A Subclass of Harmonic Functions Associated with Wright s Hypergeometric Functions
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1 lied Meis doi:0.436/.00.0 Pulised Online July 00 ://www. SiRP.or/journl/ Sulss o roni Funions ssoied wi ri s yereoeri Funions sr Gndrn Muruusundroory Klliyn Vijy Sool o dvned Sienes Vellore Insiue o Tenoloy Universiy Vellore Indi E-il: soory@yoo.o kvvi@yoo.o.in eived My 7 00; revised My 00; eed June 5 00 e inrodue new lss o olex vlued roni unions ssoied wi ri yereoeri unions wi re orienion reservin nd univlen in e oen uni dis. Furer we deine ri enerlied oeror on roni union nd invesie e oeiien ounds disorion ineuliies nd exree oins or is enerlied lss o unions. Keywords: roni Univlen Srlike Funions roni Convex Funions ri yereoeri Funions Rin-Diok Oeror Disorion ounds Exree Poins. Inroduion oninuous union = u + iv is olex-vlued roni union in olex doin G i o u nd v re rel nd roni in G. In ny sily-onneed doin D G we n wrie were nd re nlyi in D. e ll e nlyi r nd e o-nlyi r o. neessry nd suiien ondiion or o e lolly univlen nd orienion reservin in D is ' ' in D see. Denoe y e ily o unions wi re roni univlen nd orienion reservin in e oen uni dis U { : } so is norlied y Tus or we y exress were e nlyi unions nd re in e ors. e noe e ily o orienion reservin norlied roni univlen unions redues o e well known lss S o norlied univlen unions i e o-nlyi r o is idenilly ero is 0. Le e drd rodu or onvoluion o wo ower series nd in S e deined y 3 4 *. For osiive rel reers... nd... N = 3... su 0 U. Te ri s enerlied yereoeri union ; ; ; is deined y ; U.! 5 I = = nd = = we ve Coyri 00 Sis.
2 88 G. MURUGUSUNDROORTY ET L. e relionsi: 0! ; F ; ; 6 ;. N0 N {0}; U is e enerlied yereoeri union see or deils 3 were N denoes e se o ll osiive ineers nd n is e Poer syol nd y usin e enerlied yereoeri union Diok nd Srivsv 3 inrodued e liner oeror. In 4 Diok nd Rin exended e liner oeror y usin ri enerlied yereoeri union. Firs we deine union ; Le : S S ; e liner oeror deined y : ; * e oserve or o e or we ve n : were is deined y 8 9! I or onveniene we wrie ; 0 inrodued y Diok nd Rin 4. I is o ineres o noe i = =... = =... in view o e relionsi 6 e liner oeror 8 inludes e Diok-Srivsv oeror see 3 or ore deils on ese oerors see 3467 nd 8. I is ineresin o noe ri enerlied yereoeri union onins Diok-Srivsv oeror s is seil ses urer oer liner oerors e olov oeror e Crlson-Ser oeror 6 e Rusewey derivive oeror 7 e enerlied ernrdi-lier-livinson oeror e rionl derivive oeror 8 nd so on. For exle i = nd = wi en ; D * is lled Rusewey derivive o order δ δ >. Fro 8 now we deine ri enerlied yereoeri roni union o e or s nd we ll is s ri enerlied oeror on roni union. Moived y e erlier works o 59-3 on e suje o roni unions we inrodue ere new sulss S o. For 0 le S denoe e suily o srlike roni unions o e or su r euivlenly ' ' 3 were is iven y nd U. e lso le V S V were V e lss o roni unions wi vryin ruens inrodued y Jniri nd Silvern 0 onsisin o unions o e or in or wi ere exiss rel nuer φ su od 0 4 were r nd r. In is er we oin suiien oeiien ondiion or unions iven y o e in e lss S. I is sown is oeiien ondiion is neessry lso or unions elonin o e lss Coyri 00 Sis.
3 G. MURUGUSUNDROORTY ET L. Coyri 00 Sis. 89 V. Furer disorion resuls nd exree oins or unions in V re lso oined.. Te Clss S α γ e ein derivin suiien oeiien ondiion or e unions elonin o e lss S. Teore. Le e iven y. I 5 0 Ten S. Proo. e irs sow i e ineuliy 5 olds or e oeiiens o en e reuired ondiion 3 is sisied. Usin nd 3 we n wrie ' ' were ' ' nd In view o e sile sserion w i nd only i w w i is suiien o sow 0. 6 Susiuin or nd e rorie exressions in 6 we e 0. y virue o e ineuliy 5. Tis ilies S. Now we oin e neessry nd suiien ondiion or union e iven wi ondiion 4. Teore. Le e iven y nd or 0 en V i nd only i 7 Proo. Sine S V we only need o rove e neessry r o e eore. ssue V en y virure o o 3 we oin 0. ' ' Te ove ineuliy is euivlen o =. 0
4 90 G. MURUGUSUNDROORTY ET L. Tis ondiion us old or ll vlues o su = r <. Uon oosin φ ordin o 4 we us ve 8. I 7 does no old en e nueror in 8 is neive or r suiienly lose o. Tereore ere exiss oin o ro in 0 or wi e uoien in 8 is neive. Tis onrdis our ssuion V. e us onlude i is o neessry nd suiien e oeiien ound ineuliy 7 olds rue wen V. Tis olees e roo o Teore. I we u in 4 en Teore ives e k ollowin orollry. Corollry. neessry nd suiien ondiion or sisyin 7 o e srlike is r / k nd r / k k Disorion ounds nd Exree Poins In is seion we oin e disorion ounds or e unions V led o overin resul or e ily V. nd Teore 3. I V en r r r r. Proo. e will only rove e ri-nd ineuliy o e ove eore. Te ruens or e le-nd ineuliy re siilr nd so we oi i. Le V kin e solue vlue o we oin r r Tis ilies r r. r r r r r r wi eslis e desired ineuliy. s onseuenes o e ove eore nd orollry we se e ollowin orollry. Corollry. Le nd o e or e so V. Ten w: w U. For o ily e xiu or iniu o e rel r o ny oninuous liner unionl ours one o e exree oins o e losed onvex ull. Unlike ny oer lsses reried y neessry nd suiien oeiien ondiions e ily V is no onvex ily. Nevereless we y sill ly e oeiien reriion o e V o deerine e exree oins. Teore 4. Te losed onvex ull o V denoed y lo V is :. r r r 0. 8 Coyri 00 Sis.
5 G. MURUGUSUNDROORTY ET L. 9 y sein nd en or ixed e exree oins or lov re x x 9 were nd x. Proo. ny union in lo V e exressed s e i i e were e oeiien sisy e ineuliy 5. Se i e e i or = 3 riin X Y = 3... nd X X ; Y ; we e Y nd In riulr uin X Y x. y x y e see exree oins o unions in lov { }. To see is no n exree oin i o x 0 nd y 0 we will sow i n en lso e exressed s onvex liner oinions o unions in lo V. iou loss o enerliy ssue x y. Coose 0 sll enou x x so. Se nd. e y y en see o nd x y y x re in lo V nd { }. Te exrel oeiien ounds sow unions o e or 9 re e exree oins or lo V nd so e roo is olee. 4. Inlusion lion Followin vii nd Zlokiewi 9 see lso Rusewey 4 we reer o e δ-neiorood o e union deined y o e e se o unions F or wi N : F. 0 In our se le us deine e enerlied δ neiorood o o e e se N : F :. Teore 5. Le e iven y. I sisies e ondiions 0 nd 0 en N S. Proo. Le sisy nd F e iven y F wi elons o N. e oin Coyri 00 Sis.
6 9 G. MURUGUSUNDROORTY ET L.. ene or we iner S wi onludes e roo o Teore 5. Now we will exine e losure roeries o e lss V under e enerlied ernrdi-li- er-livinson inerl oeror L wi is deined y L 0 d. Teore 6. Le V. Ten L V. Proo. Fro e reresenion o L i ollows were Tereore L 0 0 d 0 d d ;.. Sine V ereore y Teore L. V Teore 7. For 0 le V nd F V. Ten * F V V. Proo. Le nd V Ten F V * F For * F V we noe nd. Now y Teore we ve. Tereore * F V V nd sine e ove ineuliy ounded y wile. 5. Conludin rks Te vrious resuls resened in is er would rovide ineresin exensions nd enerliions o ose onsidered erlier or siler roni union lsses see 03. Te deils involved in e derivions o su seiliions o e resuls resened in is er re irly sri-orwrd. 6. erenes J. Clunie nd T. Seil-Sll roni Univlen Funions nnles deie Sieniru Fennie Series I Mei Vol E. M. ri Te syoi Exnsion o e Generlied yereoeri Funion Proeedins o e London Meil Soiey Vol J. Diok nd. M. Srivsv Cerin Sulsses o nlyi Funions ssoied wi e Generlied yereoeri Funion Inerl Trnsors nd Seil Funions Vol. 4 No J. Diok nd R. K. Rin Filies o nlyi Funions ssoied wi e ri Generlied yereoeri Funion Deonsrio Mei Vol. 37 No J. M. Jniri roni Funions Srlike in e Uni Dis Journl o Meil nlysis nd liions Vol. 35 No Coyri 00 Sis.
7 G. MURUGUSUNDROORTY ET L C. Crlson nd D.. Ser Srlike nd Presrlike yereoeri Funions SI Journl on Meil nlysis Vol. 5 No S. Rusewey New Crieri or Univlen Funions Proeedins o e erin Meil Soiey Vol. 49 No M. Srivsv nd S. Ow Soe Creriion nd Disorion Teores Involvin Frionl Clulus Generlied yereoeri Funions drd Produs Liner Oerors nd Cerin Sulsses o nlyi Funions Noy Meis Journl Vol Y. vii nd E. Zlokiewi On roni Univlen Mins nnles Universiis Mrie Curie-Skłodowsk Seio Vol J. M. Jniri nd. Silvern roni Univlen Funions wi Vryin ruens Inernionl Journl o lied Meis Vol. 8 No J. M. Jniri G. Muruusundroory nd K. Vijy Srlikeness o Ruewey Tye roni Univlen Funions Journl o e Indin dey o Meis Vol. 6 No G. Muruusundroory Clss o Rusewey-Tye roni Univlen Funions wi Vryin ruens Souwes Journl o Pure nd lied Meis No K. Vijy Sudies on Cerin Sulsses o roni Funions P.D. Tesis Vellore Insiue o Tenoloy Universiy Vellore Seeer S. Rusewey Neioroods o Univlen Funions Proeedins o e erin Meil Soiey Vol. 8 No Coyri 00 Sis.
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