DOI : 10.5281/zenodo.22255701
- Open Access
- Authors : Somalatha M.T
- Paper ID : IJERTV15IS080681
- Volume & Issue : Volume 15, Issue 08 , August – 2026
- Published (First Online): 02-09-2026
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Value distribution for q-shift difference polynomial of meromorphic function with maximal deficiency sum
Somalatha M.T
Department of Mathematics, Government Science College, Nrupathunga University, Bangalore – 560 001, Karnataka, India.
Abstract
The main purpose of this paper is to investigate the relationship between the charac- teristic function of a meromorphic function f (z) involving maximal deciency sum and that of q shift dierence polynomial.We improve and generalize the results of Zhaojun Wu and S.S.Bhoosnurmath, R.S.Dyavanal et.al.
Keywords: dierence operator,dierence polynomial, maximal deciency sum, mero- morphic function,
-
Introduction(Denitions and Results)
In this paper, we shall use the standard notions in Nevanlinnas value distribution theory of meromorphic functions such as T (r, f ), N (r, f ) and m(r, f )(see [8, 17, 19]). By S(r, f ) we denote any quantity satisfying the condition S(r, f ) = o(T (r, f )) as r possibly outside of an exceptional set E of nite linear measure. A meromorphic function a a(z) is called a small function with respect to f if either a or T (r, a) = S(r, f ). We denote by S(f ) the collection of all small functions with respect to f . Clearly C {} S(f ) and S(f ) is a eld over the set of complex numbers.
For a C {}, the quantities (a, f ) and (a, f ) dened as follows
(a, f ) = 1 lim Sup
r
N (r, a; f )
T (r, f )
and
(a, f ) = 1 lim Sup
r
N (r, a; f )
T (r, f )
L
are called the deciency and the ramication index of tat for the function f . The classical second fundamental theorem of Nevanlinna theory asserts that the total deciency of any meromorphic function f (z) satises the inequality
(a, f )+ (,f ) 2
aC
The above inequality holds, then we say that f has maximal deciency sum. The Valiron Mahonko identity states that if the function R(z, f ) is rational in f and has small merorphic coecients, then
T (r, R(z, f )) = degf (R)T (r, f )+ S(r, f ) (1.1)
Certain relationship between the characteristic function of a meromorphic function f (z) with maximal deciency sum and that of derivative f t(z) plays a key role in the study of a conjecture of Nevanlinna (see [4]). The main contribution of this paper is to study the relationship between the characteristic function of a meromorphic function f (z) with maximal deciency sum and that of the exact dierence cf = f (z + c) f (z), where c /= 0 (see [5]). In 1956, Shan and Singh [6] proved the following theorem.
'E
Theorem A [6] Suppose that f (z) is a transcendental meromorphic function of nite order and
aC (a, f ) = 2. Then
T (r, f t) 2T (r, f ), r +.
After that, Edrei [7] and Weitsman [4] proved the following theorem, respectively.
Theorem B [4, 7] Suppose that f (z) is a transcendental meromorphic function of nite order with maximal deciency sum. Then
f t
and
lim
r+
T (r, f t)
T (r, f )
= 2 (,f )
lim
r+
N (r, 1 )
T (r, f t)
= 0.
Under the condition of Theorem B, Singh and Gopalakrishna [8] proved that
N (r, a)
lim = 1 (a, f )
r+ T (r, f )
holds for every a C.
Theorem C [3] Suppose that f (z) is a transcendental meromorphic function of order less than one with maximal deciency sum. Then
where
K (cf )
2(1 (,f ))
2 (,f )
cf
N (r, cf )+ N (r, 1 )
K (cf ) = lim sup
r
T (r, cf )
L
Theorem D [3] Let f (z) be a transcendental meromorphic function of order less than one and assume (,f ) = 1. Then
(a, f ) (0, cf )
aC
Later on, Zhaojun Wu[10] investigated the relationship between the Characteristic function of a meromorphic function f (z) with maximal deciency sum using dierence operator and also established an inequality on the zeros and poles for dierence operator by giving an example to show that the upperbound of the inequality is accurate.
Recently, Subhas. S. Bhoosnurmath, R.S.Dyavanal et. al[11] obtained the relationship between
c
the Characteristic function of meromorphic function having maximal deciency sum and its higher order exact dierence n(f (z)) and generalized the results of Zhaojun Wu to a great
extent.
q
Inspiring by these results,we investigate the value distribution for q-shift dierence polynomial n(f (z))P (f (qz + c)) of meromorphic function with maximal deciency sum.
-
Lemmas
( )
Lemma 2.1. [14]. Let q and c be two non-zero nite complex numbers and f (z) F with zero order
m r, f (qz + c) = S(r, f ).
f (z)
Lemma 2.2. [13]. Let f (z) F of nite order, c C and < 1, then
m (r,
f (z))
n c
f (z)
T (r, f )
( )
= o r
= S(r, f ).
Lemma 2.3. [16]. Let f (z) F of nite order and q, c c, then
T (r, f (z + c)) T (r, f (z)) + S(r, f ),
N (r, f (qz + c)) N (r, f (z)) + S(r, f ),
f (qz + c)
f (z)
N (r, 1 ) N (r, 1 ) + S(r, f ).
c
Lemma 2.4. [13]. Let f (z) F of nite order and c C, then N (r, nf (z)) = (n+1)N (r, f )+
S(r, f ).
Lemma 2.5. [15]. Let f (z) F of nite order and P (f (qz + c)) be q-shift polynomial of order
s and c C. Then
N (r, P (f (qz + c))) sN (r, f )+ S(r, f ).
-
Results:
Theorem 1.1 Suppose that f (z) is a transcendental meromorphic function of nite order with maximal deciency sum. Then
q
T (r, n(f (z)) P (f (qz + c)))
L (a, f ) lim inf f (z)
aC
(
r
n(f (z))
T (r, f )
)
T
lim sup
r, q
f (z)
P (f (qz + c))
(n + s + 1) n(,f )
and
r
T (r, f )
L
-
For n = 1 and s = 0 with maximal deciency and (a, f ) = . (1 2)
aC
L
We obtain T (r, q(f (z))) T (r, f ) as r
-
For n ?, 1 and s = 0 with maximal deciency and (a, f ) = 1and(,f ) = 1
aC
T (r, n(f (z))P (f (qz + c)))
We obtain lim
r
q = 1
T (r, f )
Proof of theorem 1.1:
q
Proof. We dene, F1 = n(f (z))P (f (qz + c)). Using the rst main theorem of Nevanlinna and Lemmas 2.2 and 2.4, we have
T (r, F1) = m(r, F1)+ N (r, F1)
= m(r, nf (z)p(f (qz + c))) + N (r, nf (z)P (f (qz + c)))
m (r,
q
q
n(f (z))f (z)
f (z))
f (z)
P (f (qz + c)))
q
q
+ N (r, nf (z)) + N (r, P (f (qz + c)))
q
m (r,
n q
f (z)
+ m(r, f (z)) + m(r, P (qz + c)) + N (r, nf (z)) + N (r, P (qz + c))
S(r, f )+ m(r, f )+ Sm(r, f )+ (n + 1)N (r, f )
T (r, f ) T (r, f )(S + 1) + nN (r, f )+ S(r, f )
lim Sup T (r, F1) (S + 1) + n lim Sup N (r, f ) + n lim Sup S(r, f )
r
T (r, f )
r
T (r, f )
r
T (r, f )
(S + 1) + n(1 (,f ))
lim Sup T (r, F1 (n + S + 1) n(,f ) (3.1)
r
T (r, f )
Let {ai}(i = 1, 2, …, q) be distinct complex numbers continuing all the nite decient values of
f (z).
Consider
L
q
(z) = 1
f ai
i=1
Weknow that T (r, f (z) ai) = T (r, f )+ O(1)
Since n is a linear operator and n of a constant is zero, it follows that
q q
n [f (z) ai] = nf (z)
q q
Using Lemma 2.2, we deduce
q
m(r, (z)F1(z))
m
r,
q
i
f (z) a
+ logq
L ( n(f (z) a )P (f (qz + c)))
i=1
i
= S(r, f )
Now the above results yields,
1
m(r, (z)) = m (r, (z)F (z) 1 )
F1(z)
!
m(r, (z)F (z)) + m (r, 1 ) + c
F1(z)
therefore
m(r, (z)) m r, 1 + S(r, f ) (3.2)
( )
F1(z)
In view of Valirmon Mahonko identity (1.1), we have
qT (r, f ) = T (r, (z)) + O(1)
= m(r, (z)) + N (r, (z)) + O(1)
m (
r, 1
F1(z)
q
)
L
+
L
i=1 q
N (r, ai)+ S(r, f )
q
Hence
T (r, F1(z)) = N (r, ai)+ S(r, f )
i=1
q lim inf T (r, F1(z)) + L lim Sup N (r, ai)
r
T (r, f )
i=1
r
q
T (r, f )
q = lim inf T (r, F1(z)) + L(1 (a ,f ))
r
T (r, f )
i
i=1
q
q lim inf T (r, F1(z)) + q L (a ,f )
(OR)
r
T (r, f )
i
i=1
q
lim inf T (r, F1(z)) L (a ,f )
Since q is arbitrary, we have
r
T (r, f )
i
i=1
r
T (r, f )
L (a, f ) lim inf T (r, F1(z))
a
(3.3)
Thus from 3.2 and 3.3 we get (1.2)
r
T (r, f )
r
T (r, f )
L (a, f ) lim inf T (r, F1) lim Sup T (r, F1)
a
L
(n + s + 1) n(,f )
Now for n = 1 and s = 0 with maximal deciency and (a, f ) = . (1 2)
a
L
We obtain T (r, F1) T (r, f ) as r
Again for n 1 and s = 0 with (a, f ) = 1 and (,f ) = 1
a
We get lim
r
T (r, F1) = 1
T (r, f )
Theorem 1.2 Suppose that f (z) is a transcendental meromorphic function of nite order.
( )
Then
q
)
N r, 1 n(f (z))P (f (qz + c))
lim sup
r
T (r,
1
q
n(f (z))P (f (qz + c))
and
1
(a, f )
'E
aC
(n + s + 1) n(,f )
L
For n ?, 1 and s = 0 with maximal deciency and (a, f ) = 1and(,f ) = 1
( )
aC
q
(
N r, 1 n(f (z))P (f (qz + c))
We obtain lim
r
1
q
T r, n(f (z))P (f (qz + c))
) = 0
q
consequently (0, n(f (z))P (f (qz + c))) = 1
Proof of theorem 1.2:
Proof. Consider
'Eq 1
i=1 f ai
As in the proof of Theorem (2.1) in Haymans[1], we have
q
m(r, (z)) L m(r, a ) qlog+ 3q log2
i
i=1
q
m(r, ai)+ N
r,
F1(z)
m(r, (z)) + N
L ( 1 )
i=1
r,
F1(z)
+ O(1)
( 1 )
)
By using the rst main theorem and eq (3.2), we get
q
Li=1
m(r, ai)+ N
r, 1
(
F1(z)
+(r, F1(z)) + S(r, f )
'Eq
m(r, ai)
N r, 1
(
F (z)
S(r, f )
)
Thus,
i=1 + 1 1+
T (r, F1(z)) T (r, F1(z) T (r, F1(z))
We derive from (1.2) that
q N (r, 1 )
L lim inf m(r, ai)
+ lim Sup F1(z)
i=1
r
T (r, F1(z) r
S(r, f )
T (r, F1(z)
1 + lim Sup
r
T (r, F1(z)
1 + lim Sup
S(r, f ) T (r, f )
Thus,
r
= 1
T (r, f ) T (r, F1(z)
q
N (r, 1 )
r
T (r, F1(z))
r
T (r, F1)
1 lim Sup F1(z) + L lim inf m(r, ai)
i=1
q
N (r, 1 )
1 lim Sup F1(z) + L lim inf m(r, ai)
lim
T (r, f )
r
It follows from (1.2) that
T (r, F1(z))
i=1
r
T (r, F1) r T (r, F1(z)
(
N r, 1
F (z)
'Eq
(ai,f )
)
1 lim Sup 1 + i=1
r
Thus, since q is arbitrary, we have
T (r, F1(z))
(n + s + 1) nn(,f )
( )
N r, 1
lim Sup F1(z) 1
(ai,f )
'E
a
r
T (r, F1(z))
(n + s + 1) nn(,f )
For n = 1 and s = 0 with maximal deciency sum
N (r, 1 )
lim
r
F1(z) = 0 (3.4)
T (r, F1(z))
'E
Now for n 0 and s = 0 with (a, f ) = 1 and (,f ) = 1
a
lim
r
N r, 1
( )
F1(z) = 0 (3.5)
T (r, F1(z))
Consequently, we have that the deciency of F1(z) with respect to 0 is 1. i.e.,
(0, F1(z)) = 1
Theorem 1.3 Suppose that f (z) is a transcendental meromorphic function of nite order with maximal deciency sum. Then
1 (,f )
[(n + s + 1) n(,f )lim Sup
r
N (r, F1(z))
T (r, F1(z))
(n + s + 1)[1 (,f )]
'E
(a, f )
a
-
For n = 1 with maximal deciency sum, we get
1 (,f )
(2 + s) (,f )
lim Sup
r
N (r, n(f (z))P (f (qz + c))
q
q
T (r, n(f (z))P (f (qz + c))
(2 + s)[1 (,f )]
(2 + s) (,f )
'E
-
For n 1 with (a, f ) = 1 and (,f ) = 1,
aC
lim Sup
r
Proof of theorem 1.3
N (r, n(f (z))P (f (qz + c))
q
q
T (r, n(f (z))P (f (qz + c))) = 0
Proof. By using Lemma(2.4), we have
N (r, F1(z)) (n + s + 1)N (r, f )+ S(r, f )
Which implies that,
N (r, F1(z)) T (r, F1(z)) (n + s + 1)N (r, f ) + S(r, f )
T (r, F1(z))
Using Theorem (1.1), we have
T (r, f )
T (r, f )
T (r, f )
L (a, f ) lim Sup N (r, F1) (n + s + 1)[1 (, 1)]
Hence,
a
r
T (r, F1)
lim Sup
r
N (r, F1(z))
T (r, F1(z))
(n + s + 1)[1 (, 1)]
'E
(a, f )
a
(3.6)
On the other hand, from lemma 2.5, we have
N (r, f ) N (r, F1(z))
which implies that,
N (r, f ) N (r, F1(z)) T (r, F1(z))
Therefore,
T (r, f ) T (r, F1(z)) T (r, f )
lim Sup N (r, f ) lim Sup N (r, F1)
lim Sup T (r, F1)
r
T (r, f )
r
T (r, F1) r
T (r, f )
using Theorem (1.1), we have
1 (,f ) lim Sup N (r, F1(z))[(n + s + 1) n(,f )]
r
T (r, F1(z))
1 (,f )
lim Sup N (r, F1(z))
(3.7)
[(n + s + 1) n(,f )Thus, from equations 3.6 and 3.7 we get (1.3)
r
T (r, F1(z))
1 (,f )
[(n + s + 1) n(,f )lim Sup
r
N (r, F1(z))
T (r, F1(z))
(n + s + 1)[1 (,f )]
'E
(a, f )
a
(3.8)
Now for n = 1 with maximal deciency sum, using 3.4 and 3.8, we get
1 (,f )
lim Sup N (r, F1(z)) (2 + s)[1 (,f )]
'E
(2 + s) (,f )
r
T (r, F1(z))
(2 + s) (,f )
Also for n 1 with (a, f ) = 1 and (,f ) = 1, using 3.5 in 3.8, we get
a
lim Sup N (r, F1(z)) = 0
r
T (r, F1)
Theorem 1.4 Suppose that f (z) is a transcendental meromorphic function of nite order with maximal deciency sum (,f ) = 1. Then
q
L (a, f ) (s + 1)(0, n(f (z))P (f (qz + c)))
aC
'E
Proof of Theorem 1.4
Proof. Suppose (a, f ) = 0, then the result is trivially true.
'E
a
Next let (a, f ) > 0
a
)
Let {ai}(i = 1, 2, …, q) be a sequence of distinct complex numbers in C containing all the nite decient values of f (z). Proceeding as i the proof of theorem 1.2, we have
q
Li=1
m(r, ai)+ N
r, 1
(
F1(z)
T (r, F1(z)) + S(r, f )
holds for any q nite complex numbers in {ai}. Therefore, we have
N (r,
1 )
'Eq
m(r, ai)
+ O(1) 1
F1(z) T (r, f ) i=1
T (r, F1(z) T (r, F1(z)) T (r, f )
Hence from (1.2), we can get
N (r,
1 )
'Eq
m(r, ai)
lim Sup + O(1)
F1(z) T (r, f ) i=1
r
T (r, F1(z)) T (r, F1(z)) T (r, f )
N (r,
1 )
'Eq
m(r, ai)
1 lim Sup + lim inf O(1)
F1(z) T (r, f ) i=1
r
T (r, F1(z)) r T (r, F1(z)) T (r, f )
)
(
N r, 1
F (z)
T (r, f )
'Eq
m(r, ai)
lim Sup 1 + lim inf lim inf i=1
r
T (r, F1(z))
r,
F (z)
lim Sup 1 + i=1
(
N
r
1 )
T (r, F1(z)) r
(ai,f )
'Eq
T (r, f )
r
T (r, F1(z))
(n + s + 1) n(,f )
L
Since q is arbitrary and (,f ) = 1, we have
(a, f ) (s + 1)(0, F1(z))
aC
-
Conclusion:
-
Using Valiron Mohonko identity, we have obtained the value distribution of q-shift dierence polynomial of meromorphic function with maximum deciency sum.
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