Some Properties of Analytic Functions Defined. by a Linear Operator

Transcription

1 It Jourl o Mth Alsis Vol 6 o Some Proerties o Alti Futios Deied b ier Oertor S R Swm Dertmet o omuter Siee d ieeri R V ollee o ieeri Msore Rod Blore Idi miltoswm@redimilom ABSTRAT The im o the reset er is to derive some ew roerties o lti utios deied b the lier oertor usi dieretil suborditios tehiques Mthemtis Subjet lssiitio: 345 Kewords: Alti utios Dieretil suborditio Rusheweh derivtive ier oertor INTRODUTION et A deote the lss o utios o the orm N {3} whih re lti i the oe uit dis {: < } I rtiulr we set A A A A d A A A A whih re well ow lsses o lti utios i The Hdmrd rodut o two utios ive b d ive b b N

2 546 S R Swm is deied s usul b b The Rusheweh derivtive o o order δ - is deied b D δ δ A δ R \ ] or equivletl b 3 D δ δ where A d δ R \ ] I rtiulr i δ l N U {} we id rom or 3 tht l d D l l { } l l! d The oertor ws itrodued b Goel d Sohi3] whih whe redues to the oertor D l itrodued b Rusheweh8] The Pohhmmer smbol λ or the shited toril is ive b λ d λ λ λ λ λ N I terms o λ we ow deie the utio φ b φ R \ { where R } Sitoh6] itrodued lier oertor whih is deied b 4 φ or equivletl b 5 where A R R / For A d δ R \ ] we obti δ 6 δ D whih esil be veriied b omri the deiitios 3 d 5 M roerties o lti utios deied b the lier oertor were studied b Di-Go Yetl ] Frsi ] d Rvihdr etl7] The mi objet o this er is to reset some ew d iteresti roerties o lti utios deied b the lier oertor ssoited with the lss A The bsi tool i rovi our result is the ollowi lemm

3 Proerties o lti utios 547 emm 4] et Ω be set i the omle le Suose tht the utio Ψ : stisies the oditio i Ψ Ω or ll d or ll rel d suh tht 7 - I is lti i d or ' ψ Ω the i MAIN RSUTS Theorem et with d < I A stisies M the where ] M Proo et ] d osider The lerl d is reulr i d Oe esil veri the idetit ' A simle omuttio usi the idetit shows tht

4 548 S R Swm ' ] ] ' ψ where 3 ψ ] ] Usi d 3 we obti } : { } : ' { ψ Ω < w w Now or ll d or ll rel d ostried b the iequlit 7we id rom 3 tht } : { i ψ Hee Ω i ψ Thus b emm i This roves our theorem i Theorem ields orollr For A d omle umber with we hve imlies For ] i Theorem we et orollr 3 et omle umber with d A The ] 8

5 Proerties o lti utios 549 imlies ] mr 4 For {} N l l Theorem o the uthor 9] ollows rom Theorem whih whe l redues to Theorem 6 o Pousm 5] mr 5 I the seil ses whe / d bove results rovide iteresti ew roerties d i the seil se whe bove results would rovide orreted versio o results roved erlier b the uthor d Preethi ] I mer similr to Theorem we rove the ollowi theorem Theorem 6et be omle umber stisi d < et A d < The wheever mr 7 For Theorem 6 rees with Theorem o the uthor ] RFRNS ] DING-GONG YANG N-NG XU AND S OWA A erti lss o lti d multivlet utios deied b mes o lier oertor JIPAM 9 Issue 8 rtile 5 ] BAFRASIN Some roerties o lti utios deied b lier oertor JIPAM 8 Issue 7 Artile 53 3] RM GO AND NSSOHI A ew riterio or -vlet utios Pro Amer MthSo

6 55 S R Swm 4] SSMIR AND PT MOANU Dieretil Suborditios: Theor d Alitios Series o Moorhs d Tet Boos i Pure d Alied Mthemtis N5 Mrel Deer New Yor d Besel 5] S PONNUSAMY O sublss o λ -sirllie utios Mtemti luj ] HSAITOH A lier oertor d its litios o irst order dieretil suborditios MthJo ] V RAVIHANDRAN N SNIVASAGAN AND HMSRIVASTAVA Some iequlities ssoited with lier oertor deied or lss o multivlet utios JIPAM4Issue 43 Artile 7 8] S RUSHWYH New riteri or uivlet utios Pro Amer MthSo ] SRSWAMY Some studies i uivlet utios PhD thesis Krt Uiverit Dhrwd Idi 99 uublished ] SRSWAMY Some iequlities ssoited with lier oertor deied or lss o lti utios JIPAM7 Issue 6 rtile 6 6 ] SRSWAMY d RSPRTHI Some roerties o lti utios deied b lier oertor J Al d omuttios 7 No 9- eived: Setember