International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.

1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 2 Issue 1 || January. 2014 || PP-41-48
On The Value Distribution of Some Differential Polynomials
1,
Subhas.S.Bhoosnurmath, 2, K.S.L.N.Prasad
1,
Department of Mathematics, Karnatak University,
Dharwad-580003-INDIA
2,
Lecturer, Department of Mathematics,
Karnatak Arts College, Dharwad-580001-INDIA
ABSTRACT: We prove a value distribution theorem for meromorphic functions having few poles, which
improves several results of C.C.Yang and others.Also we obtain a generalized form of Clunie’s result.
C. C. Yang [8] has stated the following.
Theorem A Let f(z) be a transcendental meromorphic function with N(r, f) = S(r, f).
If P[f] = f
n
 a 1  n 1 f   a 2  n 2 f   ......  a n
(1)
 i f  is a homogeneous differential polynomial in f of degree i, then
Tr, Pf   nT(r, f )  Sr, f  .
Where
each
This result is required at a later stage and we prove the above theorem on the lines of G. P. Barker and A. P.
Singh [1].
Proof we have, P[f] = f
n
 a 1  n 1 f   a 2  n 2 f   ......  a n
an 
 a  f  a  f 
 f n 1  1 n 1  2 n 2  ........ n 
fn
fn
f 

A
A 
 A
 f n 1  1  22  ........ nn 
f
f
f 

Where A i 
(2)
a i  n i f 
, i  1,2,........
n
f n i
Now, since each  n f is a homogeneous differential polynomial in f of degree n,
we have ,
 
 
  f lo f 1 l1 .......f k 
  n f  
m r, n   m r,

fn
 f 

lk




lk
1 l1
2  l 2

 k 
 r,  f   f  ..... f 

= m 
 f   f 
 

 f 



 




  f 1  l1  f 2   l2  f k   lk
 
 .....

  m r, 
 f 

  f   f 

 









  O1


= S(r, f), using Milloux‟s theorem [10].
Hence m(r, Ai) = S(r, f), for i = 1,2, . . . ,n.
Now on the circle |z | = r, let
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41 | P a g e

2. On The Value Distribution of Some Differential Polynomials
 
 
 
A rei  Max  A1 rei  | , | A 2 rei 


Then, m (r, A(z)) = S (r, f)
Let

 
E1    0,2 f rei
   and E
 2A rei
1
2
 
,......... A n rei 
..,
1
n



(3)
2
be the complementof E1 .
Then, on E1, we have
P[f ]  f n 1 
A1 A 2
A
 2  ......... nn

f
f
f
A
A
A 
n
 f 1  1  22  ..........  nn 
..
f
f
f 

1 1
1
n
 f 1   2  ..........  n 
..
2 
 2 2
1 n
 n f
2

Hence, on E 1 , log 2
Therefore,
 
P[f ]  log  f
n
n
n log  f  n log 2  log  Pf 
Therefore, n mr, f  
2
1

 n log f d
2 0

1
1


 n log f d  2 E n log f d
2 E1
1

1
n


 n log 2  log P[f ] d  2 E log 2A d
2 E1
2

1
n


 log P[f ] d  2 E log A d  O1
2 E1
2


 m r, P[f ]  n mr, A   O1
 m r, P[f ]  Sr, f  .
Thus, n m (r, f)  m (r, P([f]) + S(r, f).
(4)
Adding n N(r, f) both sides and noting N(r, f) = S(r, f), we get,
nT(r, f)  m (r, P[f] ) + S(r, f).
Or n T (r, f)  T(r, P(f) ) + S(r, f)

Now, m(r, P[f]) = m r, f
n

(5)
 A1f n 1  A 2 f n  2  ........  A n 1f  A n , by (2)


 m r, f {f n 1  A1 f n 2  ........  A n 1 }  m r, A n   O (1)


 mr, f   m r, f n 1  A1 f n  2  ........  A n 1  S r, f 
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3. On The Value Distribution of Some Differential Polynomials
Proceeding on induction, we have
mr, Pf   n mr, f   S(r, f )
Therefore, T(r, P[f] )  n m(r, f) + N(r, P[f] + S(r, f)
But, N(r, P[f] ) = S(r, f), since N (r, f) = S(r, f).
Hence, T(r, P[f] )  n T(r, f ) + S(r, f)
(6)
From (5) and (6) we have the required result.
We wish to prove the following result


Theorem 1 Let f(z) be a transcendental meromorphic function in the plane and Q1 f , Q 2 f be differential


F = P[f] Q1 f   Q 2 f  ,
polynomials in f satisfying Q1 f  0, Q 2 f  0. Let P(f) be as defined in (1).


If
Then
(7)

n   Tr, f  N r, 1   N r, P1f    


  

f
Q2



Q2


  Q2  1 Nr, f   Sr, f 
To prove the above theorem, we require the following Lemmas.
Lemma 1 [5]: If Q[f] is a differential polynomial in f with arbitrary meromorphic co-efficients
q j ,1  j  n, then
mr, Qf    Q m r, f    mr, q j   Sr, f  .
n
j1
*

  denote differential polynomials
Lemma 2 [5]: Let Q f and Q f
efficients
If
*
1
*
2
in f with arbitrary meromorphic co-
*
n
q , q ,...,q and q1, q 2 ,...,q s respectively.
P[f]
is
a
homogeneous
Pf  Q f   Qf where Q  n, then
differential
polynomials
in
f
of
degree
n
and
*


n



s

m r, Q * f    m r, q *   m r, q j  Sr, f  .
j
j1
j1
This leads to a generalised form of Clunie‟s result.
Lemma 3 [5]: Suppose that M[f] is a monomial in f. If f has a pole at z = z 0 of order m, then z0 is a pole of M[f]
order (m-1)  m  M .
Lemma 4 [5]: Suppose that Q[f] is a differential polynomial in f. Let z 0 be a pole of f of order m and not a
zero or a pole of the co-efficients of Q[f]. Then z0 is a pole of Q[f] of order atmost m  Q  Q   Q


Proof of Theorem 1
Let us suppose that n   Q 2 .
We have
F = P[f] Q1 [f] + Q2 [f]
Therefore,
F 
and
F
F
Pf Q1f   Q 2 f 
F
F



F  Pf  Q1 f   Pf  Q1 f   Q 2 f  
  * 

Hence, it follows that P f Q f  Q f
(8)
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43 | P a g e

4. On The Value Distribution of Some Differential Polynomials
where
and
Q* f  
Pf  Q f   Q f 
F
Q1 f  
1
1
F
Pf 
(9)
 F
Q f   Q 2 f   Q 2 f  .
F
(10)
We have by Lemma 2 and (8),


m r, Q * f   Sr, f 
Qf 
Q * f 

Again from (8), P f 
Therefore,
(11)

1 
mr, Pf   m r, Qf   m  r, * 
 Q f  


(12)
From (10) and Lemma 1, we have
mr, Qf    Q 2 mr, f   Sr, f  .
(13)
By the First Fundamental Theorem and (11), we get


1 
1 
m r, *   Nr, Q* f   N r, *   Sr, f  .
 Q f  
 Q f  




*
(14)

Clearly, the poles of Q f occur only from the zeros and poles of F and P[f], the poles of f and the zeros and
poles of the co efficients. Suppose that z0 is a pole of f of order m, but not a zero or a pole of the co-efficients
of P f , Q1 f and Q 2 f .





Hence, from Lemma 4, z0 is a pole of Q[f] of order atmost m  Q 2  Q 2   Q 2  1
*

*

If z0 is a pole of Q f , we have from (8), Q f =
Qf 
Pf 



and hence z0 is a pole of Q f of order atmost m Q 2  Q 2   Q 2  1  mn
*
 m(  Q 2  n )  (Q 2   Q 2  1)
Also, from (8),
1
Q f 
*

Pf 
Qf 



Hence z0 is a zero of Q*[f] of order atleast mn  m Q 2  Q 2   Q 2  1
Thus, we have,

 1 
1 
 1
Nr, Q * f   N r,
 Q * f    N r, F   N Pf    Q2   Q2  1 Nr, f 













  Q 2  n N r, f   Sr, f 

we have from (8), P f 
(15)
Qf 
Q * f 
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5. On The Value Distribution of Some Differential Polynomials
Therefore,
 Qf  
mr, Pf   m r,
 Q * f  



Therefore,

1 
mr, Pf   mr, Qf   m r,
 Q * f  



Using (14) we have,

1 
mr, Pf   mr, Qf   Nr, Q * f   N r,
 Q * f    Sr, f 



Using (4), (13) and (15) we have


1 

 1
n mr, f    Q 2 mr, f   N r,   N r,
 + Q2   Q2  1 Nr, f 
 F
 Pf  


  Q 2  n  Nr, f   Sr, f  .



1 
  Q 2   Q 2  1 Nr, f   Sr, f  ,
 Pf  


1
n    Tr, f   N r, F   N r,



Hence, we get ,
Q2





which is the required result.
f n in the above result, we get
F  f n Q1 f   Q 2 f  and
1
n   Q2 Tr, f   N r, F   N r, 1   Q2   Q2  1 Nr, f   Sr, f ,






 f
Remark (1) Putting P[f] =
which is the result of Zhan Xiaoping [9]

(2) Putting P f  a n f

n
 a n 1f n 1  .......... .  a 0 in the above result,


We get, F  P f Q1 f  Q 2 f
and
1
n   Tr, f   N r, F   N r, P1f    



  

Q2




Q2

  Q2  1 Nr, f   Sr, f 
which is the result of Hong Xun Yi[4].
Theorem 2 Let
and Q[f]
F  f n Qf  , where Q[f] is a differential polynomial in f
 0.

 1
N r,   Nr, f 
 f
If lim
 n, then  (a, F)  1
r 
T, r, f 
Proof Let F  f
n
where a  0 .

Q[f ]. Putting Q 2 [f ]  0 in the result of Zhan Xiaoping ( remark 1) we have,
 1
 1
n T(r, f)  N r,   N r,   N(r, f )  S(r, f )
 F
 f
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6. On The Value Distribution of Some Differential Polynomials
Hence
 1
N r , 
F
n  lim    n
r  T ( r , f )
Therefore,
using hypothesis.
 1
N r, 
F
lim 
0
r  T(r, f )
1 

N r,

 F  a   0 , for a  0 .
Hence, lim

r 
T(r, F)
Therefore, 1  (a, F)  0
Or
(a, F)  1 for a  0 .

Hence the result.


Theorem 3 Let F  f Q1 f  Q 2 f where Q1[f] and Q2[f] are as defined in Theorem 1.
n

If

 1
N r,   Q2   Q!  1 Nr, f 
f
lim  
 n   Q2
r 
Tr, f 


and
Q 2 f   0, t hen  a, F  1 where a  0.


Proof Let F  f Q1 [f ]  Q 2 [f ] where Q1 [f ] and Q 2 [f ] are as defined in Theorem .
n
By the result of Zhan Xiaoping [Remark 1], we have
 1
 1
(n   Q 2 )  T(r, f )  N  r,   N  r,   (Q 2   Q 2  1) N(r, f )  S(r, f ) .
 F
 f
 1
N r, 
 F   (n   ) using hypothesis
Or (n   Q )  lim
Q2
2
r  T(r, f )
Hence,
1 

N r,

 Fa   0
lim
r 
T( r , f )
which implies 1  (a, F)  0 .
Or
(a, F)  1 for a  0 .

Hence the result.
 
Theorem 4 Let F  P f Q f where P[f] is as defined in (1) and Q[f] is a differential polynomial in f such that
Q[f]
 0. If n > 1, then  F   f and  F   f

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46 | P a g e

7. On The Value Distribution of Some Differential Polynomials
Proof Let G = F – a, where a(0) is a finite complex number. Then by Theorem 1,
 1
 1
n Tr, f   N r,   N r,   Nr, f   Sr, f 
 G
 f
Obviously, the zeros and poles of f are that of F respectively
 1
 1
  Nr, f   N r,   Nr, F  Sr, f 
 f
 F
Therefore, N r ,
1 

 1
  N r,   Nr, F  Sr, f 
 Fa
 F
Thus, n T(r, f)  N r ,
 3Tr, F  Sr, f 
Therefore,
Tr, f   OTr, F as r  
Also,
Tr, F  OTr, f  as r  
Hence the theorem follows.
As an application of Theorem 1, we observe the following .
Theorem 5
No transcendental meromorphic function f can satisfy an equation of the form
a 1 P f Q f  a 2 Q f  a 3  0 ,where a 1  0, a 3  0, P f is as in (1) and Q[f] is a differential


 


polynomial in f.

Putting P f  f
n
Theorem 6
No transcendental meromorphic function f can satisfy an equation of the form
in the above theorem, we have the following.
a 1f Q[f ]  a 2 Q[f ]  a 3  0 ,where a 1  0, a 3  0, n is a positive integer and Q[f] is a differential
n
polynomial in f.
This improves our earlier result namely ,
Theorem 7: No transcendental meromorphic function f with N(r, f) = S(r, f) can satisfy an equation of the form
a1 zf z f   a 2 zf   a 3 z  0
n
(1)
i 1
where
n  1, a 1 z   0 and f   M i f    a j z  M j f  is a differential polynomial in f of degree n and

j1
each Mi(f) is a monomial in f.
(Communicated to Indian journal of pure and applied mathematics)
Acknowledgement: The second author is extremely thankful to University Grants Commission for the financial
assistance given in the tenure of which this paper was prepared
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8. On The Value Distribution of Some Differential Polynomials
REFERENCES
[1.]
[2.]
[3.]
[4.]
[5.]
[6.]
[7.]
[8.]
[9.]
Barker G. P. And Singh A. P. (1980) : Commentarii Mathematici Universitatis : Sancti Pauli 29, 183.
Gunter Frank And Simon Hellerstein (1986) : „On The Meromorphic Solutions Of Non Homogeneous Linear Differential
Equation With Polynomial Co-Efficients‟, Proc. London Math. Soc. (3), 53, 407-428.
Hayman W. K. (1964) : Meromorphic Functions, Oxford Univ. Press, London.
Hong-Xun Yi (1990) : „On A Result Of Singh‟, Bull. Austral. Math. Soc. Vol. 41 (1990) 417-420.
Hong-Xun Yi (1991) : “On The Value Distribution Of Differential Polynomials”, Jl Of Math. Analysis And Applications 154,
318-328.
Singh A. P. And Dukane S. V. (1989) : Some Notes On Differential Polynomial Proc. Nat. Acad. Sci. India 59 (A), Ii.
Singh A. P. And Rajshree Dhar (1993) : Bull. Cal. Math. Soc. 85, 171-176.
Yang C. C. (1972) : „A Note On Malmquist‟s Theorem On First Order Differential Equations, Ordering Differential Equations‟,
Academic Press, New York.
Zhan-Xiaoping (1994) : “Picard Sets And Value Distributions For Differential Polynomials”, Jl. Of Math. Ana. And Appl. 182,
722-730.
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