Applied Mathematics, 2011, 2, 1364-1368
doi:10.4236/am.2011.211191 Published Online November 2011 (http://www.SciRP.org/journal/am)
Uniqueness of Meromorphic Functions with Their
Nonlinear Differential Polynomials Share a Small Function
Harina P. Waghamore, Tanuja Adaviswamy
Department of Mat hematics, Ba ng al ore University, Bangalore, India
Received August 22, 2011; revised September 27, 2011; accepted October 2, 2011
Abstract
In this paper we deal with the uniqueness of meromorphic functions when two nonlinear differential poly-
nomials generated by two meromorphic functions share a small function. We consider the case for some
general differential polynomials [()
n
f
Pf f

] where P(f) is a polynomial which generalize some result due
to Abhijit Banerjee and Sonali Mukherjee [1].
Keywords: Entire Functions, Meromorphic Functions, Nonlinear Differential Polynomials, Uniqueness
1. Introduction
In this paper, we use the standard notations and terms in
the value distribution theory [2]. For any nonconstant
meromorphic function

f
z


on the complex plane C,
we denote by any quantity satisfying
= o

as , except possibly for a set of
r of finite linear measures. A meromorphic function
is called a small function with respect to f(z) if
. Let
(, )Srf
r
)f
(, )Srf
(, ),Trf

az
(, )(,TraSr

Sf
()az
be the set of meromorphic
function in the complex plane C which are small func-
tions with respect to f. Set
, where a
zero point with multiplicity m is counted m times in the
set. If these zero points are only counted once, then we
denote the set by

(, )Eaz f

: ()zf z

0, ()()azS f
(, ).fEa Let k be


az a positive
integer. Set
 
 





),
:0,,1,..,

 
k
ii
Eazf
zfzaziikstfz az

0
where a zero point with multiplicity m is counted m
times in the set.
Let f(z) and g(z) be two transcendental meromorphic
functions, If ,
then we say that
()()( ).azS fSg
()
((),)((), )EazfEaz g
f
zand ()
g
z share the value
CM, especially, we say that f(z) and g(z) have the same
fixed points when . If
()az
()az z(,)(, ),Eaf Eag then
we say that f(z) and g(z) share the IM. If
we say that
()az
()
))
) (),g(,()
k
f Eaz(),
k
Eaz
f
za and
()
g
za have same zeros with the same multiplicities

k.
Moreover, we also use the following notations.
We denote by


), k
Nrf the counting function for
poles of f(z) with multiplicities and by ,k


), k
Nrf
the corresponding one for which the multiplicity is not
counted. Let


,Nrf
(k be the counting function for
poles of f(z) with multiplicities and let ,k


(, k
Nrf
be the corresponding one for which the multiplicity is not
counted. Set






(2 (
,, ,,
kk
NrfrfrNNrNf .f
Similarly, we have the notations
))((
1111
,, ,, ,, ,,,
1
kk kkk
Nrr Nrr Nr
fff
NN .
f
f



Let f(z) and g(z) be two nonconstant meromorphic
functions and




1,1, .EE
f
g We denote by
1
,1
L
Nr
f




the counting function for 1-points of both
()
f
z and ()
g
z about which ()
f
z has larger multi-
plicity than ()
g
z, with multiplicity is not being counted,
and denote by 1]
1
,1 Nr
f




the counting function for
common simple 1-points of both f(z) and g(z) where
multiplicity is not counted. Similarly, we have the nota-
tion

1
,.
1
L
Nr
g





In 2002 Fang and Fang [3] and in 2004 Lin-Yi [4] in-

H. P. WAGHAMORE ET AL.1365
dependently proved the following result.
Theorem A ([3,4]). Let f and g be two nonconstant
meromorphic functions and be an integer. If
(13)n
2
(1)
n
f
f
f
and 2 )(1
n
g
g g
 share 1 CM, then f ≡ g.
In 2004 Lin-Yi [5] improved Theorem A by general-
izing it in view of fixed point. Lin-Yi [5] proved the fol-
lowing result.
Theorem B ([5]). Let f and g be two transcendental
meromorphic functions and be an integer. If
(13)n
2
(1)
n
f
f
f
and 2 )(1
n
g
g g
 share z CM, then
.fg
With the notion of weighted sharing of value recently
the first author [6] improved Theorem A as follows.
Theorem C ([6]). Let f and g be two nonconstant
meromorphic functions and
  
12 2Θ;2Θ;minΘ;,Θ;,
n
fg fg


 

2
is an integer. If

n

1
f
ff

 and

n

12
g
gg

 share
then“

1, 2” fg.
In the mean time Lahiri and Sarkar [7] also studied the
uniqueness of meromorphic functions corresponding to
nonlinear differential polynomials which are different
from that of previously mentioned and proved the fol-
lowing.
Theorem D ([7]). Let f and g be two nonconstant
meromorphic functions such that

2
1
n
f
ff

and

2
1
n
g
g g share “”, where is an in-
teger then either or . If n is an even
integer then the possibility of does not arise.

1, 2
g

n13
fg
fg
f
In 2008, Banerjee and Murkherjee [1] proved the fol-
lowing theorem.
Theorem E ([1]). Let f and g be two transcendental
meromorphic functions such that

2n
f
afbfc f


and

2n
g
agbgcg  where and 0a0bc


share “”. Then the following holds:

,2 
1) If and 0, 0bc
 


max 122Θ;2Θ;minΘ;,Θ;,
42
Θ;Θ;
n
f
gf
fg






 

g
0
be an integer, where , then

Θ;Θ;fg
.
f
g
2) If and 0, 0bc




 
max 122Θ;2Θ;
min Θ;,Θ;
nf
fg





2
g
,
the roots of the equation are distinct
and one of f and g is nonentire meromorphic function
having only multiple poles, then f ≡ g.
0azbz c
3) If 0, 0bc

 and






max 122Θ;2Θ;
min Θ;,Θ;
nf
fg
 




2
g
and the roots of the equation coincides,
then
0azbz c
.
f
g

4) If 0, 0cb

and




 
max 122Θ;2Θ;
min Θ;,Θ;
nf
fg
 




g
,
then either
f
g

or
f
g

. If n is an even integer
then the possibility
f
g

 does not arise.
Here, we obtain unicity theorem when [()
n
f
Pf f

]
and [()
n
g
Pgg

] share a small function.
Theorem 1. Let
f
and
g
be two transcendental
meromorphic functions. Let
11 0  and
1mm
f (),(0),
mm m
Pfafaf a a 
(0,1,,)
i
ai ma

 is the first nonzero coefficient from the
right, and n, m, k be a positive integer with


 
10 2Θ;2Θ;
min Θ;;Θ;.
nmfg
fg
 




n

If [()
f
Pf f

] and [()
n
g
Pgg
] share “

” then

,2 
.
f
g

2. Lemmas
In this section we present some lemmas which will be
needed in the sequel. Let f, g, F1, G1 be four nonconstant
meromorphic functions. Henceforth we shall denote by h
and H the following two functions.
22
11
f
fgg
hff gg

  
 







1111
11
11
22
and .
11
FFGG
HFG
FG



 











Lemma 2.1. ([1]) If for a positive integer k,


,0; 0
k
Nr ff


denotes the counting function of
those zeros of
f

which are not the zeros of f, where a
zero of
f

with multiplicity m is counted m times if
mk

and k times if then mk





1
,0;0,0;,;
,0;,.


 
 




k
pk
Nr fNfrfrf
fp
f
NS
N
rfr
Lemma 2.2. ([1]) Let ,
f
g be share “

” and

1, 2
0h

. Then





 
 
22 2
23
,,0;,;,0;
,;,0;,, .
TrfN rfN rfN rg
g
Nrgrp SrfNS
g




 


rg

H. P. WAGHAMORE ET AL. 1366
,.
Lemma 2.3. ([8]) Let f be a nonconstant meromorphic
function and , where
are constants and . Then

01
n
n
Pfa afaf 
0
n
a
01
,, ,
n
aa a
  
,,TrPfnTrf Srf
Lemma 2.4. Let 1
()
n
f
Pf f
F


 and 1
()
n
g
Pgg
G


,
where is a small function of f and g. Then
and .

0,  

1
,F Srf


,

,
Sr

1
,,SrG Srg
Proof. Using Lemma 2.3 we see that
 
 
1
,,,
2,,.


 
TrFnmTrfTrfSrf
nmTrfSrf
And
 



1
,, 1
,,',,
 

n
nmTrf TrfPfO
TrFTrfSrf
that is,
 


1
,2,TrFnmTrfSrf 

,,.Srf

,.
Hence
1
In the same way we can prove
SrF


1Srg,SrG,.
This proves the Lemma.
Lemma 2.5. ([9]) If and 0h









,0;,;,0;,;
lim sup1,



r
NNNrfr frgrg
Tr
N
rI then or . fg1fg
,fg Lemma 2.6. Let be two nonconstant mero-
morphic functions. Then



2, nn
fPffgPgg

 
where is an integer.

6nm

Proof. Let



2. nn
fPffgPgg

  (2.1)
Let be a 1-point of f with multiplicity .
Then is a pole of g with multiplicity such
that i.e.,
0
z
0
z
np
(1)p
1)(q
11 pnqqmq ,
2( 1)() mqnpq (2.2)
From (2.2) we get 1
n qm and again from (2.2) we
obtain

1( 1)
1
2
1
nm nnm
pnm m
 




1
.
)
i.e.,
Let be a zero of with multiplicity 1.
Then 1 is a pole of g with multiplicity , say.
So from (2.1) we get
1
z
z()Pf(1)p 1)
1(q

1
2111(2pnmqnm  
1
(3
2
nm
p)


.
Sin
f
ce a pole of ()
n
g
Pg is either a zero of or a
zero of ,
g

we have








 


 

0
0
0
,0; ,0;'
,,
2
,;,0; ,0;
13
,0;,,
2
,,0;
13
,
 

 
 
 

,; ,0;

 

 


m
m
NNfrg rg
Srf Srg
m
rfNrg Nrg
nm nm
rgSrf Srg
mm
Trgr g
nm nm
N
Sr
N
f
N

,,Srg
where
NNr rg


0,0;Nr g

denotes the reduced counting func-
those zerotion ofs of
g

which are not the zeros of
g()Pg.
As


1mm
Pfafafafa

11 mm 0

 where
a10 ,,a
 are m distinct complex num
nd fundal
,
mm
abers. Then
ament theorem of Nevanlinna we get by seco








0
,,;,0; ,0;mTrfr frfrfNNN

  



 


1
0
0
,; ,
,0;,;,;
,0;( ,)
2,(,)
13
,0; ,0;
,,.
m
j
j
m
ra fSr f
rfr fraf
rf Srf
mm
Trg Trf
nm nm
rg
N
NNN
N
Nrf
Srf Srg
N


 
 

 

 





(2.3)
Similarly, we have
 


00
2

,
,
13
,0;,0;,,.N
m
mTr g
N
Trf
nm nm
rf rgSrfSrg

 
 



(2.4)
Adding (2.3) and (2.4) we obtain
,
m
Tr g


 
24
1Tr



 

, (,)
13
,,
g Trf
nm nm
Srf Srg

 


which is a contradiction. This proves the Lemma.
mero-Lemma 2.7. Let f and g be two transcendental
morphic function and
1n
m
a
Ff
1
10
11
mm
m
a a
f f
mn mnn








 



11
10
11
nmm
mm
aa
Ggg g
mn mnn

a











H. P. WAGHAMORE ET AL.1367
where is an integer. Then n( 2)m ''
F
G im lies
that
p
F
G.
Proof. Let ''
F
G, then 
F
Gc c is a
constant 0. Then by second fundtal theo-
rem w
where
t. Leamen c
e get
 

 
 
;
,;,0;,;
1
,0;, ;,
1
2,,,, ,.
m m
m m
F
a
rf rfrf
mn
a
rgrg Srf
mn
NNN
N
TrfmTrfTrg mTrg Srf
N

 




 



 
Hence we get
,, ,0;,;(,)TrFr rFrcFSrFNNN  





1, 2,
1,,
m nTrfmTrf
mTrgSrf

 

.
(2.5)
Similarly, we have

(2.6)
Adding (2.5) and (2.6) we obtain
 

1, 2,
, ,.
m nTrgmTrg
rfSrg



1mT
  



1,,32 ,
,
mnTrf TrgmTrf
rg
 
 
32 (,),mTrg SrfS 
i.e.,





(2), ,,,nmTrfT rgS rfSrg  .
which is a contradiction. So and the Lemma is
ed.
a no
0c
prov
Lemma 2.8. ([10]) Let f benconstant meromor-
phic function. Then


,0;, ;,0;,.
k
Nrfk rfNrfNfSr
Lemma 2.9. Let
F
and be given as in Lemma
2.7 and
G
1
F
, 1 G ben byma 2.4. Then give Lem

 

where are roots of the algebraic equation
 
11
1
, 0
,;
,;,0;' ,
m
m
r b f
Nrc fNrfSrf
1) ,,;,;
,;
TrFTrFNrfNrbf
NNrc f
 



 
 
 
11
1
2) ,,,0;,;
,; ,;
,;,0;',
m
m
T rGT rGN rgN rbg
Nrb gNrcg
Nrc gNrgSrg
 




12
,,,
m
bb b
1 10
0
11 mn n

 

mm mm
aa a
zz
mn
 

and are roots of the algebraic equat on

oof. By therem
and Lemmas 2.3 we obtain
12
,,,
m
cc c i
1
10
0.
mm
mm
aza za



Pr Nevanlinna’s first fundamental theo
 
 
 
 
  



11
1
1
,,1
1
,0;,,0;'1
,,0;,0;,
,,0;,; ,;
,;, ;,0;',.
m
m
TrFTrO
F
F
Nr Fmrmr FO
F
TrFNr FNr FSrF
TrFNrfNrbfNrbf
Nrc fNrcfNrfSrf











 
 



Similarly, we have
1
,0 ; ,
F
Nr FmrO








 
 

11
12
,, ,0;,;
,; ,; ,;
,0; ,
m
m
T rGT rGNrgN rb g
NrbgNrcgNrc g
r gSrg
 
,;NrcgN







where are roots of the algebraic equation

12
,,,
m
bb b
1 10
0
11
mm mm
aa a
zz
mn mnn
 

 
 

and 12 ,,cre roots of the algebraic equation ,
m
c c a
1
10
0.
mm
mm
aza za



 
Th
is proves the Lemma.
3. Proofs of the Theorems
rPoof of Theorem 1. Let ,
F
G be defined as in Lemma
2.7 and 11 and
F
G be defined as
ollows that and
in Lemma 2.4. Then it
f
F
G

 share “

,2  ” and hence 1
F
and G1 share “


,2  ”. Suppose 0H. Then by Lem-
ma 2.2, 2.4 and (2.3) we get

 
 
 
 
 
 
  

1 1
1
1
,0;
,;
,;,0;' 2,0;2,;
,;, ;,0;
,,.
m
m
r FF
Sr
Nrcf
Nrc fNrfrgrg
NrcgN rcgN rg
Srf Srg
NN




  



(3.1)
Now from Lemma 2.3, 2.8 and 2.9 we can obtain from
(3.1) for
1 2221
21
(,), ;,0;
,;, ,
2,0; 2,;
TrFNN rN rG
Nr Gf Srg
rfrfN N

 



0  







 

1,
,0;2,0;2, ;
(, ),0;
3, (,)2,;
3,0; ,,.
mn Trf
rfr grg
mT r gNrg
mTrfTrg rf
rgSrf Srg
NN
N
N

 
 
 



(3.2)
2, 0;2,;(,)rfr fmTrf
N
NN


H. P. WAGHAMORE ET AL. 1368

.Sr
In a similar manner we can obtain
.Sr
(3.3)
From (3.2) and (3.3) we get

 


1,
2113Θ;2Θ;2
mn Trf
mgfTr




  

1,
2113Θ;2Θ;2
mn Trg
mfgTr 


 


10 2Θ;2Θ;
2
.
nmf g

 
minΘ;;Θ;
f
g T
Sr

 




r
(3.4)
Since is arbitrary, (3.4) implies a contrad-
tion. He. Since for we have

0  
nce H
ic
0

0  

  
  
( ,1)
,,;,;(,1)
(, )
2Θ;,,1,
rf
mrfNrfrfmr f
Srf
,0; ,rfTrf m
.
f
Trfmr fSrf
N


N 

 

 
te that We no

 

  
 


1
1
1
, ,;
,0;,;,;
,; ,0;,0;
,;, ;,;,0;.
282Θ;2Θ;2()
,0; .
m
m
r G
rf rcfrcf
rf rfrg
rc grcgrgrg
mfgTr
NN N
NNN
r
NN
Sr
NN
g


 



,0 ;mr fm
11 1
,; 0;,0;rFF rrGNNNN 

 





(3.5)
Also using Lemma 2.3 we get











,,1 ,,1
(, ;)
,(,;)
,,(1).
n
n
nn
n
TrFmrfmrfP ffmrf
Nrf Pff
mrfP fNrfP f
TrfPfnmTrfO
 
 




(3.6)
Similarly

,,1 ,TrGmrgnmTrgO



1. (3.7)
From (3.6) and (3.7) we get












11
max,,,,1TrFTrGn mTrmrf

 .(3.
By (3.5) and (3.8) applying Lemma 2.5 we get either
11
F
G

or 111.FG

Now from Lemma 2.6 it follows111.FG

that Again
11
F
G

implies .
F
G



So from Lem
.
4. Acknowledgements
Researce first
y, Bangalore under the project U.O. No.:
1.
. References
nction,” Archivum Mathe-
p. 41-56.
ue Distribution Theory,” Springer, Berlin,
. 44, No. 5-6,
ma 2.7 the theo-
rem follows
h work of th author is supported by Banga-
lore Universit
DEV:D2:YRB-BUIRF:2010-1
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