Some Improvements of Hölder s Inequality on Time Scales

Transcription

1 DOI: 0.55/uom An. Şt. Univ. Ovidius Constnţ Vol. 253,207, Some Improvements of Hölder s Inequlity on Time Scles Cristin Dinu, Mihi Stncu nd Dniel Dănciulescu Astrct The theory nd pplictions of dynmic derivtives on time scles hve recently received considerle interest. In this pper, we define function using the comined dimond-α integrl, investigte its monotonicity nd give some refinements for Hölder s inequlity. We present some pplictions for Z nd R. Introduction There hve een recent developments of the theory nd pplictions of dynmic derivtives on time scles. This study, which is n unifiction of the discrete theory with the continuous theory, provides n unifiction nd etension of trditionl differentil nd difference equtions. It is lso n importnt tool in mny computtionl nd numericl pplictions. Using the delt nd nl dynmic derivtives, comined dynmic derivtive, so clled α dimond-α dynmic derivtive, ws introduced s liner comintion of nd dynmic derivtives on time scles. The dimond-α derivtive reduces to the derivtive for α = nd to the derivtive for α = 0. We refer the reder to [2], [4] nd [5] for n ccount of the clculus ssocited with the dimond-α dynmic derivtives. Recently, it hs een proven complete weighted version of dimond-α of Jensen s Inequlity see [3] : Key Words: Time scles, conve function, dynmic derivtives, Hölder s inequlity. 200 Mthemtics Suject Clssifiction: Primry 26D5; Secondry 39A3. Received: Revised: Accepted: Unuthenticted Downlod Dte 0/2/8 4:52 AM

2 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 84 Theorem. [3, Theorem 2]. Let, T nd m, M R. If g C[, ] T, [m, M] nd w C[, ] T, R with wt αt > 0, then the following ssertions re equivlent: i w is n α-sp weight for g on [, ] T ; ii for every F C[m, M], R conve function, we hve F gtwt αt wt F gtwt αt αt wt. αt This version of Jensen s inequlity is complete since it is true if nd only if w is n α-sp weight for g nd this specil clss of weights llows them to tke some negtive vlues. Using Theorem, we cn otin stronger versions of mny importnt results, such s Hölder s inequlity. Theorem 2. Hölder s inequlity. Let T e time scle, < T, f, g C[, ] T, [0, + nd w C[, ] T, R such tht wg q is n α-sp weight for fg q p, where p nd q re Hölder conjugtes, tht is, p + q = nd p >. Then, we hve p wtftgt α t wtf p t α t If p <, then the inequlity 2 is ckwrd. wtg q t α t q. 2 Proof. We choose F = p in Theorem 2, nd F is conve function on [0, for p >. We get gtwt p αt wt gtp wt α t αt wt. 3 αt Since wg q is n α-sp weight for fg q p, we hve p wtgq tftg q/p t α t wtgq tftg q/p t p α t. 4 wtgq t α t wtgq t α t But p nd q re Hölder conjugtes, nd so Unuthenticted Downlod Dte 0/2/8 4:52 AM

3 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 85 p wtftgt α t wtf p t α t wtg q t α t q. 5 In section 2, we give our min results, regrding the function defined using Hölder s inequlity. Some pplictions re presented in section 3. 2 Min results Now, we will define function sed on Hölder s inequlity, for time scles. Thus, let f, g nd w s in Theorem 2. We consider the function h : T T, given y y h, y = y /p y wtf p t α t wtftgt α t. wtg q t α t 6 This difference is generted y the inequlity 2, tht gives the nonnegtivity of the function h. We will study the monotonicity properties of h, nd we will find some refinements of the inequlity 2 sed on his properties. These results generlize the ones otined in [2]. The monotonicity properties of h re given y the net theorem. Theorem 3. Let T e time scle, < T, f, g, w C[, ] T, [0, +, nd w C[, ] T, R such tht wg q is n α-sp weight for fg q p, where p nd q re Hölder conjugtes. Then, i if p >, then the function h, is monotonously decresing, while if p <, then the function h, is monotonously incresing on T, in reltion with ; ii if p >, then the function h, y is monotonously incresing, while if p <, then the function h, y is monotonously decresing on T in reltion with y; iii for ll T, < <, if p >, then: Unuthenticted Downlod Dte 0/2/8 4:52 AM

4 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 86 wtftgt α t + /p wtf p t α t wtftgt α t wtg q t α t /p /q wtf p t α t wtg q t α t, 7 nd /p wtftgt α t wtf p t α t wtg q t α t wtftgt α t 2 + wtftgt α t /p /q wtf p t α t wtg q t α t, /p wtf p t α t wtg q t α t wtftgt α t /p wtf p t α t wtg q t α t /p wtf p t α t If p <, then the inequlities 7, 8 nd 9 re ckwrds. wtg q t α s. Proof. i Suppose tht p >. Let, 2 T, < 2 <. Using the reltion =, for i =, 2 we hve p + q 9 Unuthenticted Downlod Dte 0/2/8 4:52 AM

5 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 87 /p wtf p t α t wtg q t α t = wtg q t α t i i i i wtf p /p t α t. i wtg q t α t 0 The function /p is concve nd, using Jensen s inequlity for concve functions, we hve wtg q t α t wtf p t α t wtg q t α t /p = wtg q t α t 2 wtg q t α t wtg q t α t 2 wtf p t α t 2 wtg q t α t 2 + wtg q 2 t α t wtg q t α t wtf p /p t α t 2 wtg q t α t wtg q t α t 2 wtf p /p t α t 2 2 wtg q t α t wtg q t α t 2 wtf p /p t α t. wtg q t α t Using, once gin Jensen s inequlity for concve functions on time scles, we get Unuthenticted Downlod Dte 0/2/8 4:52 AM

6 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES wtf p /p t α t = wtg q t α t = Using 0, nd 2, it follows wtg q t α t wtg q t ftgt /p p α t /p wtg q t α t wtg q t ftgt /p p /p α t wtftgt α t. wtg q t α t 2 h, = = wtf p t α t /p wtg q t α t wtftgt α t wtg q t α t 2 wtf p /p t α t 2 2 wtg q t α t 2 + wtftgt α t wtftgt α t /p wtf p t α t wtg q t α t 2 2 wtftgt α t 2 = h 2,. If 2 =, the inequlity 7 implies 3 h 2, = h, = 0 h,. 4 The result of the lst inequlity 4 is the decresing monotony of the function h, on T, in reltion with. Unuthenticted Downlod Dte 0/2/8 4:52 AM

7 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 89 2 Suppose tht p <. If 0 < p <, then the function /p is conve so tht the inequlities from 7, nd 2 re reversed. This implies tht 3 nd 4 re lso reversed, hence h, is monotonously incresing over T in reltion with. If p < 0, then 0 < q < nd using the sme rguments s for the cse 0 < p <, we otin the incresing monotony of the function h, over T in reltion with. ii Using the sme rguments s efore, we cn prove tht the function h, y is monotonous in reltion with y. iii For every T, < < nd p >, the function h, y is monotonously incresing over T in reltion with y so tht, it follows mening tht h, h, h, = 0, 5 /p wtf p t α t wtg q s α t wtftgt α t /p wtf p t α t wtg q t α t wtftgt α t 0, nd, dding wtftgt αt to ech side of the previous inequlity, we get 7. Becuse h, is monotonously decresing over T in reltion with, we hve h, h, h, = 0, 6 Unuthenticted Downlod Dte 0/2/8 4:52 AM

8 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 90 mening /p wtf p t α t wtg q t α t wtftgt α t /p wtf p t α t wtg q t α t wtftgt α t 0, nd, dding wtftgt αt to ech side of the previous inequlity, we get 8. Adding the inequlities from 7 nd 8, we otin 9. If p <, the inequlities from 7, 8 nd 9 re reversed, which implies tht the inequlities from 5 nd 6 re lso reversed. 3 Applictions If T = R nd w then Theorem 3 ecomes Theorem.2 from [2]. If T = Z, = 0, = n, α = 0 nd fi = i, gi = i, wi = i for i =,..., n, we hve the following corollry tht improves Theorem. from [2], if we note Hn = h0, n: Corollry 4. Let i > 0, i > 0, w i > 0 i =, 2,..., n; n > nd consider p, q s Hölder conjugtes. If p >, then it follows If we define C nd D y Hn Hn i 7 k /p k Ck = w i p i w i q i + n w i i i, nd n /p n k Dk = w i p i w i q i + w i i i, Unuthenticted Downlod Dte 0/2/8 4:52 AM

9 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 9 for k =, 2,..., n, then n k /p k w i i i = C C2... Cn = w i p i w i q i nd 8 n k /p k w i i i = Dn Dn... D = w i p i w i q i. nd Also, for ll m =,..., n, we hve n w i i i 2 Cm + Cn m k n w i i i 2 Dm + Dn m k If p <, then the ove inequlities re reversed. 9 /p k w i p i w i q i 20 /p k w i p i w i q i. 2 If T = Z, = 0, = n nd fi = i, gi = i, wi = w i, for i = 0,..., n, then, for ll α [0, ], we get: while, wtftgt α t = α w i i i + α n w i i i = αw αw n n n + w i i i, /p n wtf p t α t = α w i p p i + α w i i = αw 0 p 0 + αw n p n + /p i p /p Unuthenticted Downlod Dte 0/2/8 4:52 AM

10 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 92 nd /q n wtg q t α t = α w i q q i + α w i i Then, Theorem 3 proves, for every 0 < k < n tht /q = αw 0 q 0 + αw n q n + w i q i. n w i i i + α w i i i α i 0 α w i p i + α k p w i i α w i q i + α + α w i i i + α i=k /p k q w i i n w i i i /p n α w i p p i + α w i i /q n α w i q i + α w i q i, 22 Unuthenticted Downlod Dte 0/2/8 4:52 AM

11 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 93 hence, αw αw 0 n n + w i i i αw 0 p 0 + αw k p p k + w i i /p αw 0 q 0 + αw n q q n + w i i + α w i i i + α i=k n w i i i αw 0 p 0 + αw n p p n + w i i αw 0 q 0 + αw n q n + i q /p 23 nd, lso αw αw n n n + w i i i αw k k p + αw n n p + αw k k q + αw n n q + + α w i i i + α n w i i p n k w i i i αw 0 p 0 + αw n p p n + w i i w i i q /p /p /q αw 0 q 0 + αw n q n + w i q i. 24 The inequlities from 22 nd 23 re improvements of the inequlities Unuthenticted Downlod Dte 0/2/8 4:52 AM

12 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 94 from 8, 9, 20 nd 2. iv If T = 2 N with = 0, = n nd fi = 2 i, gi = 2 i, wi = 2 wi, where i, i, w i R, for i = 0,..., n, then for ll α [0, ], we get: α2 w α2 wn+n+n + 2 wi+i+i /p α2 w0+p0 + α2 w k+p k + 2 wi+pi /q α2 w0+q0 + α2 wn+qn + 2 wi+qi + α 2 wi+i+i + α i=k n 2 wiii /p α2 w0+p0 + α2 wn+pn + 2 wi+pi /q α2 w0+q0 + α2 wn+qn + 2 wi+qi 25 Unuthenticted Downlod Dte 0/2/8 4:52 AM

13 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 95 nd, lso α2 w000 + α2 wnnn + 2 wiii α2 w k+p k + α2 wn+pn + α2 w kq k + α2 wn+qn + + α 2 wiii + α k n n 2 wiii 2 wi+pi /p 2 wi+qi /p α2 w0+p0 + α2 wn+pn + 2 wi+pi /q α2 w0+q0 + α2 wn+qn + 2 wi+qi. 26 References [] R. P. Agrwl, M. Bohner, Bsic clculus on time scles nd some of its pplictions, Results Mth , [2] M. Bohner nd A. Peterson, Dynmic Equtions on Time Scles, An introduction with Applictions, Birkhäuser, Boston, 200. [3] C. Dinu, A weighted Hermite Hdmrd inequlity for Steffensen Popoviciu nd Hermite Hdmrd weights on time scles, Anlele Ştiinţifice le Universităţii Ovidius, Constnţ, Ser. Mt., [4] C. Dinu, Hermite-Hdmrd Inequlity on Time Scles, Journl of Inequlities nd Applictions, vol. 2008, Article ID , 24 pges, [5] S. Hilger, Anlysis on mesure chins- unified pproch to continuous nd discrete clculus, Results Mth , pp [6] S. Hussin nd M. Anwr, On certin inequlities improving the Hermite-Hdmrd inequlity, J. Inequl. Pure nd Appl. Mth. JI- PAM , Issue 2, Article no. 60. Unuthenticted Downlod Dte 0/2/8 4:52 AM

14 SOME IMPROVEMENTS OF HÖLDER S INEQUALITY ON TIME SCALES 96 [7] C. P. Niculescu nd L.-E. Persson, Conve functions nd their pplictions. A contemporry pproch, CMS Books in Mthemtics vol. 23, Springer-Verlg, New York, [8] M. R. S. Ammi, Rui A. C. Ferreir, nd Delfim F. M. Torres, Dimond-α Jensen s Inequlity on Time Scles nd Applictions, Journl of Inequlities nd Applictions, vol. 2008, Article ID , 3 pges, [9] T. Popoviciu, Notes sur les fonctions convees d ordre superieur IX, Bull. Mth. Soc. Roum. Sci., 43 94, [0] J. W. Rogers Jr. nd Q. Sheng, Notes on the dimond-α dynmic derivtive on time scles, J. Mth. Anl. Appl., , no., [] Q. Sheng, M. Fdg, J. Henderson nd J. M. Dvis, An eplortion of comined dynmic derivtives on time scles nd their pplictions, Nonliner Anl. Rel World Appl., , no. 3, [2] Ling-Cheng Wng, Two Mppings Relted to Hölder s Inequlity, Univ. Beogrd. Pul. Elektrotehn. Fk., Ser. Mt , Cristin DINU, Deprtment of Computer Science, University of Criov, Str. A. I. Cuz, Criov, Romni. Emil: c.dinu@yhoo.com Mihi STANCU, Deprtment of Computer Science, University of Criov, Str. A. I. Cuz, Criov, Romni. Emil: mihissss@yhoo.com Dniel DĂNCIULESCU, Deprtment of Computer Science, University of Criov, Str. A. I. Cuz, Criov, Romni. Emil: dndnciulescu@yhoo.com Unuthenticted Downlod Dte 0/2/8 4:52 AM