Advances in Pure Mathematics, 2011, 1, 322-324
doi:10.4236/apm.2011.16058 Published Online November 2011 (http://www.SciRP.org/journal/apm)
The Normal Meromorphic Functions Family Concerning
Higher Order Derivative and Shared Set by One-Way
Yi Li
School of Science, Southw est Universi t y of Scie nce a nd Te chn olo gy, Mianyang, China
Received October 7, 2011; revised October 28, 2011; accepted November 7, 2011
Abstract
Let F be a meromorphic functions family on the unit disc Δ, If for every
f
F

(the zeros of f is a multi-
plicity of at least k) and if


0fz then


k
f
zc

and


__ __
k

ff
E
SES (


,Sab), then F is nor-
mal on Δ.
Keywords: Meromorphic Function, Normality Criterion, Shared Set by One-Way, Higher Order Derivative
1. Introduction and Results
First, we introduce the following definition:
Definition: For are two distinct complex values,
we have set and
,ab

,bSa
__
 
   

__
,
:)0,
ff ES Eab
zfza fzbzD



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 the normal meromorphic functionsfamilyconcern-
ing one-way sharing set, W. H. Zhang proved the fol-
lowing result [1]:
Theorem A. Let F be a family of meromorphic func-
tions in the unit disc , a and b be two distinct nonzero
complex values. , If for every


,Sa

b
f
F, all of
whose zeros is multiple,


f
S
'f
ES E, then F is
normal on . 
W. H. Zhang continued considering the relation be-
tween normality and the shared set, and proved the next
result [2]:
Theorem B. Let
F
be meromorphic functions fam-
ily in the unit disk , a and b be two distinct nonzero
complex values. If for every , all of whose zeros
is multiplicity at least (k is a positive integer),

1
fF
k

 
S
kf
f
In 2008, F. J. Lv got following theorem in [3]:
ESE, then F is normal in . 
Theorem C. Let
F
be a family of meromorphic
function in the unit disk

, a and b is two distinct non-
zero complex values, is positive integer, k


,Sab .
If for every fF

, all of whose zeros have multiplicity
1k

at least,



kf
f
ESES, then F is normal in

.
In this paper,we continue to discuss about normality
theorem of meromorphic functions families concerning
higher order derivative and shared set by one-way, and
obtain main results as follow.
Theorem. Let F be a meromorphic functions family
on the unit disc Δ, a and b is two distinct nonzero com-
plex values, is positive integer, , If for
every
k
F

,bSa
f

, all of whose zeros have multiplicity at
least, If
k
1) exists ， 0c
such that
 

0k fzfz c

;
2) ;

 
__ __
k
ff
ESE

S
then F is normal in

.
Remark: Through the following example, all of
whose zeros have multiplicity at least is necessary. k
Example: Let


= 1,. 1 , and 1,
k
i
Si  
12
  1, 2.
a family of meromorphic function in the unit disk

be




n
F
fz there



12
,1,2,3
zz
n
fz nn

ee


k
n
,.
Obvously
 
n
f
zf z and

Y. LI323
1




11
11
ee, 1,2,3,,
lzz
ll
n
fznl k

  ,
Hence satisfy 1) and 2) and all of whose zeros
of
0z

n
f
z have multiplicity 1k

at most, Since
 
 
#
12
2
0
0
10
f
fn
f



n

,
F
is not normal at by Marty Theorem. 0z
2. Lemmas
Lemma 1 [4]. Let
F
be meromorphic functions fami-
lies in the unit disk , all of whose zeros have multi-
plicity at least. If for every

k
f
F, there exists
, such that 0A


k
f
zA when eve

0fz

. If
F
is not normal in , then for every 0k

, there
exists , 01rn zr, n
f
F

, 0
n

, such
that


nn n
n
n
fz
g




 converges locally and uni-
formly to a noncontant meromorphic function


g

,
and . where
 
##
0ggkA  

1
 
#
2
1
g
gg




 is said to be Spherical derivative of g.
Lemma 2 [5]. Let

f
z be nonconstant meromor-
phic function in C
  
__
___
1
,
,1lim,
,
r
Nrfa
af a
Trf





 
,
 

__
___ ,
,1lim
,
r
Nrf
fTrf


then .
 
,,
aC
af f




2
3. Proof of Theorem
Suppose that
F
be not normal in , then by Lemma 1
we get that there exists , and

n
z
n
fF 0
n



such that


nn n
nk
n
fz
g

  converges locally and
uniformly to a noncontant meromorphic function


g

[6], and
 
##01
g
gkc  . We claim that the
following conclusions hold.
1) zeros of

g

have multiplicity at least, and k
 
()k
0
g
gc   ;
2)




,
kk
g
ag b  ;
In fact, suppose that there exists 0

, such that


00g


, there exists 0n

,



0
nnn n
k
n
fz 


nn
g 

 for sufficiently large n,
Hence


nn nn
fz  0

.
Therefore, the following conclusions is obviously

j
 
, 2,3,,1
nn n
fz k  0, 1
nj

 
and


k
nnnn
fz c 


k
,
Hence




0, 1,2,3,,1
jj
jk
nn nn
gf j 


nnn
z 

 
and


k
nn
g
c 

,
So

 
0lim 1,2,3,,1
n
gg j



0,
jj
nn
 k

 
and


0
k
g
c 

,
Hence, zeros of


g

have multiplicity at least, k
and



k
g
0gc 

,
Therefore, conclusion (1) is hold
In what follow, we complete the proof of the claim 2):
Suppose that there exists 0

Such that


0
k
g
a 

, by Hurwitz theorem, there
exists 0n

 such that

 
nn nnnn

kk
g
fz  a


for sufficiently large n.
By conditions:

 
__ __
k
ff
ESES,

 
or
nn nn
f
za  b,
Hence
  
0limlim nn nn
nn k
nn
n
fz
gg

 
 


,
this is a contradictions for


0
k
g
a  ．
If


k
g
a 

, then


g

is polynomial of degree
k.
Because, zeros of


g

have multiplicityat least, k
then
 
0
!
k
a
gk


and
 

1
0
1!
k
a
gk 




Obviously
 



1
00
#
0
0
,1
1! 2
0
,1
11!
!
k
k
ak
k
ga
a
k
k











 

Because


0
k
g
ac 

, then

Y. LI 324






,, ,2
1
kk k
k
ag bggk

1
1! 1!
ackc
kk

 2

 

1c , Suppose is not general,
this is a contradictions for Lemma 2. The proof of Theo-
rem is completed.
therefore 1
2
kkc
, hence


#0 1
g
kc, this is a con-
contradictions for conditions of

g

. Hence


k
g
a  . Similar to prove


k
g
b  . 4. References
Therefore [1] W.-H. Zhang, “The Normality of Meromorphic Functions
Concerning One-Way Sharing Set,” Journal of Nanhua
University, Vol. 18, 2004, pp. 36-38.





__
___
1
,
,1lim
,
k
k
k
r
Nr
ga
ag Trg






 1
 and [2] W.-H. Zhang, “The Normality of Meromorphic Func-
tions,” Journal of Nanhua University, Vol. 12, No. 6,
2004, pp. 709-711.





__
___
1
,
,1lim
,
k
k
k
r
Nr
gb
bg Trg







[3] F.-J. Lv and J.-T. Li, “Normal Families Related to Shared
Sets,” Journal of Chongqing University: English Edition,
Vol. 7, No. 2, 2008, pp. 155-157.
1 
,
[4] X. C. Pang and L. Zalcman, “Normal Families and
Shared Values,” Bulletin of London Mathematical Society,
Vol. 32, No. 3, 2000, pp. 325-331.
doi:10.1112/S002460939900644X
Since




 
__ __
,,
,,1
kk
Trg Nrg
kNrg NrgkNrg


[5] W. K. Hayman, “Meromorphic Function,” Oxford Uni-
versity Press, London, 1964.
So [6] J. Clunie and W. Hayman, “The Spherical Derivative of
Integral and Mer-Functions,” Commentarii Mathematici
Helvetici, Vol. 40, 1966, pp. 117-148.
doi:10.1007/BF02564366




 
__
___ ,1
,1lim 1
11 ,
k
k
r
Nrg k
gkk
Trgk

  

Therefore