, Two mppings in connection to Hdmrd s ineulity, J. Mth. Anl. Appl., 67 99, 49 56. 3] 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.
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