Meromorphic Functions Sharing Three Values *
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1 Alied Maheaic doi:1436/a11695 Pulihed Olie Jue 11 (h://wwwscirporg/joural/a) Meroorhic Fucio Sharig Three Value * Arac Chagju Li Liei Wag School o Maheaical Sciece Ocea Uiveriy o Chia Qigdao Chia E-ail: chagjuli791@hoailco wag-83liei@163co Received March 6 11 revied Aril 1 11 acceed Aril I hi aer we rove a reul o he uiuee o eroorhic ucio harig hree value couig uliliciy irove a reul oaied y Xiaoi Li Hogxu Yi Keyword: Uiuee Meroorhic Fucio Sharig Three Value 1 Iroducio Mai Reul Le g e wo o-coa eroorhic ucio i he colex lae I i aued ha he reader i ailiar wih he ard oaio o Nevalia heory uch a Tr r Nr Nr o o which ca e oud i [1] We ue E o deoe ay e o oiive real uer o iie liear eaure o ecearily he ae a each occurrece The oaio Sr deoe ay uaiy aiyig Sr Tr r r E A eroorhic ucio i called a all ucio wih reec o rovided ha Tr Sr A eroorhic ucio i called a exceioal ucio o 1 rovided ha Nr Sr Le a e a colex uer we ay ha g hare he value a CM rovided a g a have he ae zero couig uliliciie (ee []) We ay ha g hare CM rovided ha 1 1 g hare CM Xiaoi Li Hogxu Yi rove he ollowig heore: Theore A ([3]) Le g e wo diic ocoa eroorhic ucio harig hree value 1 CM i here exi a iie colex uer a 1 uch ha a i o a Picard value o he 1 N1 r Tr Sr a * The Projec-oored y SRF or ROCS SEM 1 N1 r Tr Sr a oe o he ollowig cae will hold: 1 1 e 1 e 1 1) g wih e 1 e a1 1 1 a 1 1 a 1 e 1 e 1 ) g 1 1 e 1 e wih a 1 a 1 1 a 1 e 1 e 1 3) g 1 1 e 1 e 1 wih 1 a 1 a 1 1 1a 1 1 e 1 e 1 4) g e 1 1 e 1 wih 1 5) a1 a e 1 e 1 g e 1 1 e 1 wih a 1 a e 1 e 1 6) g e 1 1 e 1 wih 1 Coyrigh 11 SciRe
2 C J LI ET AL a 1 a where i a ocoa eire ucio are oiive ieger uch ha 1 are uually rie 1 i 1) ) 3) are uually rie 1 1 i 4) 5) 6) Xihou Hua Migliag Fag roved he ollowig heore: Theore B ([4]) Le g e wo ocoa eroorhic ucio harig hree value 1 CM i T r N r z S r z 1 i a all ucio o he oe o he ollowig hold: 1) g ) g 1 are exceioal ucio o 3) 11g1 are exceioal ucio o 4) g11 are exceioal ucio o A we all ow ay reul o coa are alo valid or all ucio alhough oe ie hey are ore diicul I hi aer we irove he aove heore oai he ollowig reul Theore 11 Le g e wo diic ocoa eroorhic ucio harig hree value 1 CM i here exi a all ucio z 1 o uch ha z i a exceioal ucio o he N r z T r S r 1 (11) N r 1 z Tr Sr (1) oe o he ollowig cae will hold: 1) 1 1 e 1 e 1 g wih e 1 e ) 1 3) e 1 e 1 g 1 1 e 1 e 1 wih e 1 e 1 g wih ( 1) ( 1 1 ) e e 1 ( ) (1 ) 4) wih 5) wih 6) 1 ( 1 ) ( 1) e 1 e 1 g (1 c) e 1 (1/ (1 c)) e c c c c c 1 1 c c e 1 e 1 g 1 e e c c c c c 1 1 c e 1 e 1 g e e 1 wih c c c c c 1 1 where i a ocoa eire ucio are oiive ieger uch ha 1 are uually rie 1 i 1) ) 3) are uually rie 1 1 i 4) 5) 6) c are coa Soe Lea Lea 1 ([4]) Le g e wo ocoa eroorhic ucio harig hree value 1 CM I g he or ay all ucio z 1 we have N r z N r z g S r 3 3 Lea ([3]) Le e a ocoa erorhic ucio a 1 a a3 e hree diic all ucio o i 1 1 Nr Nr Sr a a 1 1 Coyrigh 11 SciRe
3 7 C J LI ET AL he 1 N r T r S r a3 1 Uig he ae ehod o [3] i Lea we ge he ollowig reul: Lea 3 Le g e wo ocoa eroorhic ucio harig hree value 1 CM I i a racioal liear raoraio o g or ay all ucio z 1 he eiher i a exceioal ucio o or z 1 N r T r S r 1 Lea 4 Le are wo ieger e a ocoa eroorhic ucio z i a all ucio o i 1 he N r 1 S r where N r 1 deoe he reduced cou ig ucio o he coo zero o 1 Proo I z i a zero o 1 he we have z 1 (1) z z Fro (1) () we ge z 1 hu N r 1 Sr ice 1 Lea 5 Le () P a (3) where e i a ocoa eire ucio a are wo all ucio o are oiive ieger uch ha > 1) he ) I 1 N3 r Sr P a a aa aa 1 N r Sr P 3) I are uually rie (4) (5) (6) a a a a aa he 1 N r T r S r P Proo 1) Diereiaig iaig oai (7) (8) P wo ie eli- ro he hree euaio we P hp 1 h P 1 (9) wih Tr h Sr i 1 i ) Suoe P a zero o we have Thu (4) hold 1 N r Sr le z e P wih uliliciy he ro (3) z a z z z (1) z z ( az z azz zz z Fro (1) (11) we ge z z z z z a z a z z z (11) (1) a z z z z z (13) a z a z z z Sice have e i a ocoa eire ucio we T r S r (14) Fro (7) (1) (13) (14) we ge (6) hold 3) Le z e a zero o P wih uliliciy uig roceedig a i ) we ca ge (1) (13) O he oher h ice are uually rie here exi oe oly oe air o ieger uch ha 1 < < < < (15) Fro (1) (13) (15) we ca ge o a a a aa z i a roo Coyrigh 11 SciRe
4 C J LI ET AL 71 which ilie (5) hold ice ha wo diic exceioal ucio Lea 6 ([5]) Le 1 e wo ocoa eroorhic ucio aiyig 1 Nr i Nr Sr i 1 i The eiher 1 N r1 S r or here exi wo ieger > uch ha 1 1 where N r1 1 deoe he reduced couig ucio o 1 relaed o he coo 1-oi 1 T r T r T r S r o T r r r E oly deedig o 1 Lea 7 ([6]) Le e a ocoa ero- orhic ucio R P where Q j P a Q 1 j1 are wo uually rie olyoial i I he coeicie a z j z are all ucio o a he ax T r R T r S r 3 Proo o Theore 11 I i a racioal raoraio o g y Lea 3 we have ha eiher z i a exceioal ucio 1 o or N1 r Tr Sr which coradic wih he auio o Theore 11 Thu i o a racioal raoraio o g By Theore B we have 1 Nr Tr Sr (31) Fro (11) (31) we oai 1 N r S r (3) By Lea 1 we have 1 N r S r 3 j (33) Coiig (3) (33) we ge Noig ha have 1 N r S r (34) g hare 1 CM we 1 e e g g1 (35) where are wo eire ucio Fro (35) we ge e 1 e 1 g e 1 e 1 (36) e e 1 (37) e 1 Aue ha Tr e Sr are Picard value o e ro (36) we have are exceioal ucio o ge 1 N r T r S r 1 Noig 1 e 1 y Lea we which coradic wih he auio o Theore 11 T r e S r Thu Siilarly we have Tr e Sr S r T r e Le z e a ulile zero o u o a zero o Fro (37) we oai z z e ze z 1 (38) z z z e z z z e z Fro (38) (39) we have z z zz zz zzzzz z zzzz z z z z z e e (39) (31) Se 1 e e (311) 1 T r T r T r S r o T r r r E (31) Coyrigh 11 SciRe
5 7 C J LI ET AL Fro (35) (311) (31) we ge S r S r (313) Fro (311) (31) (313) we ge 1 Nr i Nr Sr i 1 (314) i Fro (31) (311) we have z Thu 1 1 z 1 1 N r Nr1 1 Sr (315) N r1 1 deoe he reduced couig ucio o he coo 1-oi o 1 Fro (34) (313) (315) we oai N r1 S r 1 (316) Fro (316) Lea 5 we ow here exi wo ieger > uch ha (317) 1 1 Noig T re S r T re S r Tre Sr ro (311) (317) we have e (318) Le Qz he ro (318) we ge e 1Q 1 Q (319) Noig ha z i a all ucio o we oai ha where oai c c Q z (3) i a coa Fro (319) (3) we c c e 1 1 (31) Wihou lo o geeraliy Fro (31) we ay aue ha are uually rie > Le 1 c where i a eire ucio The ro (36) (31) we oai e 1 1c e 1 e 1 g 1 1c e 1 (3) Noig ha We dicu he ollowig hree cae Cae 1 Suoe ha > > we dicu he ollowig wo ucae Sucae 11 I 1 c 1 Seig 1 le 1 e e Fro (3) (37) we ge Ad 1 1δ e 1 e 1 g δ δ e 1 e 1 1δ δ e e 1 δ e 1 (33) (34) Sice i hi ucae i a coa le (33) aue he or (1) i Theore 11 Fro he roo o Theore A we ow (1) hold wih Sucae 1 I 1 c 1 Seig Fro (3) we ge e 1 1c e 1 e 1 g 1 1c e 1 which aue he or (iv) i Theore 11 We have ro (35) (37) Sice e 1c e 1 1c e 1 (35) (36) 1 c 1 ro (35) (36) Lea 4 Lea 7 we ge T r T r e S r (37) N re 1c e 1 1c e 1 S r where N re 1c e 1 1c e 1 (38) deoe he reduced couig ucio o coo zero o e 1c e 1 1 e c 1 I Coyrigh 11 SciRe
6 C J LI ET AL c c c 1 c 1c y Lea 5 () we ge a coradicio wih (34) Thu Fro (37) (38) Lea 5 (3) we oai (1) hold wih c c c c c 1 1 Cae Suoe ha > > we dicu he ollowig wo ucae Sucae 1 I 1c 1 Seig 1 le 1 c e e Fro (3) (37) we ge δ δ e 1 e 1 g 1δ 1δ e 1 e e δ δ e δ e (39) (33) Sice i hi ucae i a coa le δ (39) aue he or o ) i Theore 11 By he roo o Theore A we ow (1) hold wih Sucae I 1c 1 Seig ro(3) we ge e 1 1c e 1 e 1 g 1 1c e 1 (331) Which aue he or 5) i Theore 11 We have ro (331) (37) e 1c e 1 1c e 1 (33) I he ae aer a Sucae 1 we ow (1) hold wih c c c c c 1 1 Cae 3 Suoe ha > we dicu he ollowig wo ucae Sucae 31 I 1 c 1 Seig 1 le 1 c δ e e Fro (3) (37) we ge δ δ e 1 e 1 g 1δ 1δ e 1 e 1 δ 1δ e e 1 1δ e 1 (333) (334) Sice i hi ucae i a coa le δ (333) aue he or (3) i Theore 11 By he roo o Theore A we ow (1) hold wih Sucae 3 I 1 c 1 1 Seig we have ro (3) e 1 1c e 1 e 1 g 1 1 e 1 c (335) which aue he or (6) i Theore 11 Fro (335) (37) we ge e 1c e 1 1c e 1 (336) I he ae aer a Sucae 1 we ge (1) hold wih c 1 c c c c Theore 11 i hu coleely roved Coyrigh 11 SciRe
7 74 C J LI ET AL 4 Acowledgee The auhor wa o exre heir ha o he aoyou reeree or hi valuale uggeio roeor Qizhi Fag or her uor 5 Reerece [1] W K Haya Meroorhic Fucio Claredo Pre Oxord 1964 [] H X Yi C C Yag Uiuee Theory o Meroorhic Fucio Sciece Pre Beijig 1995 [3] X M Li H X Yi Meroorhic Fucio Sharig Three Value Joural o he Maheaical Sociey o Jaa Vol 56 No [4] X H Hua M L Fag Meroorhoc Fucio Sharig Four Sall Fucio Idia Joural o Pure Alied Maheaic Vol 8 No [5] Q C Zhag Meroorhic Fucio Sharig Three Value Idia Joural o Pure Alied Maheaic Vol 3 No [6] A Z Moho o O he Nevalia Characeriic o Soe Meroorhic Fucio Theory o Fucio Fucioal Aalyi Their Alicaio Vol Coyrigh 11 SciRe
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