Desigualdades integrales fraccionales y sus q-análogos
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1 Desiulddes ierles rccioles y sus -áloos Suil Du Puroi Fru Uçr 2 d R.K. Ydvc 3 Dediced o Proessor S.L. Kll Revis Tecocieíic URU Uiversidd Rel Urde Fculd de eierí Nº 6 Eero - Juio 204 Depósio lel: ppi 20402ZU4464 SSN: Depre o Bsic Scieces Meics Collee o Tecoloy d Eieeri M.P. Uiversiy o Ariculure d Tecoloy Udipur-3300 Rjs di. suil puroi@yoo.co 2 Depre o Meics Uiversiy o Mrr TR Kd öy sul Turey. ucr@rr.edu.r 3 Depre o Meics d Sisics J. N. Vys Uiversiy. Jodpur di. rdydv@il.co Reciido: Acepdo: Resue El ojeo de ese rjo es eslecer lus desiulddes ue evuelve operdores ierles de Sio. Se us el clculo -rcciol pr oeer vrios resuldos e l eorí de ls desiulddes -ierles. Los resuldos ddos erioree por Puroi y Ri 203 y Suli 20 so csos especiles de los oeidos e ese rjo. Plrs clve: Desiulddes ierles operdores ierles rccioles operdores -ierles rccioles. O rciol ierl ieuliies d eir -loues Asrc Te i o is pper is o eslis soe ierl ieuliies ivolvi Sio rciol ierl operors. We e use rciol -clculus or yieldi vrious resuls i e eory o -ierl ieuliies. Te resuls ive erlier y Puroi d Ri 203 d Suli 20 ollow s specil cses o our idis. Key words: erl ieuliies rciol ierl operors rciol -ierl operors. 53
2 54 Desiulddes ierles rccioles y sus -áloos Revis Tecocieíic URU Nº 6 Eero - Juio roducio Frciol ierl ieuliies ve y pplicios e os useul oes re i eslisi uiueess o soluios i rciol oudry vlue proles d i rciol pril diereil euios. Furer ey lso provide upper d lower ouds o e soluios o e ove euios. For deiled pplicios oe y reer o e oo [] d e rece ppers [2]-[5] o e sujec. rece pper Puroi d Ri [6] ivesied ceri Ceysev ype [7] ierl ieuliies ivolvi e Sio rciol ierl operors d lso eslised e -exesios o e i resuls. Te i o is pper is o eslis severl ew ierl ieuliies or sycroous ucios re reled o e Ceysev uciol usi e Sio rciol ierl. -Exesios o e i resuls re lso eslised. Soe o e resuls due o Puroi d Ri [6] d Suli [8] ollows s specil cses o our resuls. Followi deiiios will e eeded i e seuel. Deiiio. Two ucios d re sid o e sycroous o [ ] i { x yx y} 0 or y x y Î []. Deiiio 2. A rel-vlued ucio 0 is sid o e i e spce C μ μ Î R i ere exiss rel uer p μ suc p were Î C 0. Deiiio 3. Le α 0 β Î R e e Sio rciol ierl αβ o order α or rel-vlued coiuous ucio is deied y [9] see lso [0 p. 9] []: 0 { } 2F 2 d 0 were e ucio 2 F i e ri-d side o 2 is e Gussi ypereoeric ucio deied y 2 F c 3 c! 0 d is e Pocer syol Te ierl operor 2 icludes o e Rie-Liouville d e Erd e lyi-koer rciol ierl operors ive y e ollowi reliosips: R { } 0 { } d d 0 { } { } 0 Î 0 R. 5 d 0
3 Suil Du Puroi e l. Revis Tecocieíic URU Nº 6 Eero - Juio For µ i 2 we e e ow resul [9]: α β 0 µ { } μ μ β μ β μ βμ α 6 α 0iμ μ β 0 wic sll e used i e seuel. Frciol erl euliies Te ollowi eores ivolvi Sio ierl ieuliies or e sycroous ucios will e eslised. Teore. Le d e wo sycroous ucios o [0 0 e or ll 0 α x{0 β} β < β < < 0. β β α β α β 0 { } { } { } α β α β 0 0 α β α β α β α β 0 {} 0 { } 0 {} 0 { }. 7 Proo: Usi Deiiio d 0 or ll 0 we ve wic iplies { } 0 8 Cosider. 9 F 2F α α β α α βα β α 2 2.
4 56 Desiulddes ierles rccioles y sus -áloos Revis Tecocieíic URU Nº 6 Eero - Juio Sice ec er o e ove series is posiive i view o e codiios sed wi Teore we oserve e ucio F reis posiive or ll τ Î 0 0. Muliplyi o sides o 9 y F deied ove y 0 d ieri wi respec o ro 0 o d usi 2 we e α β { } α β { } α β { } α β α β α β 0 { } 0 { } 0 { } α β α β 0 0 { }. { } Nex uliplyi o sides o y F Î 0 0 were F is ive y 0 d ieri wi respec o ro 0 o d usi orul 6 we rrive e desired resul 7. Teore 2. Le d e wo sycroous ucios o [0 d 0 e β { } { } β α β γ γ α β 0 0 α β γ α β γ α β γ 0 { } { } 0 0 { } { } 0 0 { } { } 0 α β { } γ { } α β {} γ { } α β { } γ {} or ll 0 α x{0 β} γ x{0 } β < β < < 0 < < 0. Proo: To prove e ove eore we sr wi e ieuliy. O uliplyi o sides o y 2F Î 0 0 d i ierio wi respec o ro 0 o we e γ α β α β γ 0 {} 0 { } 0 {} { } 0 α β γ α β γ α β γ 0 { } 0 { } 0 { } 0 { } 0 {} 0 { } α β γ α β γ 0 { } 0 { } 0 {} 0 { } α β γ 0 { } 0 {} wic o usi 6 redily yields e desired resul 2.
5 Suil Du Puroi e l. Revis Tecocieíic URU Nº 6 Eero - Juio Rer. y e oed e ieuliies 7 d 2 re reversed i e ucios re sycroous o [0 i.e. or y x y [0. { x y x y } 0 Rer 2. For Teore 2 iediely reduces o Teore. Teore 3. Le d e ree oooic ucios o [0 sisyi e ieuliy { } 0 e or ll 0 α x{0 β} γ x{0 } β < β < < 0 < < β β α β γ { } { } γ α β 0 0 α β γ α β γ α β γ 0 { } { } 0 0 { } { } 0 0 { } { } 0 α β γ α β γ α β γ 0 { } { } 0 0 { } { } 0 0 { } {}. 0 5 Proo: By pplyi e siilr procedure s o Teore d 2 oe c esily eslis e ove eore. Tereore we oi e deils o e proo o is eore. Oserve i we se 0 d 0 ddiiolly or Teore 2 d e use o e relio 5 Teores o 3 respecively yield e ollowi ierl ieuliies ivolvi e Erd e lyi-koer ype rciol ierl operor deied y 5: Corollry. Le d e wo sycroous ucios o [0 d 0 e α { } α { } α { } α { } α { } α { } { } 6 or ll 0 α 0 < < 0. Corollry 2. Le d e wo sycroous ucios o [0 d 0 e or ll 0 α γ 0 < x < 0 γ { } α { } α γ { } { } { } { } { } { }
6 58 { } { } { } { } { } { }. 7 Corollry 3. Le d e ree oooic ucios o [0 sisyi e ieuliy 4 e or ll { } { } { } { } { } { } { } { } { } { } { } { }. 8 Ai i we replce y d y i Teores 2 d 3 d e use o e relio 4 we oi ow resuls due o Suli [8 pp Teores 2. o 2.2]. -Exesios o Mi Resuls is secio we eslis -exesios o e resuls derived i e previous secio. We ei wi e eicl preliiries o -series d -clculus. For ore deils o -clculus d rciol -clculus oe c reer o [2] d [3]. Te -sied coril is deied or C s produc o cors y 9 d i ers o e sic loue o e ucio 20 were e - ucio is deied y [2 p. 6 e..0.] 2 We oe 22 d i < e deiiio 9 reis eiul or s covere iiie produc ive y 23 Desiulddes ierles rccioles y sus -áloos Revis Tecocieíic URU Nº 6 Eero - Juio α γ 0 < x < 0 γ α { } { } α γ 0 N α α < <.. 0 j j
7 Suil Du Puroi e l. Revis Tecocieíic URU Nº 6 Eero - Juio Also e -ioil expsio is ive y / /. 0 / ν ν y x x y ν x y x ν x 24 ν y x Le 0 R e we deie speciic ie scle see [4] d [5] d or se o coveiece we deoe { o - eive ieer} { 0} 0 < < T T y T rouou is pper. 0 Te -derivive d -ierl o ucio deied o T re respecively ive y see [2 pp. 9 22] d D 0 d Deiiio 4. Te Rie-Liouville rciol -ierl operor o ucio o order due o [5] see lso [3] is ive y 0 { } / d 0 0 < < 28 were R. 29 Deiiio 5. For 0 R d 0 < < e sic loue o e Koer rciol ierl operor c. [6] [3] is ive y 0 { } / d. 30 Deiiio 6. For 0 d R sic loue o e Sio s rciol ierl operor [7 p. 72 e. 2.] is ive or / < y / 0 { }
8 60 3 wic i view o 27 c e wrie s see [7 p. 73 e. 2.5]: 32 e seuel we sll e usi e ollowi ie orul [7 p. 73 e. 2.]: 33 Now we sll eslis ew -ierl ieuliies or e sycroous ucios ivolvi e rciol -ierl operors wic c e reed s e -loues o e ieuliies 7 2 d 5. Teore 4. Le d e wo sycroous ucios d 0 o T e { } { } { } { } 34 were Proo: By e ypoesis e ucios d re sycroous ucios o T or ll τ 0 d 0 ereore e ieuliy 9 is sisied is. Sice τ Î < < e Î 0 or Î N ereore o replci τ y i e ove ieuliy we e Desiulddes ierles rccioles y sus -áloos Revis Tecocieíic URU Nº 6 Eero - Juio /2 d β { } 0 0. { } µ µ µ µ μ μ β β α β β α 00 < < iμ μ β < < α x {0 β} β < β < < 0. { } { } { }
9 6 35 Cosider 36 Evidely uder e codiios sed wi Teore 4 we oserve e ucio H is posiive or ll vlues o N. Tereore o uliplyi o sides o 35 y H d i suios ewee e liis 0 o we e. Now o i i suio ro 0 o d e i use o e deiiio 32 we oi { } { } { } { } { } { } { } { }. 37 Nex i e ove ieuliy o replci y uliplyi o sides o y H i suios ewee e liis 0 o d 0 o d e i use o e de- iiios 32 d 33 we rrive e desired ieuliy 34. Teore 5. Le d e wo sycroous ucios o T d 0 e or ll 00 < < α x{0 β} γ x{0 } β < β < < 0 < ξ < 0 Suil Du Puroi e l. Revis Tecocieíic URU Nº 6 Eero - Juio H β Î N. 0 H 0 H 0 H. 0 H 0 H 0 H 0 H 0 H
10 62 { } { } { } { } { } { } { } { } { } { } { } { }. 38 Proo: To prove e ove eore we sr wi e ieuliy 37. O replci y d uliplyi o sides y posiive ucio F ive y 39 i suios ewee e liis 0 o d 0 o d e i use o e deiiio 32 e e ieuliy 37 leds o { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } { } 40 wic yields e desired resul y i 33 io ccou. Rer 3. Te ieuliies 34 d 38 re reversed i e ucios re sycroous o T. Rer 4. Ai we γ α β e Teore 5 leds o Teore 4. Teore 6. Le d e ree oooic ucios o [0 sisyi e ieuliy 4 e or ll 00 < < x{0 } γ x{0 } β < β < < 0 < < 0 we ve { } { } { } { } { } { } Desiulddes ierles rccioles y sus -áloos Revis Tecocieíic URU Nº 6 Eero - Juio { } { } β β α β γ γ α β F Î N. { } β β α β γ β γ { } { } { }
11 63 { } { } { } { }. 4 Proo: By pplyi e se procedure s o Teore 4 d 5 oe c eslis e ove eore. Tereore we oi e deils o e proo. Now i we se 0 d ddiiolly 0 or Teore 5 d e use o e ow resul [8 p.73 e. 2.9] ely { } { } 0 42 Teores 4 o 6 respecively reduce o e ollowi -ierl ieuliies ivolvi e Erd e lyi-koer ype rciol -ierl operors: Corollry 4. Le d e wo sycroous ucios o T d 0 e { } { } 43 or ll Corollry 5. Le d e wo sycroous ucios o T d 0 e or { } { } { } { } { } { } { } { } { } { } { } { }. 44 Corollry 6. Le d e ree oooic ucios o [0 sisyi e ieuliy 4 e or ll { } { } { } { } { } { } { } { } { } { } { } { }. 45 Suil Du Puroi e l. Revis Tecocieíic URU Nº 6 Eero - Juio { } { } { } { } { } α α α γ γ 00 < < α 0 d < < < < α γ 0 suc < x < 0 0 α γ 0 < x < 00 < < { } { } { } { }
12 64 Furer we oserve i we replce y d y d e use o e relio [7 p.73 e. 2.7] ely { } { } 46 d { } { } 47 e Teores 4 o 6 reduce o e ollowi -ierl ieuliies ivolvi e Rie-Liouville ype o rciol -ierl operors. Corollry 7. Le d e wo sycroous ucios o T d 0 e { } { } 48 or ll < 00 < d 0. Corollry 8. Le d e wo sycroous ucios o T d 0 e or { } { } { } { } { } { } { } { } { } { }. 49 Corollry 9. Le d e ree oooic ucios o [0 sisyi e ieuliy 4 e or ll { } { } { } { } { } { } { } { } { } { }. 50 Specil Cses We ow riely cosider soe o e coseueces o e resuls derived i e previous secios. we le d use e lii oruls: Desiulddes ierles rccioles y sus -áloos Revis Tecocieíic URU Nº 6 Eero - Juio { } { } { } { } { } α α 0 0 < < α γ 0 0 α γ 00 < < α γ α γ { } { } { } { } α γ α γ { } { } { } { }
13 Suil Du Puroi e l. Revis Tecocieíic URU Nº 6 Eero - Juio Li 5 d Li 52 e resuls o Secio 3 correspod o e resuls oied i Secio 2. Ai i view o e ove liii cses Corollries 8 d 9 provide respecively e -exesios o e ieuliies due o Suli [8 pp Teores 2. o 2.2]. Filly i we cosider e ucio s cos 0 e Teores 2 4 d 5 d Corollries 7 d 8 provide respecively e ow resuls due o Puroi d Ri [6] d Öğüez d Öz [4]. Acowledees Te uors re ul o e reeree or very creul redi d vlule suesios ledi o e prese iproved or o e pper. Reereces. G.A. Asssiou Advces o Frciol euliies Sprier Bries i Meics Sprier New Yor Z. Deo A.S. Vsl Moooic ierive eciue or ii syse o olier Rie-Liouville rciol diereil euios Opuscul Meic S.L. Kll d Al Ro O Gru ss ype ieuliy or ypereoeric rciol ierls Le Meice V. Lsi d A.S. Vsl Teory o rciol diereil ieuliies d pplicios Cou. Appl. Al J.D. Rírez A.S. Vsl Moooic ierive eciue or rciol diereil euios wi periodic oudry codiios Opuscul Meic S.D. Puroi d R.K. Ri Ceysev ype ieuliies or e Sio rciol ierls d eir -loues J. M. eul P.L. Ceysev Sur les expressios pproxiives des ierles deiies pr les ures prises ere les êes liies Proc. M. Soc. Crov W.T. Suli Soe ew rciol ierl ieuliies J. M. Al M. Sio A rer o ierl operors ivolvi e Guss ypereoeric ucios M. Rep. Kyusu Uiv V.S. Kiryov Geerlized Frciol Clculus d Applicios Pi Res. Noes M. Ser. 30 Lo Scieiic & Tecicl Hrlow R.K. Ri Soluio o Ael-ype ierl euio ivolvi e Appell ypereoeric ucio erl Trsors Spec. Fuc G. Gsper d M. R Bsic Hypereoeric Series Cride Uiversiy Press Cride 990.
14 66 Desiulddes ierles rccioles y sus -áloos Revis Tecocieíic URU Nº 6 Eero - Juio M.H. Ay d Z.S. Msour -Frciol Clculus d Euios Lecure Noes i Meics 2056 Sprier-Verl Berli Heideler H. Öğüez d U.M. Öz Frciol uu ierl ieuliies J. eul. Appl. Volue 20 Aricle D pp. 5. R.P. Arwl Ceri rciol -ierls d -derivives Proc. C. Pil. Soc W.A. Al-Sl Soe rciol -ierls d -derivives Proc. Edi. M. Soc M. Gr d L Ccli -Aloue o Sio s rciol clculus operors Bull. M. Al. Appl
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