A Hilbert-type fractal integral inequality and its applications

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1 Liu ad Su Joural of Ieualiies ad Alicaios 7) 7:83 DOI.86/s R E S E A R C H Oe Access A Hilber-e fracal iegral ieuali ad is alicaios Qiog Liu ad Webig Su * * Corresodece: swb5@63.com Dearme of Sciece ad Iformaio Sciece, Shaoag Uiversi, Shaoag, 4, P.R. Chia Absrac B usig hefracal heor ad he mehods of weigh fucio, a Hilber-e fracal iegral ieuali ad is euivale form are give. Their cosa facors are roved beig he bes ossible, ad heir alicaios are discussed briefl. Kewords: fracal se; Hilber-e fracal iegral ieuali; weigh fucio Iroducio If f, g, saisfig < f ) d <,< g ) d <, he here is he followig basic Hilber-e iegral ieuali ad is euivale form dd <4 f ) d g ) d}, ) ma, } d <6 f ) ma, } d f ) d, ) where he cosas are oimal. Ieualiies ) ad) are imora i he aalsis ad arial differeial euaios,. I 4 ad 6, resecivel, )ad) were geeralized ad imroved b iroducig a ideede arameer λ ad wo arameers λ, λ 3, 4. I rece ears, he fracal heor has bee develoed raidl, ad i has bee widel used i he fields of sciece ad egieerig. Some researchers have used he fracal heor o discuss ad geeralize some classical ieualiies o fracal ses 5, 6, bu he research io he Hilber-e iegral ieuali o he fracal se is sill o ivolved. I his aer, b usig he fracal heor ad he mehod of weigh fucio o make a meaigful aem, a Hilber-e iegral ieuali ad is euivale form o a fracal se are esablished. Prelimiaries Defiiio. 7) A o-differeiable fucio f : R R < ), f ) is called local fracioal coiuous a if for a ε >,hereeissδ >suchha f ) f ) < ε wheever < δ.iff ) is local fracioal coiuous o he ierval a, b), we deoe f ) C a, b). The Auhors) 7. This aricle is disribued uder he erms of he Creaive Commos Aribuio 4. Ieraioal Licese h://creaivecommos.org/liceses/b/4./), which ermis uresriced use, disribuio, ad reroducio i a medium, rovided ou give aroriae credi o he origial auhors) ad he source, rovide a lik o he Creaive Commos licese, ad idicae if chages were made.

2 Liuad Su Joural of Ieualiies ad Alicaios 7) 7:83 Page of 8 Defiiio. 7) The local fracioal derivaive of f ) oforder < ) a is defied b f ) d f ) d f ) f )) lim, ) where Γ z) e u u z du z >)8. If for all I R, hereeissf k+) ) k+ }} D D f ), he we deoe f D k+)i), where k,,,... Lemma. 9) Suose ha f ) C a, b) ad f ) D a, b). The, for <, we havea -differeial form d f )f ) ) d. Lemma. 5) Le I be a ierval, f, g : I R R I is he ierior of I) such ha f, g D I ). The he followig differeiaio rules are valid: i) d f )±g)) d ii) d f )g)) d f ) ) ± g ) ); f ) )g)+f )g ) ); iii) d f ) g) d f ) )g) f )g ) ) g) ); g ) iv) d Cf )) d Cf ) ), where C is a cosa; v) If )f g)), he d ) d f ) g))g ) )). Defiiio.3 7) Le f ) C a, b). The he local fracioal iegral is defied b ai b f ) b a f )d) lim λ N f i ) i ), wih i i i i,...,n) adλ ma i N i },ada < < < N b is ariio of ierval a, b. Here, i follows ha a Ib ifa b, aib f ) bia f )ifa < b. Lemma.3 7) ) Suose ha f )g ) ) C a, b), he we have i ai b f )gb) ga); ) Suose ha f ), g) D a, b), ad f ) ), g ) ) C a, b), he we have ai b f )g) )f )g) b a ai b f ) )g). Lemma.4 7) For f ) γ γ >),we have he followig euaios: d γ ) Γ + γ ) d Γ + γ ) γ ; b a γ d) Γ + γ ) b γ + a γ +). Γ + γ + )

3 Liuad Su Joural of Ieualiies ad Alicaios 7) 7:83 Page 3 of 8 Lemma.5 7, ) If f, g ) C a, b), F, G, h ) C S β) ), >, +,Sβ) is afracalsurface, he we have i) Hölder s ieuali o he fracal se b f )g)d) a b f )d) a b ii) Hölder s weighed ieuali o he fracal se h, )F, )G, )d) d) Γ +) S β) h, )F, )d) d) Γ +) S β) h, )G, )d) d). Γ +) S β) a g )d) ; The ieuali kees he form of euali, he here eis cosas A ad B such ha he are o all zero ad AF, )BG, ) a.e. o S β). Lemma.6 Suose ha >, +,<, ad weigh fucios are defied b ω,, ): ω,, ): ma, } ma, } d), d),, +),, +). The ω,, )η) ), ω,, )η) ), where η) + Γ + ). 3) Proof Se u,hed) du). Noe he followig echage iegral, le u,ad u s,wehavedu) d) ad u du) ds).thewehave ω,, ) ma, } d) ) ma, u } ma, } u du) ) u du) + u 3 du)

4 Liuad Su Joural of Ieualiies ad Alicaios 7) 7:83 Page 4 of 8 ) u du) + ) ) u du) ) + ) ds) + Γ + ) ) η) ). d) Similarl, we obai ω,, )η) ). Lemma.7 Suose ha >, +,<, ad ε >is small eough, le us defie he real fucios as follows: f ) he we have,, ), ε,, ), g),, ), ε, J ε I ) f ) ) I ) g ) ) ε h ε I I ma, } ε >, ),, 4) η) ) o) ε + ). 5) Proof Noe he roeries of local fracal sace 5, 7: a + b) a + b ad a) a, we easil obai J ε I ) f ) ) I ) g ) ) ε I +ε) ) I +ε) ) ε. Le ε u, + ε v,adfrom ε ) ε d) du), ε ) 3 + ε d) dv),wewrie I ) I ε ) ) ) d) Γ ε d) +) ) + ε ) ε d) dv) ) Γ +) ε ) Γ +) ε ) Γ +) ε ) Γ +). 3 + ε d) dv)

5 Liuad Su Joural of Ieualiies ad Alicaios 7) 7:83 Page 5 of 8 Furher, le, ad b Lemma.6,wehave h ε I I ε ma, } I ε ) I ε I +ε) ) I ma, } ε ma, } ) ε ) ε I +ε) ) I ε ) + ma, I ε ) ε ) } ma, I ε } ma, } I +ε) ) I ε ) + I 3 ε ) I ε ma, } I +ε) ) I ε ) + I + ε ) I ε ) ε > ε ) Γ +) + + ε ) Γ +) I η) + o ) I η) + o ) I η) + o ) +ε) ) I ) I ε ε ) Γ +) ε ) ε ε ) ε +ε) ) I ) ε ε ) ε η) ) o) ε + ). 3 Mai resuls ad alicaios Iroducig he mark: I I F, ) Γ +) F, )d) d) see 7). Theorem 3. If >, +,<, f, g > ) C, ), ad < I ) f )) <,< I ) g )) <, he I I < η) I ma, } ) f ) )} I ) g ) ), 6) where he cosa facor η) defied i 3) is he bes ossible. Proof B Hölder s weighed ieuali o he fracal se ad Lemma.6,weobai I I ma, } Γ +) Γ +) Γ +) ma, } d) d) ma, } ma, } d) d) d) d)

6 Liuad Su Joural of Ieualiies ad Alicaios 7) 7:83 Page 6 of 8 Γ +) ma, } d) d) ω,, )f )d) Γ +) ω,, )g )d) Γ +) η) ) f )d) ) g )d) η) I ) f ) ) I ) g ) ). 7) Now assume ha euali holds i 7), here eis wo ozero cosas A ad B such ha A f ) B a.e. i,), ), he here is cosa C suchha A ) f ) B ) g ) C a.e. i,), ). Assumig ha A,wehave ) f ) C A a.e. i,). Because Γ +) C C AΓ +) A d) is diffuse, which coradics he fac ha < I ) f )) <,husieuali7)issric. If he cosa facor η)i6) is o oimal, he here eiss osiive K < η)such ha ieuali 6) is sill valid if we relace η) bk. Heceb4) ad5), we have η) o)) < K. Leig ε +,wegek η), which coradics he fac ha K < η), herefore η) i 6)ishebesossible. Theorem 3. Uder he codiios of Theorem 3., we have I ) ) I f ) } < η ) ma, I } ) f ) ), 8) where he cosa facor η ) is he bes ossible, ad ieuali 8) is euivale o ieuali 6). Proof Defie f ) : mi, f )}. Sice< I ) f )) <, hereeiss N such ha < I ) f ) )< ). Seig g ): ) ) I f ) ma, } < <, ), whe,b6), we fid < I ) g )) ) g )g )d) f ) I ma, } Γ +) < η) I ) f ) ) ) ) I f ) g ) ma, } d) d) ) I f ) d) ma, } ) g )). 9)

7 Liuad Su Joural of Ieualiies ad Alicaios 7) 7:83 Page 7 of 8 Moreover, b 9) wehave < I ) g )) I ) ) I f ) } ma, } < η ) I ) f ) ) <. ) For, i follows ha < I ) g )) <,ad< I ) f )) <,b 6), boh 9) ad) sill kee he form of sric ieualiies. Hece we have ieuali 8). O he oher had, b Hölder s ieuali o he fracal se ad 8), we fid I I Γ +) ma, } ) ) ) ) ma, } d) d) ) g )d) f ) ) ma, } d) ) g) d) f ) ma, } d) < η) I ) f ) ) I ) g ) ). d) The above ieuali is 6), herefore ieuali 8)is euivale o ieuali 6). If he cosa facor i 8) is o oimal, he b 8) we ca ge a coradicio ha he cosa facor i 6) is o he oimal oo. Thus he cosa facor η ) i8) is he bes ossible. 4 Simle alicaios Selecig values i 6) ad8), ad usig mahemaics sofware o calculae, some Hilber-e fracioal iegral ieualiies ad heir euivale forms are obaied. Eamle Leig,,ocalculaeformula3), we ge η) 4, he we obai ieualiies )ad). Eamle Leig.5,,ocalculaeformula3), we ge η.5) 4 π.suose ha f, g > ) C.5, ), < I.5f )) <,< I.5g )) <, he we have he followig euivalece ieualiies: I.5 I.5 I.5 I.5 ma, } f ) ma, } where he cosa facors 4 π, 3 are he bes values. π <4 I.5 f ) ) I.5 g ) ), π ) < 3 π I.5 f ) ), )

8 Liuad Su Joural of Ieualiies ad Alicaios 7) 7:83 Page 8 of 8 Eamle 3 Leig.,,ocalculaeformula3), we fid η.) Γ π ) π csc π ) Suosehaf, g > ) C., ), < I.f )) <, < I.g )) <, he we have he followig euivalece ieualiies: I. I. I. I. ma.,. } f ) ma., } < η.) I. f ) )} I. g ) ), 3) < η.) I. f ) ), 4) where he cosa facors η.), η.) are he bes values. 5 Coclusios I he aer, based o he local fracioal calculus heor, a Hilber-e fracioal iegral ieuali ad is euivale form are eaivel researched. The resuls show ha some mehods ad skills of he Hilber-e iegral ieuali ca be raslaed o he research of Hilber-e fracioal iegral ieuali, which rovides a ew direcio ad field o research Hard-Hilber s iegral ieualiies. Comeig ieress The auhors declare ha he have o comeig ieress. Auhors coribuios The wo auhors coribued euall o his work. The all read ad aroved he fial versio of he mauscri. Ackowledgemes The auhors are eremel graeful o he reviewers for a criical readig of he mauscri ad makig valuable commes ad suggesios leadig o a overall imroveme of he aer. This work was suored b he Naioal Naural Sciece Foudaios of Chia No. 78) ad Scieific Suor Projec of Hua Provice Educaio Dearme of Chia No. C86). Publisher s Noe Sriger Naure remais eural wih regard o jurisdicioal claims i ublished mas ad isiuioal affiliaios. Received: December 6 Acceed: 7 Aril 7 Refereces. Hard, GH, Lilewood, JE, Póla, G: Ieualiies. Cambridge Uiversi Press, Cambridge 95). Miriović, DS, Pečarić, JE, Fik, AM: Ieualiies Ivolvig Fucios ad Their Iegrals ad Derivaives. Kluwer Academic, Boso 99) 3. Yag, BC: O he eeded Hilber s iegral ieuali. Chi. J. Eg. Mah. 54), -8 4) i Chiese) 4. Liu, Q, Zhag, XJ: A Hard-Hilber s e ieuali wih wo arameers ad he bes cosa facor. J. Na. Sci. Hua Norm. Uiv. 93), 5-8 6) i Chiese) 5. Same, E, Mehme, ZS: Geeralized Pomeiu e ieualiies for local fracioal iegrals ad is alicaios. Al. Mah. Comu. 74,8-9 6) 6. Mo, H, Sui, X, Yu, D: Geeralized cove fucios o fracal ses ad wo relaed ieualiies. Absr. Al. Aal. 4, Aricle ID ) 7. Yag, XJ: Advaced Local Fracioal Calculus ad Is Alicaios. World Sciece Publisher, New York ) 8. Huag, ZS, Guo, DR: A Iroducio o Secial Fucio. Beijig Press, Beijig ) i Chiese) 9. Yag, J, Baleau, D, Yag, XJ: Aalsis of fracal wave euaios b local fracioal Fourier series mehod. Adv. Mah. Phs. 3, Aricle ID ). Che, GS: Geeralizaios of Hölder s ad some relaed iegral ieualiies o fracal sace. arxiv:9.5567v mah.gm )