Advances in Pure Mathematics, 2011, 1, 95-98
doi:10.4236/apm.2011.13021 Published Online May 2011 (http://www.scirp.org/journal/apm)
Normality of Meromorphic Functions Family and
Shared Set by One-way
Yi Li
School of Science, Southw est Universi ty of Scie nce a nd Te chn olo gy, Mianyang, Chin a,
Received March 16, 201 1; revised April 5, 2011; accepted April 10, 2011
Abstract
We studied the normality criterion for families of meromorphic functions which related to One-way sharing
set, and obtain two normal criterions, which improve the previous results.
Keywords: Meromorphic Function, Normality Criterion, Shared Values, Shared Set by One-way
1. Introduction
For Shared values, Schwick proved the following result
[1]:
Theorem A Let F be a family of meromorphic func-
tions in the domain D, 1 a, 2 a and 3 a be three finite
complex numbers. If for every


 
,1,2,3
fi fi
fFEa Eai


then F is normal in D.
In 2000, Pang Xue-cheng and Zalcman generalized the
Schwick’s result [2]:
Theorem B Let F be meromorphic functions family in
the domain D, and 1
a, 2 a be two complex number. If
for every
  
,1,2
fi fi
fFEa Eai


then F is normal in D.
Definition For ,ab are two distinct complex values,
we have set

,Sab and
 
 



,
:0,
ff
ES Eab
z
fzafzbz D


If
 
fg
ES ES, we call that f and g share S in D;
If
 
fg
ES ES, we call that f and g share S by
One-way in D.
For shared set, W. H. Zhang obtained impor tant results
[3]:
Theorem C Let F be a family of meromorphic func-
tions in the unit di sc , a and b be two distinct nonzero
complex value,


,Sab, If for every
f
F, all of
whose zeros is multiple,


ff
ES ES
, then F is
normal on

.
W. H. Zhang continued considering the relation be-
tween normality and the shared set, and proved the next
result [4]:
Theorem D Let
F
be meromorphic functions family
in the unit disk

, a and b be two distinct nonzero com-
plex values. If for every fF, all of whose zeros is
multiplicity 1k

at least (k is a positive integer),





kf
f
ESES, then F is normal in .
For shared set by One-way, Lv Feng-jiao got follow-
ing theorem in [5]:
Theorem E Let F be a family of meromorphic func-
tion in the unit disk

, a and b is two distinct nonzero
complex values, kis positive integer,


,Sab. If for
every
f
F

, all of whose zeros have multiplicity 1k

at least,





kf
f
ESES, then F is normal in

.
In 2007, Pang Xue-cheng proved the following im-
portant results in [6]:
Theorem F Let F be meromorphic functions family in
D,


123
,,Saaa. If for every
f
F



ff
ES ES
,
then F is normal on D.
To promote the results of Pang Xue-cheng, we con-
tinue to discuss about normality theorem of meromorphic
functions families concerning shared set and shared set
by one-way, and obtain our main results as follow.
Theorem 1 Let
F
be meromorphic functions fami-
lies in D,


123 4
,, ,i Saaaaa

 (i = 1, 2, 3).
If for every
f
F

,
 
ff
ES ES
, and 4
f
a


whenever 4
f
a

, then F is normal on D.
Theorem 2 Let
F
be meromorphic functions fami-
lies in D,


12 3
,,SaaaC

. If for every
f
F

,




ff
ES ES
, and 3
f
a


, whenever 3
f
a

, then

Y. LI 96
F is normal on D.
2. Lemmas
Lemma 1 [7] Let
F
be meromorphic functions families
in the unit disk , all of whose zeros have multiplicity
kat least, and 0A. If for every
f
F,


f
zA


whenever

0.fz If
F
is not normal in , then for
every 01

, there exists
1) a positive number ,0 1rr,
2) complex sequence ,
nn
zz r,
3) Functions sequence n
f
F,
4) and positive sequence 0
n


,
such that


nnnnn
gfz





 converges locally
and uniformly to a noncontant meromorphic function

g

, and
 
##
01ggkA  . Where
 

#
2.
1
g
gg




Lemma 2 [8] Let
f
be meromorphic function with
finite order on the open plane C, and 123 ,,aaa be three
finite complex values. If


f
z have only finite zero,
and
  
123
0,,
f
zfzSaaa

 
then
f
is a rational function.
3. Proof of Theorem 1
Suppose that
F
be no t normal in , then by Lemma 1
we have that there exists
,and0
nn n
fFz 

 ,
such that





14nnnnn
gfzag
 

 con-
verges locally and uniformly to a noncontant meromor-
phic function

g

. We claim that the following con-
clusions hold.
 
04
10
g
ga 

 ;
 
0## 4
201gga  ;

0
3
g
S 
;
It is not difficult to prove claims 00 1,2, in what fol-
low, we complete the proof of the claim 0
3. Suppose
that there exists 0C

 such that

0i
g
a 
. Obvi-
ously,

i
g
a 


, in fact, if

0i
g
ac  
, it is a
contradictions for 0
1. Thus from Hurwitz Theorem, we
know that there exists a point sequence 0n

 , such
that

0
nn i
ga 
 , for sufficiently large n, that is

nn nni
f
za 
 .
Obviously,


14,
nnn i
gaa 


 as n.
Thus


0
g 

, this is a contradiction. Hence, claim
0
3 holds.
From claim 0 3 we have that

 
1,2,3
i
gai 

So


g


is identical in nonconstant. Again because
claim 0 1, we know

 
40
ga

  and

40
4 #20
40
1
011
12
a
a
ga









.
Clearly, this is a contradictions for claim 0 2. There-
fore, F is normal in D. The proof of Theorem 1 is com-
pleted.
4. Proof of Theorem 2
Suppose that
F
is not normal in , by Lemma 1 there
exists ,
nn
fFz

 and 0
n


, such that


n
g





nn n
fz g

  converges locally and uni-
formly to a noncontant meromorphic function


g

with finite orders, there
 
##
0gg  .
We asserts that


0gSg 


.
In fact, suppose that there exists 0C

, such that


0
g
S 

, thus there exists

1, 2
i
ai such that


0i
g
a 

.
From Hurwitz Theor em and

i
g
a  

, we have there
exists 0n

  such that

nni
g
a  , that is


nn
g



nn ni
f
za 

 for sufficiently large n. Thus in
contrast with conditions of Theorem, we get

nn n
f
zS 


. Obviously,

nn nn
f
zA 

. So
we get


00g



.
Since


g

is a nonconstan t entire function, without
loss of generality, we assume that

1
g
a   have zero
on C for 1 a, and consider function sequence


n
G

:




1 1nnn n
nnn
f
za ga
G
 

 

Obviously,


n
G is not normal in zero of


1
g
a 

.
In fact，if 0

is zero of

1
g
a  , then


0n
G



0
00
nn n
fz 

. With conditions of Theorem,
we get

0nn n
f
zA 


 and

0n
GA 
. There-
fore,


n
G is not normal in zero of

1
g
a  . So there
exists n G, n


 and 0
n


, such that




 
1
11
nnnnn
nnnn
FG
gaF
 
 



 

converges locally and uniformly to a noncontant and
meromorphic function


F

with finite order, and
0
1 the number of zeros of

F

is finite,

Y. LI 97
  
03
20
F
FSa 
  ,
 
0
30FSF 
 ,
  
0
00
410FF




In fact, suppose that 0

is the zero of


1
g
a 

with order k. If there exists 1k

distinct 121 ,k
 


at least, such that

0,1,2, ,1
j
F
jk   .
By Hurwitz Theorem, it is certainly that there exist a
positive integer N, such that

0, 1,2, ,1
j
nn
F
jk   
as nN. Thus,

10
j
nn nn
ga  .
Since

0
j
nnn n
  , 1, 2,,1jk, we
deduce that 0

is a zero of

1
g
a   with 1k

orders, this is a contradictions for suppose. Therefore
zeros numbers of

F

is finite.
Suppose that 0

is a zero of

00F


. For

0F  

and Hurwitz theorem, we know that there
exists sequence 0n

, such that
 

1
1
0,
nnnn nn
nn nn
nnnn nn
fz a
F
fz a
 

 
  


  

Thus, we get

3nnnn nn
fzS a  

 

 and
subsequence n fF such that

nnnn nni
fz a  

 

,
thus
 
03
lim nnnn nn
n
F
fzS a 
 



,
for

3i
aS a.
If there exists 0

such that

0
F
S 
, that is,
there exists i aS such that

0i
F
a 
. Since

i
F
a 


, by Hurwitz theorem, there exists 0n


such that
 
nnn nnnnni
F
fz a 




.
Hence,

nnnn nn
fz S     

,

0
F
S  .
If there exists N such that

1for , nnnn nn
fza nN  
  

we get



1
0lim nnnnnn
nnn
fz a
F
 


  


This contradicts

0i
F
a 
. Thus exists subsequence
n
f, such that

1nnnn nn
fz a  
  

for every n.
Therefore,
 
1
0lim 0
nnnnnn
nnn
fz a
F
 


  



Now we prove that


0
010FF






 


 .
Since


31 31
131
1
0
()
nnn
nnnn
nn
nn n
FaaGaa
gaaa




 



 
there exists 0n

, such that

31
10
n
nn
Faa



,
we get


3nnnn nn
f
za  




,
thus

3nnnn nn
f
za  




,
that is,


3nn
F
a 


. Therefore,





0
0
0
22
1lim 0
nn
nn
F F
FFF



 





  


 
 
 
.
So far, we give complete proofs of all assertion. Next
we will complete the proof of theorem 2 using assertion
00
1~4.
By Lemma 2 and assertion 0 2, we get that


F

is
a rational function. Again by assertion 0 4, it is clear that
the pole of F be multiple. If n G is not normal at 0

,
thus 0

be zero of


1
g
a 

. By the isolation of zero,
we have that n G are holomorphic functions at 0

for
sufficiently large n. We get that



1
nnnnn
FG
 


are holomorphic functions in R   for sufficiently
large R, thus


F

be nonconstant holomorphic
functions in C. Therefo re

F

be a polynomial. Let
it, s order is


0pp. Thus,



 
,1lnTrF pr
 ,






,lnand, 1NrFprSrFO


Therefore,








21lnln 1,prprOr

. We
get 02p

 easily.
If 1p

, thus

 
010
0Fccc 

, by 0 2 and
0
3, we find that there exists i a for every

, such that


i
F
a 


. Therefore

be an zero of


F

. But


F

have only a zero, this is a contradiction.
If 2p

, thus

Y. LI 98
   
00 1001
0,Fc c
 
 
As a result,
 
001
2Fc


. Obviously ze-
ros of

i
F
a 
 are

00 010
2
i
ac cc   . Hence
we get that

F

have three zeros, this still is a con-
tradiction from
   
00 1001
0,Fc c

 
and the proof of theorem 2 is completed.
5. References
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[3] W. H. Zhang. “The Normality of Meromorphic Func-
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[8] X. J. Liu and X. C. Pang, “Shared Values and Nor-
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