Generalization of Uniqueness Theorems for Entire and Meromorphic Functions

Abstract

In this paper, we deal with the uniqueness problems on entire and meromorphic functions con- cerning differential polynomials that share fixed-points. Moreover, we generalise and improve some results of Weichuan Lin, Hongxun Yi, Meng Chao, C. Y. Fang, M. L. Fang and Junfeng xu.

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Waghamore, H. and Shilpa, N. (2014) Generalization of Uniqueness Theorems for Entire and Meromorphic Functions. Applied Mathematics, 5, 1267-1274. doi: 10.4236/am.2014.58118.

1. Introduction

In this paper, the term “meromorphic” will always mean meromorphic in the complex plane C. Let a be a complex number and be a meromorphic function such that. We say f and g share the value CM, if and assume the same zeros with the same multiplicities; if and assume the same zeros with the same multiplicities, then we say and share CM, especially we say that and have the same fixed-points when. It is assumed that the reader is familiar with the notations of the Nevanlinna theory that can be found, for instance, in [1] . We denote by any function satisfying

as, possibly outside of finite measure.

Set

It is well known that if f and g share four distinct values CM, then f is a fractional transformation of g. In 1997, corresponding to one famous question of Hayman, C. C. Yang and X. H. Hua showed the similar conclusions hold for certain types of differential polynomials when they share only one value. They proved the following result.

Theorem A ([2] ). Let f and g be two non-constant meromorphic functions, be an integer and. If and share the value a CM, then either for some root of unity d or and, where and are constants and satisfy

In 2001, M. L. Fang and W. Hong obtained the following result.

Theorem B ([3] ). Let f and g be two transcendental entire functions, an integer. If and share the value 1 CM, then.

Recently, W. C. Lin and H. X. Yi extended the above theorem with respect to fixed point. They proved the following results.

Theorem C ([4] ). Let and be two transcendental meromorphic functions, an integer. If and share z CM, then either or

where h is a nonconstant meromorphic function.

Theorem D ([4] ). Let and be two transcendental meromorphic functions, an integer. If and share z CM, then.

We generalise the above results and prove the following Theorem.

Theorem 1.1 Let and be two transcendental meromorphic functions, an integer. If and share CM then

For, we get Theorem C.

For, we get Theorem D.

One may ask the following question, can the nature of the fixed point z be relaxed to IM in the above theorems?

In 2008, Meng Chao answered to the above question and proved the following theorems.

Theorem E ([5] ). Let and be two transcendental meromorphic functions, an integer. If and share z IM, then either or

where h is a nonconstant meromorphic function.

Theorem F([5] ). Let and be two transcendental meromorphic functions, an integer. If and share z IM, then.

We generalise the above results and prove the following Theorem.

Theorem 1.2 Let and be two transcendental meromorphic functions, an integer. If and share IM then

For, we get which improves Theorem E.

For, we get, we get Theorem F.

In 2002, Fang and Fang [6] proved that there exists a differential polynomial d such that for any pair of nonconstant entire functions f and g we can get, if and share one value CM.

Theorem G ([6] ). Let and be two nonconstant entire functions, be a positive integer. If and share 1 CM, then.

In 2004, Lin-Yi [7] and Qiu-Fang [8] proved that Theorem G remains valid for.

Theorem H ([7] [8] ). Let and be two nonconstant entire functions, be a positive integer. If and share 1 CM, then.

We generalise the above results and prove the following theorem.

Theorem 1.3 Let f and g be two transcendental entire functions, an integer. If and share z CM then

For, we get Theorem H.

For, , we get new result.

Fang-Fang discussed Theorem H by replacing CM with IM and proved the following Theorem.

Theorem I ([6] ). Let f and g be two nonconstant entire functions, n be a positive integer. If and share 1 IM and, then.

We generalise the above results and prove the following Theorem.

Theorem 1.4 Let f and g be two transcendental entire functions, an integer. If and share z IM then

For, which improves Theorem I.

For, , we get new result.

2. Some Lemmas

Lemma 2.1 ([9] ) Let f be a nonconstant meromorphic function, n be a positive integer. where ai is a meromorphic function satisfying . Then

Lemma 2.2 ([10] ) Let f be a non-constant meromorphic function k be a positive integer, then

where denotes the counting function of the zero’s of where a zero of multiplicity m is counted m times if and p times if. Clearly.

Lemma 2.3 ([11] [12] ) Let F and G be two nonconstant meromorphic functions sharing the value 1 IM. Let

If, then

Lemma 2.4 ([5] ) Let f and g be two nonconstant meromorphic functions, , positive integers, denotes as in section 1 and, and let

if F and G share IM, then

Lemma 2.5 ([13] ) Let H be defined as above. If and

where and I is a set with infinite linear measure, then or.

Lemma 2.6 ([14] ) Let then

where , which are distinct respectively.

3. Proofs of the Theorems

In this section, we present the proofs of the main results.

Proof of Theorem 1.2.

Lemma 2.4 implies that.

Let

(1)

(2)

and

(3)

(4)

where

Thus we obtain that F and G share the value 1 IM. Moreover, by Lemma 2.1, we have

(5)

(6)

Noting that, we deduce

(7)

and by the First Fundamental Theorem,

(8)

Note that,

(9)

where are distinct roots of the algebraic equation

and

(10)

Since F and G share 1 IM, by Lemma 2.3, we have

(11)

Obviously, we have

(12)

(13)

So, we have

(14)

From (5) to (14), we have

(15)

Therefore, that is

(16)

By integration, we have

(17)

where and B are constants. Thus

(18)

Since,

(19)

we note that,

(20)

and

(21)

Similarly, we have

(22)

From (19) to (22) and applying Lemma 2.5, we get

or.

We discuss the following cases.

Case (i) Suppose that.

As in the proof of Theorem 1, in [5] we arrive at a contradiction.

Case (ii), thus, that is,

Set, we substitute in the above, it follows that

(23)

If h is not a constant, using Lemma 2.6 and (23), we conclude that

By (23), we get

where

Using Lemma 2.6, we get

where which are pairwise distinct.

This implies that every zero of has a multiplicity of at least n. By the Second Fundamental Theorem, we obtain that, which is again a contradiction.

Therefore h is a constant. We have from (23) that, which imply, and hence.

Proof of Theorem 1.1. Let F and G be given by (1) and (2). Suppose H is given as in Lemma 2.3, and. Proceeding as in the proof of Theorem 1.2 we can obtain (3) to (10). Since F and G share 1 CM, by Lemma 2.3, we have

Hence from (3) to (10) and (12) to (14), we get

Proof of Theorem 1.4. Let F and G be given by (1) and (2). Suppose H is given as in Lemma 2.3, and. Proceeding as in the proof of Theorem 1.2 we can obtain (3) to (10) and (11). Since F and G share 1 IM, by Lemma 2.3, obviously we have,

therefore (14) reduces to

Hence, which contradicts that. Proceeding in the same way as in Theorem 1.2 we get.

Proof of Theorem 1.3. Let F and G be given by (1) and (2). Suppose H is given as in Lemma 2.3, and. Proceeding as in the proof of Theorem 1.2 we can obtain (3) to (10). Since F and G share 1 CM, by Lemma 2.3, we have

Hence from (3) to (10) and (12) to (14), we get

Hence, which contradicts that. Proceeding in the same way as in Theorem 1.2, we get.

Acknowledgements

The author would like thank Prof. S. S. Bhoosnurmath for his valuable suggestions for the improvement of the paper. This research was supported by the UGC under MRP(S).

Conflicts of Interest

The authors declare no conflicts of interest.

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