THE UNIQUENESS OF MEROMORPHIC FUNCTIONS WHOSE DIFFERENTIAL POLYNOMIALS SHARE SOME VALUES

Abstract

In this article, we deal with the uniqueness problems of meromorphic functions concerning differential polynomials and prove the following theorem. Let f and g be two nonconstant meromorphic functions, n ≥ 12 a positive integer. If fⁿ(f³ - 1)f′ and gⁿ(g³ - 1)g′ share (1, 2), f and g share ∞ IM, then f ≡ g. The results in this paper improve and generalize the results given by Meng (C. Meng, Uniqueness theorems for differential polynomials concerning fixed-point, Kyungpook Math. J. 48(2008), 25-35), I. Lahiri and R. Pal (I. Lahiri and R. Pal, Nonlinear differential polynomials sharing 1-points, Bull. Korean Math. Soc. 43(2006), 161-168), Meng (C. Meng, On unicity of meromorphic functions when two differential polynomials share one value, Hiroshima Math.J. 39(2009), 163-179).

1. Introduction, definitions and results

Let f be a nonconstant meromorphic function defined in the open complex plane C. Set E(a, f) = {z : f(z) − a = 0}, where a zero point with multiplicity m is counted m times in the set. If these zeros points are only counted once, then we denote the set by (a, f). Let f and g be two nonconstant meromorphic functions. If E(a, f) = E(a, g), then we say that f and g share the value a CM; if , then we say that f and g share the value a IM. We assume that the reader is familiar with the notations of Nevanlinna theory that can be found, for instance, in [4] or [14].

Let m be a positive integer or infinity and a ∈ C ∪ {∞}. We denote by Em)(a, f) the set of all a-points of f with multiplicities not exceeding m, where an a-point is counted according to its multiplicity. Also we denote by (a, f) the set of distinct a-points of f with multiplicities not greater than m. We denote by Nk)(r, 1/(f − a)) the counting function for zeros of f − a with multiplicity ≤ k, and by (r, 1/(f − a)) the corresponding one for which multiplicity is not counted. Let N(k(r, 1/(f − a)) be the counting function for zeros of f − a with multiplicity at least k and (r, 1/(f − a)) the corresponding one for which multiplicity is not counted. Set

By the above definition, we have

Definition 1.1 ([17]). Let F and G be two nonconstant meromorphic functions such that F and G share the value 1 IM. Let z0 be a 1-point of F with multiplicity p, a 1-point of G with multiplicity q. We denote by the counting function of those 1-points of F and G where p > q, by the counting function of those 1-points of F and G where p = q = 1 and by the counting function of those 1-points of F and G where p = q ≥ 2, each point in these counting function being counted only once.

We also require the following notion of weighted sharing which was introduced by I. Lahiri.

Definition 1.2 ([5,6]). For a complex number a ∈ C ∪ {∞}, we denote by Ek(a, f) the set of all a-points of f where an a-point with mutiplicity m is counted m times if m ≤ k and k + 1 times if m > k. For a complex number a ∈ C ∪ {∞}, such that Ek(a, f) = Ek(a, g), then we say that f and g share the value a with weight k.

The definition implies that if f, g share a value a with weight k then z0 is a zero of f − a with multiplicity m(≤ k) if and only if it is a zero of g − a with multiplicity m(≤ k) and z0 is a zero of f − a with multiplicity m(> k) if and only if it is a zero of g − a with multiplicity n(> k), where m is not necessarily equal to n. We write f, g share (a, k) to mean that f, g share the value a with weight k. Clearly if f, g share (a, k) then f, g share (a, p) for all integer p, 0 ≤ p < k. Also we note that f, g share a value a IM or CM if and only if f, g share (a, 0) or (a,∞) respectively. We call f and g share (z, k) if f − z and g − z share (0, k).

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 1.3 ([13]). Let f and g be two nonconstant meromorphic functions, n ≥ 11 an integer and a ∈ C − {0}. If fnf′ and gng′ share the value a CM, then either f = dg for some (n + 1)th root of unity d or g(z) = c1ecz and f(z) = c2e−cz, where c, c1 and c2 are constants and satisfy (c1c2)n+1c2 = − a2.

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

Theorem 1.4 ([3]). Let f and g be two transcendental entire functions, n ≥ 11 an integer. If fn(f − 1)f′ and gn(g − 1)g′ share the value 1 CM, then f ≡ g.

In 2004, W.C. Lin and H.X. Yi extended the above theorem in view of the fixed-point. They proved the following result.

Theorem 1.5 ([8]). Let f and g be two transcendental meromorphic functions, n ≥ 13 an integer. If fn(f − 1)2f′ and gn(g − 1)2g′ share z CM, then f ≡ g.

In 2008, the first author relaxed the nature of fixed-point to IM and proved

Theorem 1.6 ([10]). Let f and g be two transcendental meromorphic functions, n ≥ 28 an integer. If fn(f − 1)2f′ and gn(g − 1)2g′ share z IM, then f ≡ g.

Some works have already been done in this direction [?],[9]. In 2006, I. Lahiri and R. Pal proved the following result.

Theorem 1.7 ([7]). Let f and g be two nonconstant meromorphic functions and let n(≥ 14) be an integer. If E3)(1, fn(f3 − 1)f′) = E3)(1, gn(g3 − 1)g′), then f ≡ g.

Naturally, we consider the following question: Can the nature of the sharing value be relaxed in the above theorem?

In 2009, the first author gave a positive answer to the above Question and proved

Theorem 1.8 ([11]). Let f and g be two nonconstant meromorphic functions such that fn(f3−1)f′ and gn(g3−1)g′ share (1, l), where n be a positive integer such that n + 1 is not divisible by 3. If (1) l = 2 and n ≥ 14, (2) l = 1 and n ≥ 17, (3) l = 0 and n ≥ 35, then f ≡ g.

In this paper, we study the uniqueness problems of meromorphic functions concerning differential polynomials and prove the following results

Theorem 1.9. Let f and g be two nonconstant meromorphic functions, n ≥ 12 a positive integer. If fn(f3 − 1)f′ and gn(g3 − 1)g′ share (1, 2), f and g share ∞ IM, then f ≡ g.

Theorem 1.10. Let f and g be two nonconstant meromorphic functions, n ≥ 19 a positive integer. If fn(f3 − 1)f′ and gn(g3 − 1)g′ share 1 IM, f and g share ∞ IM, then f ≡ g.

 

2. Some Lemmas

In this section, we present some lemmas which will be needed in the sequel. We will denote by H the following function:

where F and G are two meromorphic functions.

Lemma 2.1 ([12]). Let f be a nonconstant meromorphic function, and let a1, a2,…, an be finite complex numbers, an ≠0. Then

Lemma 2.2 ([1]). If F and G share (1, 2) and (∞, k), where 0 ≤ k ≤ ∞, then one of the following cases holds.

the same inequality holds for T(r,G), (2) F ≡ G, (3) FG ≡ 1. Here (r,∞, F,G) is the reduced counting function of those a-points of F whose multiplicities differ from the multiplicities of the corresponding a-points of G.

Lemma 2.3 ([16]). Let f be a nonconstant meromorphic function. Then

Lemma 2.4 ([7]). Let f and g be two nonconstant meromorphic functions. Then fn(f3 − 1)f′gn(g3 − 1)g′ ≢ 1, where n is a positive integer.

Lemma 2.5 ([7]). Let , where n(≥ 2) is an integer. If F∗ ≡ G∗, then f ≡ g.

Lemma 2.6 ([18]). Suppose that two nonconstant meromorphic function F and G share 1 and ∞ IM. Let H be given as above. If H ≢ 0, then

Lemma 2.7 ([15]). Let H be defined as above. If H ≡ 0 and

where I is a set with infinite linear measure and T(r) = max{T(r, F), T(r,G)}, then FG ≡ 1 or F ≡ G.

 

3. Proof of Theorem 1.9

Let

and

Thus we obtain that F and G share (1, 2). If the case (1) in Lemma 2.2 occur, that is

Moreover, by Lemma 2.1, we have

Since (F∗)′ = F, we deduce

and by the first fundamental theorem

Note that

It follows from (6) − (8) that

It follows from (1) that

From (2), (9), (10) and (11) we obtain

By Lemma 2.3 we have

We have from (12) and (13) that

In the same manner as above, we have

Therefore by (14) and (15), we obtain that n ≤ 11, which contradicts n ≥ 12. Thus by Lemma 2.2, we get F ≡ G or FG ≡ 1. If FG ≡ 1, that is

By Lemma 2.4, we get a contradiction. If F ≡ G, that is

where c is a constant. It follows that T(r, f) = T(r, g) + S(r, f). Suppose that c ≠0, by the second fundamental theorem, we have

which contradicts the assumption. Therefore F∗ ≡ G∗. Thus by Lemma 2.5, we have f ≡ g. This completes the proof of Theorem 1.9.

 

4. Proof of Theorem 1.10

Let

and

Thus we obtain that F and G share 1 IM. If possible, we suppose that H ≢ 0. Thus, by Lemma 2.6, we have

Also we have

We get from (19) and (20) that

It’s obvious that

Combining (21), (22) and (23), we deduce

Moreover, by Lemma 2.1, we have

Since (F∗)′ = F, we deduce

and by the first fundamental theorem

Note that

It follows from (24), (28), (29) and (30) that

We have from (25) and (31) that

In the same manner as above, we have

Therefore by (32) and (33), we obtain that n ≤ 18, which contradicts n ≥ 19.

Therefore H ≡ 0. That is

By integration, we have from (34)

where A(≠0) and B are constants. Thus

From (18), we have

Note that

and

From (37) − (39), we apply Lemma 2.7 and get F ≡ G or FG ≡ 1. Proceeding as in the proof of Theorem 1.9, we get the conclusion. This completes the proof of Theorem 1.10.