On a New Subclass of Multivalant Functions Defined by Al-Oboudi Differential Operator

Transcription

1 Glol Jourl o Pure d Alied Mthetics. ISSN Volue 4 Nuer Reserch Idi Pulictios htt:// O New Suclss o Multivlt Fuctios eied y Al-Ooudi ieretil Oertor r.m.thirucher 2 T.Stli P.G. d Reserch ertet o Mthetics L.N. Govt. College Poeri-624-Mdrs Uiversity Chei Idi. 2 E-il: drrhirucher@gil.co; goldstlee@gil.co Astrct The i oect o this er is ivestigtig ew suclss o orlied ultivlet lytic uctio i the oe uit disk U which is deied y Al- Ooudi ieretil Oertor. We oti the co-eiciet iequlity d extree oits d itegrl es o iequlities or this clss re give. Keywords: Multivlet lytic uctios dieretil oertors geerlied Al-Ooudi dieretil oertor.. INTROUCTION AN EFINITIONS The study o Geoetric uctio theory the ultivlet uctio is ocl re s we c see i recet yers; y ew rticles re writte y eiet uthors i this re. Now oertors o orlied lytic uctios ecoe very oulr ely or ieretil d Itegrl. My rticles discuss o oertors d ew geerlitio o vrious uthors. Perhs Ruscheweyh [] ws ledig the wy i the dieretil oertor who itroduced o 975. It ollowed y Slge [2] i 983 givig other versio o dieretil d itegrl oertor. My roerties hve ee discussed d studied y reserchers or this two oertors. I 24 Al-Ooudi [3] geerlied Slge oertor ollowed y S.Suer Eker d S.Ow d S.Suer d Bill Sekr [4] [5]. I this study we use these oertors to id other tye o ieretil Oertors d oti co-eiciet iequlities d extree oits d itegrl es o iequlities.

2 734 r. M.Thirucher T.Stli eiitio. A deote the clss o uctios orlied y. 2 Which re lytic i the oe uit disc U {C : }. For A Al-Ooudi [3] itroduces the ollowig oertor..2.3 N N N..5 I the we get Slge [2] dieretil oertor. eiitio.2 A deote the clss o uctio o o - vlt lytic uctio.6 Which re lytic i the oe uit disc U {C : }. For A Al-Ooudi [3] itroduces the ollowig oertor..7.8 N N N I the we get Slge [2] dieretil oertor...

3 O New Suclss o Multivlt Fuctios eied y Al-Ooudi ieretil Oertor 735 eiitio:.3 N deoted the su clss o A cosistig o uctios which stisy the iequlity Re For soe C N N d ll U.. 2. COEFFICIENT INEQUALITIES Theore 2. A stisies 2 For soe The N. C N N. 2. Where Proo N.For Suose tht 2 is true or soe N C A the deie the uctio F y F = F the It is suices to rove tht F 2.3 F - =

4 736 r. M.Thirucher T.Stli = = F ] 2 [ 2.5 ] 2 [ F F 2 = = 2 2 F F

5 O New Suclss o Multivlt Fuctios eied y Al-Ooudi ieretil Oertor Hece 2 This coletes the theore. 3. EXTREME POINTS I we deie the ew suclss N N which cosists o the uctio 3. Whose Tylor-Mcluri coeiciet stisy the iequlity 2. Now we deterie the extree oits o the suclss N. Theore 3.

6 738 r. M.Thirucher T.Stli d The N i d oly i it c e exressed i the or 3.3 Where d. Proo Suose tht 2 The ro Which is shows tht stisies the coditio 2. N. Coversely Suose tht N. Sice

7 O New Suclss o Multivlt Fuctios eied y Al-Ooudi ieretil Oertor d The we oti This is coletes the roo o theore 3. Corollry 3.2 The extree oits o re uctios N d INTEGRAL MEANS OF INEQUALITIES eiitio 4. two uctios d g re lytic i U we sy tht is suordite to g i there exist uctio w lytic i U stisyig w d w such tht g w It is deoted y g. Le 4.2 g is uivlet i U. The g i d oly i g d U g U.[7] Theore 4.3 I d g re lytic i U with g the d g d i or d re r. Littlewood [6] Theore 4.4 N d suose tht is deied y 2 2

8 74 r. M.Thirucher T.Stli I there exist lytic uctio w give y w 2 the or r re i d re g d re i i 2 2. Proo We ust show tht d d By the hel o Littlewood suorditio theore its suices to shoe tht 2 w 2 Thereore w 2 Hece w. Furtherore usig 2. w 2 2. Hece theore coleted. REFERENCES [] St.Ruscheweyh975 New criteri or Uivlet uctios Proc.Aer.Mth.Soc

9 O New Suclss o Multivlt Fuctios eied y Al-Ooudi ieretil Oertor 74 [2] G.Slge 983 Suclsses O Uivlet Fuctios Lecture Notes I Mths 3Sriger-VerlgBerli [3] F.M.Al-Ooudi O uivlet uctios deied y geerlied Slge oertor Itertiol Jourl O Mthetics Ad Mtheticl Scieces vol.24no [4] S.Suer Eker d S.Ow New lictios o clsses o lytic uctios ivolvig the Slge oertor i roceedigs o the itertiol syosiu o colex uctio theory d lictios.2-34 Trsilvi Uiversity o Pritig House Brsov Roi Seteer 26 [5] S.Suer Eker d Bill Sekr O clss o Multivlt Fuctios deied y Slge Oretor Geerl MtheticsVol.5Nr [6] J.E. Littlewood O iequlities i the theory o uctios Proceedigs o society vol.23 No [7] P.L.ure Uivlet uctios Sriger New York NY USA 983.

10 742 r. M.Thirucher T.Stli