We study the uniqueness of meromorphic functions and differential polynomials sharing one value with weight and prove two main theorems which generalize and improve some results earlier given by M. L. Fang, S. S. Bhoosnurmath and R. S. Dyavanal, and so forth.

1. Introduction and Results

Let f be a nonconstant meromorphic function defined in the whole complex plane ℂ. It is assumed that the reader is familiar with the notations of the Nevanlinna theory such as T(r, f), m(r, f), N(r, f), and S(r, f), that can be found, for instance, in [1–3].

Let f and g be two nonconstant meromorphic functions. Let a be a finite complex number. We say that f and g share the value a CM (counting multiplicities) if f − a and g − a have the same zeros with the same multiplicities, and we say that f and g share the value a IM (ignoring multiplicities) if we do not consider the multiplicities. When f and g share 1 IM, let z₀ be a 1-point of f of order p and a 1-points of g of order q; we denote by N₁₁(r, 1/(f − 1)) 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.

is the counting function of those 1-points of both f and g, where p > q. In the same way, we can define N₁₁(r, 1/(g − 1)),

, and

. If f and g share 1 IM, it is easy to see that

(1.1)

Let f be a nonconstant meromorphic function. Let a be a finite complex number and k a positive integer; we denote by N_(k)(r, 1/(f − a)) (or

) the counting function for zeros of f − a with multiplicity ≤k (ignoring multiplicities) and by N_(k)(r, 1/(f − a)) (or

) the counting function for zeros of f − a with multiplicity at least k (ignoring multiplicities). Set

(1.2)

We further define

(1.3)

In 2002, C. Y. Fang and M. L. Fang [4] proved the following result.

Theorem A (see [4].)Let f(z) and g(z) be two nonconstant entire functions, and let n(≥8) be a positive integer. If [fⁿ(z)(f(z) − 1)]f^′(z) and [gⁿ(z)(g(z) − 1)]g^′(z) share 1 CM, then f(z) ≡ g(z).

Fang [5] proved the following result.

Theorem B (see [5].)Let f(z) and g(z) be two nonconstant entire functions, and let n, k be two positive integers with n > 2k + 8. If and share 1 CM, then f(z) ≡ g(z).

In [6], for some general differential polynomials such as , Liu proved the following result.

Theorem C (see [6].)Let f(z) and g(z) be two nonconstant entire functions, and let n, m, k be three positive integers such that n > 5k + 4m + 9. If and share 1 IM, then either f(z) ≡ g(z) or f and g satisfy the algebraic equation R(f, g) ≡ 0, where .

The following example shows that Theorem A is not valid when f and g are two meromorphic functions.

Example 1.1. Let f = (n + 2)(h − hⁿ⁺²)/(n + 1)(1 − hⁿ⁺²), g = (n + 2)(1 − hⁿ⁺¹)/(n + 1)(1 − hⁿ⁺²), where h = e^z. Then [fⁿ(z)(f(z) − 1)]f^′(z) and [gⁿ(z)(g(z) − 1)]g^′(z) share 1 CM, but f(z)≢g(z).

Lin and Yi [7] and Bhoosnurmath and Dyavanal [8] generalized the above results and obtained the following results.

Theorem D (see [7].)Let f(z) and g(z) be two nonconstant meromorphic functions with Θ(∞, f) > 2/(n + 1), and let n(≥12) be a positive integer. If [fⁿ(z)(f(z) − 1)]f^′(z) and [gⁿ(z)(g(z) − 1)]g^′(z) share 1 CM, then f(z) ≡ g(z).

Theorem E (see [8].)Let f(z) and g(z) be two nonconstant meromorphic functions satisfying Θ(∞, f) > 3/(n + 1), and let n, k be two positive integers with n > 3k + 13. If and share 1 CM, then f(z) ≡ g(z).

Naturally, one may ask the following question: is it really possible to relax in any way the nature of sharing 1 in the above results?

To state the next result, we require the following definition.

Definition 1.2 (see [9].)Let k be a nonnegative integer or infinity. For one denotes by E_k(a, f) the set of all a-points of f, where an a-point of multiplicity m is counted m times if m ≤ k and k + 1 times if m > k. If E_k(a, f) = E_k(a, g), one says that f, g share the value a with weight k.

We write that 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 integers p with 0 ≤ p ≤ k. Also, we note that f, g share a value a IM or CM if and only if they share (a, 0) or (a, ∞), respectively.

Recently, with the notion of weighted sharing of values, Xu et al. [10] improved the above results and proved the following theorem.

Theorem F (see [10].)Let f(z) and g(z) be two nonconstant meromorphic functions, and let n, k be two positive integers with n > 5k + 11. If Θ(∞, f) + Θ(∞, g) > 4/n, , and share 1 (1,2), then f(z) ≡ g(z).

Theorem G (see [10].)Let f(z) and g(z) be two nonconstant meromorphic functions, and let n, k be two positive integers with n > 7k + 23/2. If Θ(∞, f) + Θ(∞, g) > 4/n, , and share 1 (1,1), then f(z) ≡ g(z).

Remark 1.3. The proof of Theorem E contains some mistakes: for example, one cannot get formulas (6.9) and (6.10) in [8]. Therefore, the last inequality in page 1203 of [8] does not hold. So, Theorem E will not stand. Similarly, in [10], the proof of Case 1 in Theorem F is incorrect (see page 63 of paper [10]). Hence, the conclusion of Theorems F and G will not stand.

Now one may ask the following questions which are the motivations of the paper.

(i)
What happens if the conclusions of Theorems E, F, and G can not stand?
(ii)
Can the value n be further reduced in the above results?

In the paper, we investigate the solutions of the above questions. We improve and generalize the above related results by proving the following theorems.

Theorem 1.4. Let f(z) and g(z) be two nonconstant meromorphic functions, and let n, k be two positive integers with n > 3k + 11. If Θ(∞, f) > 2/n, , and share 1 (1,2), then f(z) ≡ g(z) or .

Theorem 1.5. Let f(z) and g(z) be two nonconstant meromorphic functions, and let n, k be two positive integers with n > 5k + 14. If Θ(∞, f) > 2/n, and share 1 (1,1), then f(z) ≡ g(z) or .

2. Some Lemmas

For the proof of our result, we need the following lemmas.

Lemma 2.1 (see [1].)Let f be nonconstant meromorphic function, and let a₀, a₁, …, a_n be finite complex numbers such that a_n ≠ 0. Then

(2.1)

Lemma 2.2 (see [1].)Let f(z) be a nonconstant meromorphic function, k a positive integer, and c a nonzero finite complex number. Then

(2.2)

Here N₀(r, 1/f^(k+1)) is the counting function which only counts those points such that f^(k+1) = 0 but f(f^(k) − c) ≠ 0.

Lemma 2.3 (see [11].)Let f be a nonconstant meromorphic function, and let k, p be two positive integers. Then

(2.3)

Clearly

Lemma 2.4 (see [1].)Let f(z) be a transcendental meromorphic function, and let a₁(z), a₂(z) be two meromorphic functions such that T(r, a_i) = S(r, f), i = 1,2. Then

(2.4)

Lemma 2.5. Let f and g be two nonconstant meromorphic functions, and let k(≥1), l(≥1) be two positive integers. Suppose that f^(k) and g^(k) share (1, l).

(i)
If l = 2 and
(2.5)
then either f^(k)g^(k) ≡ 1 or f ≡ g.
(ii)
If l = 1 and
(2.6)
then either f^(k)g^(k) ≡ 1 or f ≡ g.

Proof. Let

(2.7)

Suppose that h≢0.

If z₀ is a common simple 1-point of f^(k) and g^(k), substituting their Taylor series at z₀ into (2.7), we see that z₀ is a zero of h(z). Thus, we have

(2.8)

By our assumptions, h(z) have poles only at zeros of f^(k+1) and g^(k+1) and poles of f and g, and those 1-points of f^(k) and g^(k) whose multiplicities are distinct from the multiplicities of correspond to 1-points of g^(k) and f^(k), respectively. Thus, we deduce from (2.7) that

(2.9)

here N₀(r, 1/f^(k+1)) has the same meaning as in Lemma 2.2.

By Lemma 2.2, we have

(2.10)

Since f^(k) and g^(k) share (1,0), we get

(2.11)

We obtain from (2.8)–(2.11) that

(2.12)

(i)
If l ≥ 2, it is easy to see that
(2.13)

Since

(2.14)

combining with (2.12), (2.13), and (2.14), we obtain

(2.15)

Without loss of generality, we suppose that there exists a set I with infinite measure such that T(r, g) ≤ T(r, f) for r ∈ I:

(2.16)

for r ∈ I and 0 < ε < Δ₁ − (k + 7), that is,

(2.17)

that is, Δ₁ ≤ k + 7, which contradicts hypothesis (2.5).

Therefore, we have h ≡ 0, that is,

(2.18)

By solving this equation, we obtain

(2.19)

where a(≠0), b are two constants. Next, we consider three cases.

Case 1 (b ≠ 0, −1). For more details see the following subcases.

Subcase 1.1 (a − b − 1 ≠ 0). Then, by (2.19), we have .

By Lemma 2.2 and Lemma 2.3, we get

(2.20)

that is, T(r, f) ≤ (k + 8 − Δ₁)T(r, f) + S(r, f). Thus, by (2.5) we deduce that T(r, f) ≤ S(r, f) for r ∈ I, a contradiction.

Subcase 1.2 (a − b − 1 = 0). Then, by (2.19), we have g^(k) = (b + 1)f^(k)/(bf^(k) + 1).

Therefore .

By Lemma 2.2 and Lemma 2.3, we get

(2.21)

that is, T(r, f) ≤ (k + 8 − Δ₁)T(r, f) + S(r, f). Thus, by (2.5) we deduce that T(r, f) ≤ S(r, f) for r ∈ I, a contradiction.

Case 2 (b = −1). For more details see the following subcases.

Subcase 2.1 (a + 1 ≠ 0). Then by (2.19), we get a/((a + 1) − f^(k)) = g^(k). So, we have . We can deduce a contradiction as in Case 1.

Subcase 2.2 (a + 1 = 0). Then by (2.19), we get f^(k)g^(k) ≡ 1.

Case 3 (b = 0). For more details see the following subcases.

Subcase 3.1 (a − 1 ≠ 0). Then by (2.19), we get (f^(k) + a − 1)/a = g^(k). So, we have . We can deduce a contradiction as in Case 1.

Subcase 3.2 (a − 1 = 0). Then by (2.19), we get f^(k) ≡ g^(k). From this, we obtain f = g + P(z), where P(z) is a polynomial, and so T(r, f) = T(r, g) + S(r, f). If P(z)≢0, then, by Lemma 2.4, we get

(2.22)

Thus, by (2.5) we deduce that T(r, f) ≤ S(r, f) for r ∈ I, a contradiction. Therefore, we conclude that P(z) ≡ 0, that is, f ≡ g.

(ii) If l = 1, it is easy to see that

(2.23)

By Lemma 2.3, we can get

(2.24)

Combining with (2.12), (2.24), and (2.14), we obtain

(2.25)

Without loss of generality, we suppose that there exists a set I with infinite measure such that T(r, g) ≤ T(r, f) for r ∈ I:

(2.26)

for r ∈ I and 0 < ε < Δ₂ − (2k + 9), that is,

(2.27)

that is, Δ₂ ≤ 2k+9, which contradicts hypothesis (2.6).

Therefore, we have h ≡ 0, that is,

(2.28)

By solving this equation, we obtain

(2.29)

where a(≠0), b are two constants. Using the argument in (i), we can obtain f^(k)g^(k) ≡ 1 or f ≡ g. We here omit the details.

The proof of Lemma 2.5 is completed.

3. Proof of Theorem 1.4

Proof. Let F(z) = fⁿ(f − 1) and G(z) = gⁿ(g − 1). We have

(3.1)

Since

(3.2)

similarly,

(3.3)

Next, by Lemma 2.1, we have

(3.4)

Similarly

(3.5)

From (3.2)–(3.5), we get

(3.6)

Since n > 3k + 11, we get Δ₁ > k + 7. Considering that F^(k) and G^(k) share (1,2), then, by Lemma 2.5, we deduce that either F^(k)G^(k) ≡ 1 or F ≡ G.

Next, we consider the following two cases.

Case 1. F^(k)G^(k) ≡ 1, that is, .

Case 2. F ≡ G, that is,

(3.7)

Suppose that f≢g; then we consider following two cases.

(i)
Let h = f/g be a constant. Then from (3.7) it follows that h ≠ 1, hⁿ ≠ 1, hⁿ⁺¹ ≠ 1 and g = (1 − hⁿ)/(1 − hⁿ⁺¹) = constant, which leads to contradiction.
(ii)
Let h = f/g be not a constant. Since f≢g, we have h≢1, and hence we deduce that g = (1 − hⁿ)/(1 − hⁿ⁺¹) and g = ((1 − hⁿ)/(1 − hⁿ⁺¹))h = (1 + h + h² + ⋯+hⁿ⁻¹)h/(1 + h + h² + ⋯+hⁿ), where h is a nonconstant meromorphic function. It follows that
(3.8)

On the other hand, by the second fundamental theorem, we get

(3.9)

where α_j(≠1)(j = 1,2, …, n) are distinct roots of the algebraic equation hⁿ⁺¹ = 1.

So we have

(3.10)

which contradicts the assumption that Θ(∞, f) > 2/n. Thus f ≡ g. This completes the proof of Theorem 1.4.

4. Proof of Theorem 1.5

Proof. From (3.2)–(3.5), we get

(4.1)

Since n > 5k + 14, we get Δ₂ > 2k + 9. Considering that F^(k) and G^(k) share (1,2), then, by Lemma 2.5, we deduce that either F^(k)G^(k) ≡ 1 or F ≡ G.

Next, by using the argument in Theorem 1.4, we obtain the conclusion of Theorem 1.5. We here omit the details.

Acknowledgments

This research was supported by the NSF of Key Lab of Geomathematics of Sichuan Province of China. The author wishes to thank the referee for his/her valuable comments and suggestions.

References

1 Hayman W. K., Meromorphic Functions, 1964, Clarendon Press, Oxford, UK, Oxford Mathematical Monographs, 0164038, ZBL0141.07802.
Google Scholar
2 Yang L., Value Distribution Theory, 1993, Springer, Berlin, Germany, 1301781, ZBL1210.03034.
Google Scholar
3 Yi H. X. and Yang C. C., Uniqueness Theory of Meromorphic Functions, 1995, Science Press, Beijing, China.
Google Scholar
4 Fang C.-Y. and Fang M.-L., Uniqueness of meromorphic functions and differential polynomials, Computers & Mathematics with Applications. (2002) 44, no. 5-6, 607–617, 1925805, https://doi.org/10.1016/S0898-1221(02)00175-X, ZBL1035.30018.
10.1016/S0898-1221(02)00175-X
Web of Science® Google Scholar
5 Fang M.-L., Uniqueness and value-sharing of entire functions, Computers & Mathematics with Applications. (2002) 44, no. 5-6, 823–831, 1925824, https://doi.org/10.1016/S0898-1221(02)00194-3, ZBL1035.30017.
10.1016/S0898-1221(02)00194-3
Web of Science® Google Scholar
6 Liu L., Uniqueness of meromorphic functions and differential polynomials, Computers & Mathematics with Applications. (2008) 56, no. 12, 3236–3245, 2474578, https://doi.org/10.1016/j.camwa.2008.09.020, ZBL1165.30360.
10.1016/j.camwa.2008.09.020
Web of Science® Google Scholar
7 Lin W.-C. and Yi H.-X., Uniqueness theorems for meromorphic function, Indian Journal of Pure and Applied Mathematics. (2004) 35, no. 2, 121–132, 2040726, ZBL1056.30031.
Web of Science® Google Scholar
8 Bhoosnurmath S. S. and Dyavanal R. S., Uniqueness and value-sharing of meromorphic functions, Computers & Mathematics with Applications. (2007) 53, no. 8, 1191–1205, 2327673, https://doi.org/10.1016/j.camwa.2006.08.045, ZBL1170.30011.
10.1016/j.camwa.2006.08.045
Web of Science® Google Scholar
9 Lahiri I., Weighted sharing and uniqueness of meromorphic functions, Nagoya Mathematical Journal. (2001) 161, 193–206, 1820218, ZBL1025.30027.
10.1017/S0027763000027215
Web of Science® Google Scholar
10 Xu H.-Y., Yi C.-F., and Cao T.-B., Uniqueness of meromorphic functions and differential polynomials sharing one value with finite weight, Annales Polonici Mathematici. (2009) 95, no. 1, 51–66, 2466013, https://doi.org/10.4064/ap95-1-4, ZBL1169.30010.
10.4064/ap95-1-4
Web of Science® Google Scholar
11 Zhang T. and Lü W., Uniqueness theorems on meromorphic functions sharing one value, Computers & Mathematics with Applications. (2008) 55, no. 12, 2981–2992, 2401445, https://doi.org/10.1016/j.camwa.2007.11.029, ZBL1142.30330.
10.1016/j.camwa.2007.11.029
Web of Science® Google Scholar

Citing Literature

All articles

Uniqueness of Meromorphic Functions and Differential Polynomials

Abstract

1. Introduction and Results

2. Some Lemmas

3. Proof of Theorem 1.4

4. Proof of Theorem 1.5

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Uniqueness of Meromorphic Functions and Differential Polynomials

Abstract

1. Introduction and Results

2. Some Lemmas

3. Proof of Theorem 1.4

4. Proof of Theorem 1.5

Acknowledgments

References

Citing Literature

References

Related

Information