On Hadamard and Fejér-Hadamard inequalities for Caputo k-fractional derivatives

Transcription

1 In J Nonliner Anl Appl 9 8 No, 69-8 ISSN: 8-68 elecronic hp://dxdoiorg/75/ijn8745 On Hdmrd nd Fejér-Hdmrd inequliies for Cpuo -frcionl derivives Ghulm Frid, Anum Jved Deprmen of Mhemics, COMSATS Universiy Islmd, Aoc Cmpus, Pisn Communiced y M Eshghi Asrc In his pper we will prove cerin Hdmrd nd Fejér-Hdmrd inequliies for he funcions whose n h derivives re convex y using Cpuo -frcionl derivives These resuls hve some relionship wih inequliies for Cpuo frcionl derivives Keywor: Hdmrd inequliy; convex funcions; Fejér-Hdmrd inequliy; Cpuo frcionl derivives MSC: Primry 6A5; Secondry 6A33, 6D, 6D5 Inroducion Frcionl clculus is he generlizion of clssicl clculus which is minly concerned wih operions of inegrion nd differeniion of non ineger frcionl order Since 9h cenury, he heory of frcionl clculus developed rpidly, mosly s founion for numer of pplied disciplines which include frcionl geomery, frcionl differenil equions nd frcionl dynmics The pplicions of frcionl clculus re very wide nowdys Almos no discipline of modern engineering nd science remins unouched y he ools nd echniques of frcionl clculus Frcionl clculus hs wide pplicions in rheology, viscoelsiciy, cousics, opics, chemicl nd sisicl physics, rooics, conrol heory, elecricl nd mechnicl engineering, ioengineering, ec see 7 for deils The hisory of frcionl clculus is s old s he hisory of differenil clculus Frcionl clculus is nurl exension of sndrd mhemics Frcionl clculus lso hs lo of pplicions in he Corresponding uhor Emil ddresses: fridphms@homilcom, ghlmfrid@cii-ocedup Ghulm Frid, jvednum38@gmilcom Anum Jved Received: Mrch 7 Revised: Jnury 8

2 7 Frid, Jved fiel of science couning rheology, fluid flow, diffusive rnspor, elecricl newors, elecromgneic heory nd proiliy see 3 Fourier, Ael, Lcroix, Leiniz, Leniov nd Grunwld conriued lo in his sujec see 6, 8, 9 nd references here in We give some preliminries h we will use for our resuls For his we will define convex funcions, Hdmrd inequliy for convex funcions, Fejér Hdmrd inequliy for convex funcions, Cpuo frcionl derivives nd finlly Cpuo frcionl derivives Definiion A funcion f : I R is convex if he following inequliy f λx + λy λfx + λfy, hol for ll x, y I nd λ, If reverse of he ove inequliy hol, hen f is sid o e concve funcion Theorem Le f : I R e convex funcion defined on inervl I of rel numers wih, I nd < Then he following inequliy hol + f f + f fxdx I is well nown in he lierure s he Hdmrd inequliy following weighed generlizion of he Hdmrd inequliy In 5, Fejér eslished he Definiion 3 Le f : I R e convex funcion defined on inervl I of rel numers wih, I nd < Then he following inequliy hol + f gxdx fxgxdx f + f gxdx, 3 where g : I R is nonnegive, inegrle nd symmeric o + I is well nown in he lierure s he Fejér Hdmrd inequliy Definiion 4 Le > nd / {,, 3, }, n = +, f AC n,, he spce of funcions hving nh derivives soluely coninuous The lef sided nd righ sided Cpuo frcionl derivives of order re defined s follows: nd C D +fx = Γn x f n x > 4 x n+ C D fx = n f n x < 5 Γn x x n+ If = n {,, 3, } nd usul derivive f n x of order n exiss, hen Cpuo frcionl derivive C D n +fx coincides wih f n x wheres C D n fx coincides wih f n x wih excness o consn muliplier n In priculr we hve where n = nd = C D +fx = C D fx = fx, 6

3 On Hdmrd nd Fejér-Hdmrd inequliies for Cpuo 9 8 No, For furher deils see 6 Frcionl inegrl inequliies ply very imporn role in eslishing he uniqueness of soluions for cerin frcionl pril differenil equions They provide oun for he soluion of frcionl oundry vlue prolems Due o hese considerions mny reserchers explore cerin exensions nd generlizions of severl in of inequliies y involving frcionl clculus operors see,,, 3, 4, 6 nd references herein In his pper, in Secion we define Cpuo -frcionl derivives nd uilize hem o give he Hdmrd inequliy for funcions whose nh derivives re convex We lso find he ound of difference of his inequliy In Secion 3 we derive he Fejér Hdmrd inequliy vi Cpuo frcionl derivives nd find oun of difference of his inequliy We lso deduce some reled resuls Hermie Hdmrd inequliies for Cpuo -frcionl derivives Firs we give definiion of he lef sided nd he righ sided Cpuo -frcionl derivives Definiion Le >, nd / {,, 3, }, n = +, f AC n, The righ sided nd lef sided Cpuo frcionl derivives of order re defined s follows: nd + fx = fx = Γ n x n Γ n x where Γ is he Gmm funcion defined s: lso Γ = f n x > x n+ f n x <, x n+ e, Γ + = Γ If = n {,, 3, } nd usul derivive f n x of order n exiss, hen Cpuo frcionl derivive C D+fx n, coincides wih f n x wheres C D n, fx coincides wih f n x wih excness o consn muliplier n In priculr we hve C D+fx, = C D, fx = fx, 3 where n, = nd = For =, Cpuo frcionl derivives give he definiion of Cpuo frcionl derivives In he following we give he Hdmrd inequliy for funcions whose n h derivives re convex vi Cpuo frcionl derivives Theorem Le f :, R e funcion such h f AC n,, < If f n is convex on,, hen he following inequliy for Cpuo frcionl derivives hol + f n Γ n + n + f + n f f n + f n 4

4 7 Frid, Jved Proof Since f n is convex, so x + y f n f n x + f n y 5 Le x, y,, such h x = +, y = + where, Then from 5 we hve + f n f n + + f n + 6 Muliplying oh sides of ove inequliy wih n nd inegring over,, we ge + f n n f n + + n+ By chnge of vriles, we hve + f n Γ n + n Also convexiy of f n gives f n + n+ + f + n f 7 f n + + f n + f n + f n 8 Muliplying oh sides of ove inequliy wih n nd inegring over,, we ge f n + n+ + Now y chnge of vriles we ge Γ n + n f n + n+ f n + f n + f + n f From inequliies oined in 7 nd 9 we ge inequliy in 4 n f n + f n 9 Corollry 3 If we e =, we ge he following inequliy for Cpuo frcionl derivives 4 + f n Γn + C D +f + n C D f f n + f n n For nex resul we need he following lemm Lemm 4 Le f :, R e funcion such h f AC n+,, < If f n+ is convex on,, hen he following equliy for Cpuo frcionl derivives hol f n + f n = Γ n + n + f + n f n n f n+ +

5 On Hdmrd nd Fejér-Hdmrd inequliies for Cpuo 9 8 No, Proof Consider he righ hnd side = n n f n+ + n f n+ + n f n+ + Now we compue he firs nd he second erms of ls expression s follows respecively nd = = = f n n f n+ + n f n + + n f n n Γ n + n = = f n x n f Hence he required equliy cn e eslished n+ f n x dx f n + n+ n f n+ + f n n n+ f n x x dx Γ n + n + f Using he ove lemm we eslish he oun of difference of 4 Theorem 5 Le f :, R e funcion such h f AC n+,, < If f n+ is convex on,, hen he following inequliy for Cpuo frcionl derivives hol f n + f n Γ n + + f + n f n + n n f n+ + f n+ Proof Using convexiy of f n+ nd Lemm 4, we hve f n + f n Γ n + n + f + n f n n f n+ + n n f n+ + f n+

6 74 Frid, Jved Now we hve = + n n f n+ + f n+ n n f n+ + f n+ n n f n+ + f n+ = f n+ n n + + f n+ n + n = f n+ n + n + n + n + + f n+ n + n + n + Similrly n n f n+ + f n+ = f n+ n + n + n + + f n+ n + n + n + n + Therefore ecomes f n + f n Γ n + n + f + n f f n+ n + n + n + n + + f n+ n + n + n + + f n+ n + n + n + + f n+ n + n + n + n + From which fer lile compuion we ge he required resul Corollry 6 If we e =, we ge he following inequliy for Cpuo frcionl derivives 4 f n + f n Γn + C D +f + n C D f n f n+ + f n+ n + n 3 Fejér Hdmrd inequliies for Cpuo frcionl derivives In his secion Fejér Hdmrd nd Fejér Hdmrd ype inequliies for Cpuo frcionl derivives re given

7 On Hdmrd nd Fejér-Hdmrd inequliies for Cpuo 9 8 No, Lemm 3 For < λ nd <, we hve λ λ λ In his secion we use g n = sup x, g n x nd he convoluion f g of funcions f nd g for Cpuo frcionl derivives s follows nd + f gx = f gx = Here firs we prove he following lemm Γn x n Γn x f n g n x > 3 x n+ f n g n x < 3 x n+ Lemm 3 Le g :, R e funcion such h g AC n,, < If g n is symmeric o +, hen + g = n g = C D, + g + n g Proof By definiion we hve + g = Γ n Susiuing x y + x in he ove inegrl we hve + g = Γ n g n xdx x n+ g n + xdx x n+ By symmericiy of g n we hve g n + x = g n x, herefore nd we hve + g = Γ n g n xdx x n+ Hence he required equliy cn e oined + g = n g Using ove lemm we prove he following resuls Theorem 33 Le f :, R e funcion such h f AC n,, < Also le f n e convex funcion on, nd g :, R e such h g AC n, If g n is nonnegive, inegrle nd symmeric o +, hen he following inequliy for Cpuo frcionl derivives hol + f n + g + n g + f g + n f n + f n f g + g + n g 33

8 76 Frid, Jved Proof Since f n is convex muliplying oh sides of inequliy 6 wih gn + n+ inegring he resuling inequliy over,, we hve + f n g n + n+ f n + g n + n+ + Puing x = +, we ge + f n g n x dx n x n+ f n + xg n x dx + n x n+ f n xg n + x = dx + n x n+ f n xg n x = dx + n x n+ f n + g n + n+ f n xg n x x n+ dx f n xg n x x f n xg n x dx x n+ n+ dx By using Lemm 3 we ge he firs inequliy of 33 For he second inequliy of 33 muliplying oh sides of inequliy 8 wih gn + nd inegring he resuling inequliy over, we ge n+ f n + g n + f n + g n + + n+ n+ f n + f n g n + n+ from which fer using chnge of vriles nd lile compuion we ge he required resul Corollry 34 If we e = in Theorem 33, we ge he following resul for Cpuo frcionl derivives 4 + f n C D +g + n C D g C D+f g + n C D f g f n + f n C D +g + n C D g Nex we need he following lemm Lemm 35 Le f :, R e funcion such h f AC n+,, < Also le f n+ e convex on, nd g :, R e funcion such h g AC n, Then he following equliy for Cpuo frcionl derivives hol f n + f n = Γ n + g + n g s n+ s + f g + n f g n+ f n+ nd 34

9 On Hdmrd nd Fejér-Hdmrd inequliies for Cpuo 9 8 No, Proof We noe h Γ n s n+ = Γ n s By simple clculion we ge s = s nd n+ n+ n+ s f n+ + f n+ f n n+ f n+ f n g n n+ = Γ n f n + g + f g s n+ = Γ n f n + g + n g C D, + f g s = s n+ n+ f n+ f n f n g n n+ = Γ n f n + g + n g n f g Hence y ddiion 34 is eslished f n+ Theorem 36 Le f :, R e funcion such h f AC n+,, < If f n+ is convex on, nd g :, R is such h g AC n, If g n is symmeric o +, hen he following inequliy for Cpuo -frcionl derivives hol f n + f n + g + n g + f g + n f g n + g n n + Γ n + f n+ + f n+ n 35 Proof Since f n+ is convex, so we hve where, From symmericiy of g n, we hve = s n+ f n+ f n+ + f n+, 36 + g n + s = s n+ + s n+

10 78 Frid, Jved This gives s n+ s n+ + = s n+ + g n s s, n+ g + n s s, n+, + +, 37 By virue of Lemm 35 nd inequliies 36, 37 we hve f n + f n + g + n g + + Γ n s n s n+ g n + Γ n + n n + n n + We hve + nd + n n + f g + n f n+ + f n+ f n+ + f n+ f n+ + f n+ = + = n + n + f g 38 f n+ + f n+ n n n + n + n + = n n + n = n + n + Using 39 nd 3 in 38, we ge he required resul n n + n Corollry 37 In Theorem 36 if we pu =, we ge he following resul for Cpuo frcionl derivives 4 f n + f n C D +g + n C D g C D+f g + n C D f g n + g n f n+ + f n+ n + Γn + n

11 On Hdmrd nd Fejér-Hdmrd inequliies for Cpuo 9 8 No, Theorem 38 Le f :, R e funcion such h f AC n+,, < Also le f n+ q, q e convex on, nd g :, R e such h g AC n, If g n is symmeric o +, hen he following inequliy for Cpuo frcionl derivives hol f n + f n + g + n g + f g + n f g n + g n n + Γ n + q Γ n n n + f n+ q + f n+ q n + n n + n f n+ q + f n+ q q 3 Proof By Using Lemm 35, power men inequliy, inequliy 37 nd convexiy of f n+ q respecively we hve f n + f n + g + n g + f g + n f g + Γ n q s n+ + s n+ f n+ q q + + Γ n g n s s n+ + g n q s + + s n+ + + g n s s n+ f n+ q + g n q s + + s n+ f n+ q gn n + q n n q From which fer lile compuion we hve he required resul Theorem 39 Le f :, R e funcion such h f AC n+,, < Also le f n+ q, q > e convex on, nd g :, R e such h g AC n, If g n is symmeric o +, hen he following inequliies for Cpuo frcionl derivives hold i f n + f n + g + n g + f g + n f g p n + g n np p + p Γ n + p f n+ q + f n+ q q p np ii f n + f n + g + n g + f g + n f g n + g n f n+ q + f n+ q q np p + p Γ n +, wih <, where p + q =

12 8 Frid, Jved Proof i By Using Lemm 35, Hölder s inequliy, inequliy 37 nd convexiy of f n+ q, we hve f n + f n + g + n g + f g + n f g Γ n Γ n s p n+ s n+ f n+ q + f n+ q f n + f n gn Γ n Now gives + p p f n+ q q + + q + s n+ p p + g + n g + f g + n f g n n f n+ q + f n+ q n n for, + nd n n n p n + q + A B q A q B q A B, n n n n p p p np n p p np n p n n n n p p 3 p np n 33 p p np n 34 p for +, Using 33 nd 34 in inequliy 3 nd hen solving, we ge inequliy i ii Here one cn use inequliy 3 nd Lemm 3 in order o prove inequliy ii Corollry 3 If we e = in ove heorem, we ge he following resul for Cpuo frcionl derivives 4 i f n + f n p n + g n np p + p Γn + C D +g + n C D g C D +f g + n C D f g p f n+ q + f n+ q np p q

13 On Hdmrd nd Fejér-Hdmrd inequliies for Cpuo 9 8 No, ii f n + f n C D +g + n C D g C D+f g + n C D f g n + g n f n+ q + f n+ q q, np p + p Γn + wih <, where p + q = Acnowledgmens This reserch wor is suppored y he Higher Educion Commission of Pisn under NRPU 6 References GA Ansssiou, Advnces on frcionl inequliies, World Scienific, Springer New Yor Dordrech Heidelerg London, GA Ansssiou, Advnced inequliies, World Scienific, Singpore, 3 M Dlir nd M Bshour, Applicions of frcionl clculus, Appl Mh Sci, G Frid, S Nqvi nd A Jved, Hdmrd nd Fejér Hdmrd inequliies nd reled resuls vi Cpuo frcionl derivives, Bull Mh Anl Appl, L Fejér, die Fourierreihen, II, Mh Nurwiss Anz Ungr Ad, Wiss, In Hungrin 6 AA Kils, HM Srivsv nd JJ Trujillo, Theory nd pplicions of frcionl differenil equions, Norh Hollnd Mh Sud 4, Elsevier, New Yor London, 6 7 M Lzrević, Advnced opics on pplicions of frcionl clculus on conrol prolems, Sysem siliy nd modeling, WSEAS Press, Belgrde, Seri, 8 KS Miller nd B Ross, An inroducion o he frcionl clculus nd frcionl differenil equions, John Wiley nd Sons, Inc, New Yor, K Oldhm nd J Spnier, The frcionl clculus heory nd pplicions of differeniion nd inegrion o rirry order, Acdemic Press, New Yor London 974 J Pečrić, F Proschn nd YL Tong, Convex funcions, pril orderings nd sisicl pplicions, Acdemic Press, New Yor, 99