Univalence of Integral Operators Involving Mittag-Leffler Functions

Transcription

1 Appl. Mah. If. Sci. 11, No. 3, ) 635 Applied Mahemaics & Iformaio Scieces A Ieraioal Joural hp://dx.doi.org/ /amis/1131 Uivalece of Iegral Operaors Ivolvig Miag-Leffler Fucios H. M. Srivasava 1,2,, B. A. Frasi 3,4 Virgil Pescar 5 1 Deparme of Mahemaics Saisics, Uiversiy of Vicoria, Vicoria, Briish Columbia V8W 3R4, Caada. 2 Deparme of Medical Research, Chia Medical Uiversiy Hospial, Chia Medical Uiversiy, Taichug 442, Taiwa, Republic of Chia. 3 Deparme of Mahemaics, Faculy of Sciece, Islamic Uiversiy of Madiah, Al Madiah Al Muawwarah, Mediah 42351, Kigdom of Saudi Arabia. 4 Deparme of Mahemaics, Faculy of Sciece, Al al-bay Uiversiy, P. O. Box 1395, Mafraq, Jorda. 5 Deparme of Mahemaics Compuer Sciece, Faculy of Mahemaics Iformaics, Trasilvaia Uiversiy of Braşov, Iuliu Maiu 5, R-591 Braşov, Romaia. Received: 2 Nov. 216, Revised: 18 Ja. 217, Acceped: 16 Mar. 217 Published olie: 1 May 217 Absrac: I his paper, we firs iroduce a ew family of iegral operaors ivolvig Miag-Leffler fucios. We he fid sufficie codiios for uivalece of hese iegral operaors prese he geeralized versios of he well-kow Ahlfors Becker s uivalece crieria. Fially, we derive several iequaliies for he ormalized Miag-Leffler fucios. Keywords: Aalyic fucios; Uivale fucios; Iegral operaors; Miag-Leffler fucios; Uivale crieria; Fracioal differeial fracioal iegro-differeial equaios. 1 Iroducio, Defiiios Prelimiaries The familiar Miag-Leffler fucio E α z) iroduced by Miag-Leffler 2] is geeralizaio E α,β z) iroduced by Wima see 29] 3]) are defied by E α z) := = z Γα+1) =: E α,1z) equaio, rom walks, Lévy flighs, super-diffusive raspor problems i he sudy of complex sysems. Several properies of he Miag-Leffler fucios E α z) E α,β z), ogeher wih heir geeralizaios, ca be foud i a umber of rece works 1], 13] o 19] 25] o 27]. Le A deoe he class of he ormalized fucios of he form: fz)=z+ =2 a z, 2) E α,β z) := = z Γα+β) 1) which are aalyic i he ope ui disk D={z : z C z <1}. z,α,β C; Rα)> ). The above-defied fucios E α z) E α,β z), as well as heir various furher geeralizaios, arise aurally i he soluio of fracioal differeial equaios fracioal iegro-differeial equaios which are associaed wih for example) he kieic We also deoe by S he subclass of he ormalized aalyic fucio class A which are uivale or schlich) i D. Sice he Miag-Leffler fucio E α,β i 1) does o belog o he class A, we choose o cosider here he followig ormalizaio of he Miag-Leffler fucio: Correspodig auhor harimsri@mah.uvic.ca

2 636 H. M. Srivasava e al.: Uivalece of iegral operaors... E α,β z) := Γβ)zE α,β z) = z+ =2 Γβ) Γ α 1)+β ) z 3) z,α,β C; Rα)>; β, 1, 2, ). Whils he defiiio 3) holds rue for complex-valued parameers α β z C, ye for he purpose of his paper) we shall resric our aeio o he case of real-valued parameers α β z D. We observe ha he ormalized Miag-Leffler fucio E α,β i 3) coais such well-kow fucios as is special cases as give below: E 2,1 = zcosh z), E 2,2 = zsih z), E 2,3 = 2 cosh z) 1 ] E 2,4 = 6sih z) z] z. Geomeric properies icludig sarlikeess, covexiy close-o-covexiy for he Miag-Leffler fucio E α,β were recely ivesigaed by Basal Prajapa 2]. Răducau 24], o he oher h, ivesigaed he raio of he ormalized Miag-Leffler fucio E α,β defied by 3) o is sequece of parial sums give by E α,β ) m z)=z+ m =2 α > ; β > ; m N), Γβ) Γ α 1)+β ) z 4) wheren deoes he se of posiive iegers. Recely, may auhors derived he uivalece crieria of several iegral operaors which preserve he ormalized uivale fucio class S see, for example, 3] o 12]). I paricular, Breaz Güey 5] obaied various sufficie codiios for he uivalece of he followig families of iegral operaors: F λ1,,λ,ζz)= ζ ζ 1 ) ] f j ) 1/λ 1/ζ j d 5) F γ z)= γ γ 1 e f)) ] d, 6) where he fucios f 1,, f f belog o he ormalized aalyic fucio class A he parameers λ 1,,λ, ζ γ are complex umbers such ha he iegrals i 5) 6) exis. Here, hroughou i his paper, every may-valued fucio is ake wih he pricipal brach. I his paper, we are maily ieresed i some iegral operaors of he ype 5) 6) which ivolve he ormalized Miag-Leffler fucio E α,β. More precisely, we propose o show ha, by usig some iequaliies for he ormalized Miag-Leffler fucio, he uivalece of some iegral operaors ivolvig Miag-Leffler fucios ca be derived easily via some well-kow uivalece crieria. By appropriaely specializig our resuls, we obai simple sufficie codiios for some iegral operaors which ivolve he hyperbolic sie he hyperbolic cosie fucios. I he proofs of our mai resuls, we eed each of he followig uivalece crieria. Lemma 1. see Pescar 22]) Le ζ c be complex umbers such ha Rζ)> c 1 c 1). If he fucio h A saisfies he followig iequaliy: c z 2ζ + 1 z 2ζ) zh z) ζh z) 1 for all z D, he he fucio F ζ A defied by 1/ζ d] F ζ z) := ζ ζ 1 h ) 7) is i he ormalized uivale fucio class S id. Lemma 2. see Pascu 21]) Le λ C such ha Rλ)>. Also le h A saisfy he followig iequaliy: 1 z 2Rλ) zh z) Rλ) h z) 1 for all z D. The, for all ζ C such ha Rζ) Rλ), he fucio F ζ z) defied by 7) is i he ormalized uivale fucio class S i D. Lemma 3. see Pescar 23]) Le γ C λ R such ha Rγ) 1, λ > 1 2λ γ 3 3. If h A saisfies he he followig iequaliy: zh z) λ for all z D, he he fucio F γ :D C, defied by F γ z)= γ γ 1 e h)) ] d is i he ormalized uivale fucio class S id. I our prese ivesigaio, we also eed he followig resul which is based maily upo he rece work 2].

3 Appl. Mah. If. Sci. 11, No. 3, ) / Lemma 4. Le α 1 β 1. The he followig iequaliies hold rue for all z D: ze α,β z) E α,β z) 1 2β + 1 β 2 8) β 1 ze α,β z) β 2 + 3β+ 2 β 2. 9) Proof. I is kow from 2] ha, for all α 1, β 1 z D, we have E α,β z) E α,βz) z 2β + 1 β 2 E α,β z) z β 2 β 1 β 2. Combiig hese las iequaliies, we immediaely fid ha 8) holds rue. Nex, i is kow from 24] ha, for all α 1, β 1 z D, we have E α,β z) β 2 + 3β+ 2 β 2, which obviously implies he iequaliy 9). 2 Uivalece of Iegral Operaors Ivolvig Miag-Leffler Fucios Our firs mai resul Theorem 2 below) is a applicaio of Lemma 1 coais sufficie codiios for a iegral operaor of he ype 5) whe he fucios f j are he ormalized Miag-Leffler fucios wih various parameers. Theorem 1. Le α 1,,α 1 β 1,,β ) cosider he ormalized Miag-Leffler fucios defied by Le wih z)= Γβ j )z z). 1) β = mi{β 1,,β } ζ C Rζ)> c C c 1). Suppose also ha λ 1,,λ are ozero complex umbers ha all hese he aforemeioed umbers saisfy he followig iequaliy: c + 2β + 1 β 2 β 1 The he fuciof α,λ j,ζ defied by F α,λ j,ζz) := ζ ζ 1 ζλ j 1. ) 1/λ ) j d is i he ormalized uivale fucio S id. Proof. Defie he fuciof α,λ j by F α,λ j z)= ) 1/λ ) j d. 1/ζ Sice, for all j {1,,}, we have A, ha is, )=E α ) 1=, i follows readily haf α,λ j A, ha is, F α,λ j )=F α,λ j ) 1=. O he oher h, i is easy o see ha F α,λ j z)= zf α,λ j z) F α,λ j z) = 1 λ j ) ) 1/λ j ze ) α z) z) 1. Now, by usig he iequaliy 8) for each β j j {1,,}), we obai zf α,λ j z) F z) = 1 ze α z) α,λ j λ j z) 1 1 2β j + 1 λ j β j 2 β j 1 2β+ 1 λ j β 2 β 1 for all z D β 1,,β ). Here we have used he easily verifiable fac ha he fucio 1+ ) 5 ϕ :, R, 2

4 638 H. M. Srivasava e al.: Uivalece of iegral operaors... defied by ϕx)= 2x+1 x 2 x 1, is decreasig. Therefore, for all j {1,,}, we have 2β j + 1 β 2 j β j 1 2β + 1 β 2 β 1. Thus, by usig he riagle iequaliy he hypohesis of Theorem 1, we obai c z 2ζ +1 z 2ζ ) zf α,λ j z) ζf α,λ j z) c + 2β + 1 β 2 β 1 ζλ j 1, which, i view of Lemma 1, implies ha F α,λ j,ζ S. This evidely complees he proof of Theorem 1. By seig λ 1 = =λ = λ i Theorem 1, we deduce he followig resul. Corollary 1. Le he umbers ζ, c, α 1,,α, β β 1,,β be as i Theorem 1 le λ be a ozero complex umber. Suppose also ha he fucios A j {1,,}) are as i Theorem 1 ha he followig iequaliy: c + 1 2β + 1) 2 λ ζ β 2 β 1 1 is valid. The he fuciof α,λ j,ζ defied by F α,λ j,ζz)= ζ ζ 1 ) 1/λ ) d is i he ormalized uivale fucio class S i D. If we pu = 1 i Theorem 1 or i Corollary 1, we immediaely obai he followig resul. Corollary 2. wih Le α 1, β ) ζ C Rζ)>, c C c 1) λ C λ ). 1/ζ Suppose also ha hese umbers saisfy he followig iequaliy: c + 1 2β + 1 λ ζ β 2 β 1 1. The he fuciof α,β,λ,ζ defied by F α,β,λ,ζ z)= ) ζ ζ 1,β ) 1/λ 1/ζ d] is i he ormalized uivale fucio class S id. The followig illusraive example provides several uivale fucios i D: Example 1. i) If c + 5 λ ζ 1, he he fuciof 2,2,λ,ζ defied by F 2,2,λ,ζ z)= is uivale i D. ii) If ) 1/λ ] 1/ζ sih ) ζ ζ 1 d c λ ζ 1, he he fuciof 2,3,λ,ζ defied by F 2,3,λ,ζ z)= is uivale i D. iii) If ) 1/λ 1/ζ 2cosh ) 1] ζ d] ζ 1 c λ ζ 1, he he fuciof 2,4,λ,ζ defied by F 2,4,λ,ζ z)= is uivale i D. ) 1/λ 1/ζ 6sih ) ] ζ d] ζ 1 3/2 The followig resul coais aoher se of sufficie codiios for iegrals of he ype 5) o be uivale i he ui disk D. The key ools i he proof of Theorem 2 are Lemma 2 he iequaliy 8).

5 Appl. Mah. If. Sci. 11, No. 3, ) / Theorem 2. Le α 1,,α 1 β 1,,β ). By choosig = 1 i Theorem 2, we obai he followig resul. Corollary 3. Le Cosider he ormalized Miag-Leffler fucios E αi,β i defied by 1). Le β = mi{β 1,,β } λ C Rλ)> ) suppose ha hese umbers saisfy he followig iequaliy: λ 1 β 2 ) β 1 Rλ). 2β + 1 The he fuciof αi,β i,λ, defied by F αi,β i,λ,z)= λ + 1) i,β i ) ) λ d ] 1/λ+1) is i he ormalized uivale fucio class S i D. Proof. Le us cosider he auxiliary fucio F αi,β i,λ defied by F αi,β i,λz)= We he observe ha ha is, ha F αi,β i,λ A, ) i,β i ) λ d. F αi,β i,λ)=f α i,β i,λ) 1=. O he oher h, by usig 8) he fac ha, for all i {1,,}, 2β i + 1 βi 2 β i 1 2β + 1 β 2 β 1, we fid, for all z D, ha 1 z 2Rλ) zf α i,β i,λ z) Rλ) F α i,β i,λ z) λ ze α Rλ) i,β i z) E αi,β i z) 1 λ Rλ) 2β + 1 β 2 β 1 1. Now, sice Rλ + 1) > Rλ) sice he fucio F αi,β i,λ, ca be rewrie i he form: F αi,β i,λ,z)= λ + 1) λ by applyig Lemma 2, we have F αi,β i,λ,z) S, which complees he proof of Theorem 2. ) i,β i ) λ ] 1/λ+1) d, F 2,4,λ z)= λ + 1) α 1 β ) cosider he ormalized Miag-Leffler fucios E αi,β i defied by 1). Also le he parameer λ C be so cosraied ha Rλ)> λ The he fuciof α,β,λ defied by F α,β,λ z)= λ + 1) is uivale id. β 2 ) β 1 Rλ). 2β + 1,β ) ) λ d ] 1/λ+1) I paricular, we are led o he followig uivale fucios id. Example 2. i) If λ 1 5 Rλ), he he fuciof 2,2,λ defied by F 2,2,λ z)= λ + 1) is uivale id. ii) If sih ) ] λ d ] 1/λ+1) λ 5 7 Rλ), he he fuciof 2,3,λ defied by F 2,3,λ z)= λ + 1) is uivale id. iii) If λ 11 9 Rλ), he he fuciof 2,4,λ defied by is uivale id. 2cosh ) 1] ) λ d ] 1/λ+1) ) λ ] 1/λ+1) 6sih ) ] d

6 64 H. M. Srivasava e al.: Uivalece of iegral operaors... Fially, by applyig Lemma 3 he iequaliy 9), we ca easily prove Theorem 3 below. Theorem 3. Le γ C, α 1 β 1. Cosider he ormalized Miag-Leffler fucioe α,β defied by 1). If Rγ) 1 γ he he fuciof α,β,γ defied by 3 3β 2 2β 2 + 6β+ 4, F α,β,γ z)= γ γ 1 e E α,β )) ] d is uivale id. Example 3. i) If he parameer γ C is such ha 3 Rγ) 1 γ 4, he he fuciof 2,1,γ defied by F 2,1,γ z)= γ γ 1 ) ] e cosh ) d is uivale id. ii) If he parameer γ C is such ha Rγ) 1 γ 3 2, he he fuciof 2,2,γ defied by F 2,2,γ z)= γ γ 1 e sih ) ] ) d is uivale id. iii) If he parameer γ C is such ha Rγ) 1 γ , he he fuciof 2,3,γ defied by F 2,3,γ z)= γ γ 1 ) ] e 2cosh ) 1] d is uivale id. iv) If he parameer γ C is such ha Rγ) 1 γ 4 3 5, he he fuciof 2,4,γ defied by F 2,4,γ z)= γ γ 1 is uivale id. e 6sih ) ] ) γ ] 1/γ d Refereces 1] A. A. Aiya, Some applicaios of Miag-Leffler fucio i he ui disk, Filoma 3 216), ] D. Basal J. K. Prajapa, Cerai geomeric properies of he Miag-Leffler fucios, Complex Var. Ellipic Equ ), ] Á. Baricz B. A. Frasi, Uivalece of iegral operaors ivolvig Bessel fucios, Appl. Mah. Le ), ] D. Breaz, N. Breaz H. M. Srivasava, A exesio of he uivale codiio for a family of iegral operaors, Appl. Mah. Le ) ] D. Breaz H. Ö. Güey, O he uivalece crierio of a geeral iegral operaor, J. Iequal. Appl ), Aricle ID 72715, ] S. Bulu, Uivalece preservig iegral operaors defied by geeralized Al-Oboudi differeial operaors, A. S. Uiv. Ovidius Cosaa 17 29), ] E. Deiz, O he uivalece of wo geeral iegral operaors, Filoma ), ] E. Deiz, Uivalece crieria for a geeral iegral operaor, Filoma ), ] E. Deiz, H. Orha H. M. Srivasava, Some sufficie codiios for uivalece of cerai families of iegral operaors ivolvig geeralized Bessel fucios, Taiwaese J. Mah ), ] B. A. Frasi, Some sufficie codiios for cerai iegral operaors, J. Mah. Iequal. 2 28), ] B. A. Frasi, Sufficie codiios for iegral operaor defied by Bessel fucios, J. Mah. Iequal. 4 21), ] B. A. Frasi, Uivalece crieria for geeral iegral operaor, Mah. Commu ), ] M. Garg, P. Maohar S. L. Kalla, A Miag-Leffler-ype fucio of wo variables, Iegral Trasforms Spec. Fuc ), ] R. Goreflo, F. Maiardi H. M. Srivasava, Special fucios i fracioal relaxaio-oscillaio fracioal diffusio-wave pheomea, i Proceedigs of he Eighh Ieraioal Colloquium o Differeial Equaios Plovdiv, Bulgaria; Augus 18 23, 1997) D. Baiov, Edior), pp , VSP Publishers, Urech Tokyo, ] A. A. Kilbas, H. M. Srivasava J. J. Trujillo, Theory Applicaios of Fracioal Differeial Equaios, Norh- Holl Mahemaical Sudies, Vol. 24, Elsevier Norh- Holl) Sciece Publishers, Amserdam, Lodo New York, ] V. Kiryakova, Geeralized Fracioal Calculus Applicaios, Pima Research Noes i Mahemaics Series, Vol. 31, Logma Scieific Techical, Harlow, 1994 copublished i he Uied Saes wih Joh Wiley Sos, New York, 1994). 17] V. Kiryakova, Muliple muliidex) Miag-Leffler fucios relaios o geeralized fracioal calculus, i Higher Trascedeal Fucios Their Applicaios H. M. Srivasava, Edior), J. Compu. Appl. Mah ), ] V. Kiryakova, The muli-idex Miag-Leffler fucios as a impora class of special fucios of fracioal calculus, Compu. Mah. Appl ),

7 Appl. Mah. If. Sci. 11, No. 3, ) / ] F. Maiardi R. Goreflo, O Miag-Leffler-ype fucios i fracioal evoluio processes, i Higher Trascedeal Fucios Their Applicaios H. M. Srivasava, Edior), J. Compu. Appl. Mah ), ] G. M. Miag-Leffler, Sur la ouvelle focio Ex), C. R. Acad. Sci. Paris ), ] N. Pascu, A improveme of Becker s uivalece crierio, Proceedigs of he Commemoraive Sessio Simio Soilow Brasov, 1987), ] V. Pescar, A ew geeralizaio of Ahlfors Becker s crierio of uivalece, Bull. Malaysia Mah. Soc. Ser. 2) ), ] V. Pescar, Uivalece of cerai iegral operaors, Aca Uiv. Apulesis Mah. Iform ), ] D. Răducau, O parial sums of ormalized Miag-Leffler fucios, Prepri 216; arxiv: mah.cv]. 25] H. M. Srivasava, O a exesio of he Miag-Leffler fucio, Yokohama Mah. J ), ] H. M. Srivasava, Some families of Miag-Leffler ype fucios associaed operaors of fracioal calculus, TWMS J. Pure Appl. Mah ), ] H. M. Srivasava Ž. Tomovski, Fracioal calculus wih a iegral operaor coaiig a geeralized Miag-Leffler fucio i he kerel, Appl. Mah. Compu ), ] L. F. Saciu, D. Breaz H. M. Srivasava, Some crieria for uivalece of a cerai iegral operaor, Novi Sad J. Mah. 43 2) 213), ] A. Wima, Über de Fudameal saz i der Theorie der Fuckioe Ex), Aca Mah ), ] A. Wima, Über die Nullsellu der Fuckioe Ex), Aca Mah ), Basem Aref Frasi has held he posiio of Professor i he Deparme of Mahemaics a Al al- Bay Uiversiy i Jorda sice 213, havig joied he faculy here i 22. He eared his Ph.D. degree i Complex Aalysis Geomeric Fucio Theory) from he Naioal Uiversiy of Malaysia Uiversii Kebagsaa Malaysia) i 22. His research areas iclude Special Classes of Uivale Fucios, Special Fucios Harmoic Fucios. Virgil Pescar is currely a he Trasilvaia Uiversiy of Braşov i Romaia, havig reired here o Ocober 1, 213 from his posiio as Professor i he Deparme of Mahemaics Compuer Sciece Faculy of Mahemaics Iformaics). Bor o 15 Jue 1948, he received he degree of Docor i Mahemaics Ph.D.) i he year 199 from he Uiversiy of Babeş-Bolyai i Cluj Romaia). He has published may papers especially i he area of Uivale Fucios i Geomeric Fucio Theory of Complex Aalysis. H. M. Srivasava For he auhor s biographical oher professioal deails icludig he liss of his mos rece publicaios such as Joural Aricles, Books, Moographs Edied Volumes, Book Chapers, Ecyclopedia Chapers, Papers i Coferece Proceedigs, Forewords o Books Jourals, e ceera), he ieresed reader should look io he followig Web Sie: hp://