International Journal of Mathematical Archive-4(12), 2013, Available online through ISSN

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1 Internatonal Journal o Mathematcal Archve-(2, 203, 7-52 Avlable onlne throuh ISSN ON VALUE SHARING OF MEROMORPHIC FUNCTIONS Dbyendu Banerjee* and Bswajt Mandal 2 Department o Mathematcs, Vsva -Bharat, Santnketan-73235, West Benal, Inda. 2 Buna Danal Hh School, P.O. Buna, Labpur-73303, West Benal, Inda. (Receved on: 2-0-3;Revsed & Accepted on: ABSTRACT In ths pape we ntroduce a new concept o value sharn called addtve sharn to prove some unqueness theorems or meromorphc unctons. Key words: Meromorphc unctons, Orde Addtve sharn. AMS Subject Classcaton: 32A20.. INTRODUCTION AND DEFINITIONS Let and be two non-constant meromorphc unctons dened n the open complex plane C and let a C { }. We say that and share the value a CM (countn multplctes or IM (norn multplctes provded a and a have same eros CM or IM respectvely and, share CM or IM provded that and share 0 CM or IM. It s assumed that the reader s amlar wth the standard notatons and dentons o Nevanlnna's theory as ound n [5]. In 979, Gundersen [] proved the ollown theorems. Theorem: A [] I and share our values { a } IM and, then outsde a set E o nte lnear measure: ( alm ; ( blm lm 2, (, where a a ; a ; or,2,3,. Theorem: B [] I and share three values IM, then outsde a set E o nte measure, (, (, lm sup 3 and (, (, lm sup 3. Correspondn author: Dbyendu Banerjee* and Bswajt Mandal 2 Department o Mathematcs, Vsva -Bharat, Santnketan-73235, West Benal, Inda. Internatonal Journal o Mathematcal Archve- (2, Dec

2 Dbyendu Banerjee* and Bswajt Mandal 2 / On Value Sharn O Meromorphc Functons/ IJMA- (2, Dec In 989, Brosch [3] mproved Theorem B by provn the ollown result. Theorem: C [3] I and share three values CM, then 3 8 T( o( T T 8 3 o ( + ( ( ( + ( as r ( r E. Recently Banerjee and Dutta [] ntroduced a new dea o value sharn known as relatve sharn whch runs as ollows. Let and be two non-constant meromorphc unctons and a C { }. We say that, share a CM(IM relatvely wth respect to a meromorphc uncton h, provded the unctons F and G share a CM(IM respectvely where F and G. h h Usn ths dea o relatve sharn o values o meromorphc unctons Banerjee and Dutta proved the ollowns. Theorem: D [2] Let and be two meromorphc unctons. I there s a uncton h wth T( h o( T( and T( h o( T( such that F, G share our values { a } IM, then outsde a set E o nte lnear measure, ( alm ; ( blm lm 2, (, where a a ; F a ; G or,2,3,. Theorem: E [2] Let and be two meromorphc unctons. I there s a uncton h wth T( h o( T( and T( h o( T( such that F, G share three values IM, then outsde a set E o nte measure, T( T( r, lm sup 3 and lm sup 3. T( r, T( Theorem: F [] Let and be two non-constant meromorphc unctons. I there s a uncton h wth T( h o( T( and T( h o( T( such that F, G share { a } 3 IM, then ρ ρ where F and G and ρ denotes the order o. h h In ths pape we ntroduce another noton o value sharn called `addtve sharn' and prove parallel results o Banerjee and Dutta {[], [2]} usn the dea o addtve sharn. Frst we ntroduce the ollown denton. Denton: Let and be two non-constant meromorphc unctons and a C { }. We say that, share a CM(IM addtvely wth respect to a meromorphc uncton h, provded that F and G share a CM(IM respectvely where F + h and G + h. Throuhout the paper we assume, etc. are non-constant meromorphc unctons dened n the open complex plane C and S( any quantty satsyn (, ( (, (, S r o r r E. 203, IJMA. All Rhts Reserved 8

3 Dbyendu Banerjee* and Bswajt Mandal 2 / On Value Sharn O Meromorphc Functons/ IJMA- (2, Dec THEOREMS In ths secton we prove the mn results o the paper. Theorem: Let and be two non-constant meromorphc unctons. I there s a uncton h wth T( h o( T( and T( h o( T( such that F, G share three values IM then ρ ρ where F + h and G + h. ( Proo: We have T( h S( o T( Now F h as r, r E (a set o nte lnear measure. +, so T( F T( T( h O( + + [ o(] Tr (,. + ( On the other hand, F h ves T( T( F + T( h + O( ( + o( T( F + o T T F. (2.e., ( ( ( ( Hence rom ( and (2, ( ( o T( T( F +. (3 Consequently, ρ ρ. ( F Applyn smlar aruments we can also prove that Further snce FG, share three values IM, by Theorem B So, ρ ρ. (5 203, IJMA. All Rhts Reserved 9 G ( + o( T( r, G T( r, F 3 ( + o( T( r, G. 3 ρ ρ. (6 F G Combnn (, (5 and (6, we et the result. Example: Let ( e e, ( 3 3e and h ( e. Then F( e and G( 3 2e share, 2, CM. Here T( h o( T( and T( h o( T( + and h( e. Then F( e CM. Here T( h o T( and T( h o T( ρ ρ Example: 2 Let (, ( e e 0,,, Theorem: 2 Let ( ( ( and. but ρ ρ. and G ( e share and be two non-constant meromorphc unctons. I there s a uncton h wth (, o T( and T( h o T( such that F, G share our values { a } T r h E o nte lnear measure, ( alm ; ( blm lm 2, where a a ; F a ; G ( or,2,3, and F + h and G + h. Proo: By Second Fundamental theorem, as r outsde a set o nte lnear measure, ( 3+ o( T( rf, ra, + rf,. IM, then outsde a set

4 Dbyendu Banerjee* and Bswajt Mandal 2 / On Value Sharn O Meromorphc Functons/ IJMA- (2, Dec Usn (3 and rf, T( rf,, we et at once ( 2 o( T( ra, +. (7 2 o T( + o( ra, Smlarly or, o N ra. (8 2 (, + ( (, Thereore, r, a r,0; F G So outsde a set E o nte measure, N F G T r (, F G (, (, ( o( T( r T( r T r F + T r G + O [+ ](, +,, usn (3 (9 ( o( ra, +, usn (7 and (8. (, + T( lm N ra (,. Let there s a sequence rn such that T( rn, c < N r, a 2 ( n and (, T r n (, N r a n c where c s a constant. Then r n, a c < 2, T( rn, whch contradcts (8. lm lm Hence 2. Ths proves( b. From (9, we have.e., ra, [+ o( ]( T( + T( ( + o( ra,. e.,lm r ( (, ( 203, IJMA. All Rhts Reserved 50 r (,, [ + o(][ + ] [ + o(],,, a + lm lm a

5 Dbyendu Banerjee* and Bswajt Mandal 2 / On Value Sharn O Meromorphc Functons/ IJMA- (2, Dec e.,.e., (, (, 2 + lm 2 (, (, lm. Ths proves( a. Ths completes the proo o the Theorem 2. Example: 3 Let ( e, ( e and h( and T( h o T(. Also ~. (. Then, Example: Let (, ( e e+ and ( T( h o T(. Also ~/~. ( FG share 0,-,,. An T( h o( T( h e. Then, FG share 0, -,,. An Theorem: 3 Let and be two non-constant meromorphc unctons. I there s a uncton h wth T( h o( T( and T( h o( T( such that F, G share three values IM, then outsde a set E o nte measure, (, (, lm sup 3 and (, (, lm sup 3, where F + h and G + h. Proo: Snce F,G share three values IM, so rom Theorem B, outsde a set E o nte measure, T( rf, T( rg, lm sup 3 and lm sup 3. T rg, T rf, (.e., F 3[ + o(] G ( < and G < 3[ + o(] F. Now usn (3, T( < 3[ + o(] T( T(.e., 3 o( T( r, < + T( and hence lm sup 3. T( r, Smlarly lmsup 3. Ths proves the Theorem e 3e + 3 e 3, and h ( 3 3e 3e. Then, but ~ 3T (. Example: 5 Let ( ( e CM and IM. An T( h o T( REFERENCES ( FG share three values 0,. Banerjee, D. and Dutta, R. K., Relatve sharn and order o meromorphc unctons, J. Indan Acad. Math., 29(2 (2007, pp , IJMA. All Rhts Reserved 5

6 Dbyendu Banerjee* and Bswajt Mandal 2 / On Value Sharn O Meromorphc Functons/ IJMA- (2, Dec Banerjee, D. and Dutta, R. K., On relatve sharn o Meromorphc unctons, accepted n The Mathematcs Educaton. 3. Brosch, G., Endeutketssäte ür meromorphc unktonen, Thess, Techncal Unversty o Aachen, Gundersen, G. G., Meromorphc unctons that share three or our values, J. London Math. Soc. 20(2 (979, pp Hayman, W. K., Meromorphc Functons, The Clarendon Press, Oxord, 96. Source o support: Nl, Conlct o nterest: None Declared 203, IJMA. All Rhts Reserved 52