In this paper, we study the uniqueness problems of entire and meromorphic functions concerning differential polynomials sharing fixed point and obtain some results which generalize the results due to Subhas S. Bhoosnurmath and Veena L. Pujari [1].
Let
note by
finite linear measure.
Let a be a finite complex number and k a positive integer. We denote by
counted. Set
Let
a-points of both
is not counted. Similarly, we have notation
We say that f and g share a CM (counting multiplicity) if
In 2004, Lin and Yi [
Theorem A. Let f and g be two transcendental meromorphic functions,
where h is a non constant meromorphic function.
Theorem B. Let f and g be two transcendental meromorphic functions,
In 2013, Subhas S. Bhoosnurmath and Veena L. Pujari [
Theorem C. Let f and g be two non constant meromorphic functions,
where h is a non constant meromorphic function.
Theorem D. Let f and g be two non constant meromorphic functions,
Theorem E. Let f and g be two non constant entire functions,
In this paper, we generalize theorems C, D, E and obtain the following results.
Theorem 1. Let f and g be two non constant meromorphic functions,
For
For
Theorem 2. Let f and g be two non constant entire functions,
Lemma 2.1 (see [
where
Lemma 2.2 (see [
Lemma 2.3 (see [
Lemma 2.4 (see [
Lemma 2.5 (see [
let
Lemma 2.6 (see [
The following lemmas play a cardinal role in proving our results.
Lemma 2.7 Let f and g be two non constant meromorphic functions. If
Proof. Applying Nevanlinna’s second fundamental theorem (see [
By first fundamental theorem (see [
We know that,
Therefore, using Lemma 2.3, (2) becomes
Using
since
This completes the proof of Lemma 2.7.
Lemma 2.8 Let f and g be two non constant entire functions. If
Proof. Since f and g are entire functions, we have
Proof of Theorem 1. By assumption,
Then, H is a meromorphic function satisfying
By (3), we get
Therefore,
From (6), we easily see that the zeros and poles of H are multiple and satisy
Let
Then,
We have,
Therefore,
and thus
Now, we discuss the following three cases.
Case 1. Suppose that neither
Using (8), we note that
since,
But
Using (14) and (15) in (13), we get
Since f and g share ¥ IM, we have
Using this with (8), we get
If
Similarly,
Let
By Lemma 2.6, we have
Since
By the first fundamental theorem, we have
we have
where
From (16)-(21), we get
Using Lemma 2.3, we get
Let
Then
Note that
Simplifying, we get
or
Combining (23) and (24), we get
By
If
On integrating, we get
Since
Substituting this in
that is,
From (9), we obtain
Applying Lemma 2.2, to the above equation, we get
Note that,
Using (29), we get
By, Lemmas 2.3, 2.4 and (30), we have
We obtain
Case 2. Suppose that
Applying Lemma 2.2 to the above equation, we have
By Lemmas 2.3, 2.4 and (32), we have
Using Lemma 2.7, we get
Since
Therefore,
On integrating, we get
We claim that
We have,
similarly,
Using Lemma 2.4, we have
Thus,
similarly,
Therefore, (36) becomes,
which contradicts
Let
If h is not a constant, then with simple calculations we get contradiction (refer [
Case 3. Suppose that
Applying Lemma 2.2 to above equation, we have
Using Lemmas 2.4, 2.3 and (42), we have
Using Lemma 2.7, we get
Since
Therefore
Hence,
Let
Hence,
Let
Let
In the same manner as above, we have similar results for zeros of
By Nevanlinna’s second fundamental theorem, we have from (45), (46) and (49) that,
Similarly,
From (50) and (51), we get
since
This completes the proof of Theorem 1.
Proof of Theorem 2. By the assumption of the theorems, we know that either both f and g are two transcendental entire functions or both f and g are polynomials. If f and g are transcendental entire functions, using
where k is a non-zero constant. Suppose that
Apply Lemma 2.2 to above equation, we have
Since f is a polynomial, it does not have any poles. Thus, we have
Therefore,
Using Lemmas 2.4, 2.3 and (54), we have
Using Lemma 2.8, we get
since
On Integrating, we get
We claim that
Proceeding as in Theorem 1,
we get
Harina P. Waghamore,Sangeetha Anand, (2016) Generalization of Uniqueness of Meromorphic Functions Sharing Fixed Point. Applied Mathematics,07,939-952. doi: 10.4236/am.2016.79084