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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