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Uniqueness of meromorphic functions concerning differential polynomials share one value
Journal of Inequalities and Applications volume 2011, Article number: 133 (2011)
Abstract
In this paper, we study the uniqueness of meromorphic functions whose differential polynomial share a non-zero finite value. The results in this paper improve some results given by Fang (Math. Appl. 44, 828-831, 2002), Banerjee (Int. J. Pure Appl. Math. 48, 41-56, 2008) and Lahiri-Sahoo (Arch. Math. (Brno) 44, 201-210, 2008).
2010 Mathematics Subject Classification: 30D35
1 Introduction and main results
In this paper, by meromorphic functions, we will always mean meromorphic functions in the complex plane. We adopt the standard notations in the Nevanlinna theory of meromorphic functions as explained in [1–3]. It will be convenient to let E denote any set of positive real numbers of finite linear measure, not necessarily the same at each occurrence. For a non-constant meromorphic function h, we denote by T(r, h) the Nevanlinna characteristic of h and by S(r, h) any quantity satisfying S(r, h) = o{T(r, h)}, as r → ∞, r ∉ E.
Let f and g be two non-constant meromorphic functions and let a be a finite complex value. We say that f and g share a CM, provided that f - a and g - a have the same zeros with the same multiplicities. Similarly, we say that f and g share a IM, provided that f - a and g - a have the same zeros ignoring multiplicities. In addition, we say that f and g share ∞ CM, if 1/f and 1/g share 0 CM, and we say that f and g share ∞ IM, if 1/f and 1/g share 0 IM (see [3]). Suppose that f and g share a IM. Throughout this paper, we denote by the reduced counting function of those common a-points of f and g in |z| < r, where the multiplicity of each such a-point of f is greater than that of the corresponding a-point of g, and denote by the counting function for common simple 1-point of both f and g. In addition, we need the following three definitions:
Definition 1.1 Let f be a non-constant meromorphic function, and let p be a positive integer and a ∈ C ∪ {∞}. Then by Np)(r, 1/(f - a)), we denote the counting function of those a-points of f (counted with proper multiplicities) whose multiplicities are not greater than p, by we denote the corresponding reduced counting function (ignoring multiplicities). By N(p(r,1/(f - a)), we denote the counting function of those a-points of f (counted with proper multiplicities) whose multiplicities are not less than p, by we denote the corresponding reduced counting function (ignoring multiplicities), where and what follows, mean , and , respectively, if a = ∞.
Definition 1.2 Let f be a non-constant meromorphic function, and let a be any value in the extended complex plane, and let k be an arbitrary nonnegative integer. We define
where
Remark 1.1. From (1) and (2), we have 0 ≤ δ k (a, f) ≤ δk- 1(a, f) ≤ δ1(a, f) ≤ Θ(a, f) ≤ 1.
Definition 1.3 Let f be a non-constant meromorphic function, and let a be any value in the extended complex plane, and let k be an arbitrary nonnegative integer.
We define
Remark 1.2. From (3), we have 0 ≤ Θ(a, f) ≤ Θk)(a, f) ≤ Θk-1)(a, f) ≤ Θ1)(a, f) ≤ 1.
Definition 1.4 Let k be a positive integer. Let f and g be two non-constant meromorphic functions such that f and g share the value 1 IM. Let z0 be a 1-point of f with multiplicity p, and a 1-point of g with multiplicity q. We denote by the reduced counting function of those 1-points of f and g such that is defined analogously.
It is natural to ask the following question:
Question 1.1 What can be said about the relationship between two meromorphic functions f,g when two differential polynomials, generated by f and g, respectively, share certain values?
Regarding Question 1.1, we first recall the following result by Yang and Hua [4]:
Theorem A. Let f(z) and g(z) be two non-constant meromorphic functions, n ≥ 11 an integer and a ∈ C - {0}. If fnf' and gng' share the value a CM, then either f = tg for a constant t with tn+1= 1 or g(z) = c1eczand f(z) = c2e-cz, where c, c1and c2 are constants satisfying (c1c2)n+1c2 = -a2.
Considering k th derivative instead of 1st derivative Fang [5] proved the following theorems.
Theorem B. Let f(z) and g(z) be two non-constant entire functions, and let n, k be two positive integers with n > 2k + 4. If [fn](k)and [gn](k)share 1 CM, then either f = tg for a constant t with tn= 1 or f(z) = c1eczand g(z) = c2e-cz, where c, c1andc2 are constants satisfying ( -1)k(c1c2)n(nc)2k= 1.
Theorem C. Let f(z) and g(z) be two non-constant entire functions, and let n, k be two positive integers with n ≥ 2k + 8. If [fn(z)(f(z) - 1)](k)and [gn(z)(g(z) - 1)](k)share 1 CM, then f(z) ≡ g(z).
In 2008, Banerjee [6] proved the following theorem.
Theorem D. Let f and g be two transcendental meromorphic functions, and let n, k be two positive integers with n ≥ 9k + 14. Suppose that [fn](k)and [gn](k)share a non-zero constant b IM, then either f = tg for a constant t with tn= 1 or f(z) = c1eczand g(z) = c2e-cz, where c, c1and c2 are constants satisfying ( -1)k(c1c2)n(nc)2k= b2.
Recently, Lahiri and Sahoo [7] proved the following theorem.
Theorem E. Let f and g be two non-constant meromorphic functions, and be a small function of f and g. Let n and m(≥ 2) be two positive integers with n > max{4, 4m + 22 - 5Θ(∞, f) - 5Θ(∞, g) -min[Θ(∞, f), Θ(∞, g)]}. If fn(fm- a)f' and gn(gm- a)g' share α IM for a non-zero constant a, then either f ≡ g or f ≡ -g.
Also, the possibility f ≡ -g does not arise if n and m are both even, both odd or n is even and m is odd.
One may ask, what can be said about the relationship between f and g, if we relax the nature of sharing values of Theorem D and Theorem E ? In this paper, we prove:
Theorem 1.1. Let f(z) and g(z) be two non-constant meromorphic functions, and let n(≥ 1), k(≥ 1) and m(≥ 0) be three integers. Let [fn(f - 1)m](k)and [gn(g - 1)m](k)share the value 1 IM. Then, one of the following holds:
-
(i)
When m = 0 and n > 9k + 14, then either f(z) = c 1eczand g(z) = c 2e-cz, where c,c 1 andc 2 are constants satisfying (-1)k(c 1 c 2)n(nc)2k= 1 or f = tg for a constant t with t n= 1.
-
(ii)
When m = 1, n > 9k + 18 and , then f ≡ g.
-
(iii)
When m ≥ 2, n > 4m + 9k + 14, then f ≡ g or f and g satisfies the algebraic equation R(x, y) = x n(x - 1)m- y n(y - 1)m= 0.
Theorem 1.2. Let f(z) and g(z) be two non-constant meromorphic functions, and let m, n(≥ 2) and k be three positive integers such that n > 4m + 9k + 14. If [fn(fm- a)](k)and [gn(gm- a)](k)share the value 1 IM, where a(≠0) is a finite complex number, then either f ≡ g or f ≡ -g.
The possibility f ≡ -g does not arise if n and m are both odd or if n is even and m is odd or if n is odd and m is even.
Remark 1.3. If m = 0, m = 1, then the cases become Theorem 1.1 (i) (ii).
Theorem 1.3. Let f(z) and g(z) be two non-constant entire functions, and let n(≥ 1), k(≥ 1) and m(≥ 0) be three integers. Let [fn(f - 1)m](k)and [gn(g - 1)m](k)share the value 1 IM. Then, one of the following holds:
-
(i)
When m = 0 and n > 5k + 7, then either f(z) = c 1eczand g(z) = c 2e-cz, where c,c 1 andc 2 are constants satisfying ( -1)k(c 1 c 2)n(nc)2k= 1 or f = tg for a constant t with t n= 1.
-
(ii)
When m ≥ 1, n > 4m + 5k + 7, then f ≡ g or f and g satisfies the algebraic equation R(x, y) = x n(x - 1)m- y n(y - 1)m= 0.
Theorem 1.4. Let f(z) and g(z) be two non-constant entire functions, and let m, n(≥ 1) and k be three positive integers such that n > 4m + 5k + 7. If [fn(fm- a)](k)and [gn(gm- a)](k)share the value 1 IM, where a(≠0) is a finite complex number, then either f ≡ g or f ≡ -g.
The possibility f ≡ -g does not arise if n and m are both odd or if n is even and m is odd or if n is odd and m is even.
Remark 1.4. If m = 0, then the cases becomes Theorem 1.3 (i).
2 Some lemmas
Lemma 2.1. (See [2, 3].) Let f(z) be a non-constant meromorphic function, k a positive integer and let c be a non-zero finite complex number. Then,
where is the counting function, which only counts those points such that f(k+1)= 0 but f(f(k)-c) ≠0
Lemma 2.2. (See [8].) Let f(z) be a non-constant meromorphic function, and let k be a positive integer.
Suppose that , then
Lemma 2.3. (See [9].) Let f(z) be a non-constant meromorphic function, s, k be two positive integers, then
Clearly, .
Lemma 2.4. (See [10].) Let f, g share (1,0). Then
-
(i)
,
-
(ii)
.
Lemma 2.5. Let f(z) and g(z) be two non-constant meromorphic functions such that f(k)and g(k)share 1 IM, where k be a positive integer. If
then either f(k)g(k)≡ 1 or f ≡ g.
Proof. Let
Clearly m(r, Φ) = S(r, f) + S(r, g). We consider the cases and Φ(z) ≡ 0.
Let , then if z0 is a common simple 1-point of f(k)and g(k), substituting their Taylor series at z0 into (5), we see that z0 is a zero of Φ(z). Thus, we have
Our assumptions are that Φ(z) has poles, all simple only at zeros of f(k+1)and g(k+1)and poles of f and g, and 1-points of f whose multiplicities are not equal to the multiplicities of the corresponding 1-points of g. Thus, we deduce from (5) that
here has the same meaning as in Lemma 2.1. From Lemma 2.1, we have
Since
Thus, we deduce from (6)-(9) that
From the definition of , we see that
The above inequality and Lemma 2.2 give
Substituting (11) in (10), we get
According to Lemma 2.3,
Therefore,
similarly,
Combining the above inequality, Lemma 2.4 and (12), we obtain
Without loss of generality, we suppose that there exists a set I with infinite measure such that T(r, f) ≤ T(r, g) for r ∈ I. Hence,
for ∈ I and 0 < ε < Δ - (4k +11)
Therefore, we can get T(r, g) ≤ S(r, g),r ∈ I, by the condition, a contradiction.
Hence, we get Φ(z) ≡ 0. Then, by (5), we have
By integrating two sides of the above equality, we obtain
where a(≠0) and b are constants. We consider the following three cases:
Case 1. b ≠0 and a = b
-
(i)
If b = -1, then from (14), we obtain that f (k) g (k)≡ 1.
-
(ii)
If b ≠-1, then from (14), we get
(15)
From (15), we get
Combing (13) (16) and Lemma 2.1, we have
From (17), we get
By the condition, we get a contradiction.
Case 2. b ≠0 and a ≠b.
-
(i)
If b = -1, then a ≠0, from (14) we obtain
(18)
From (18), we get
From (19) and Lemma 2.1 and in the same manner as in the proof of (17), we get
Using the argument as in case 1, we get a contradiction.
-
(ii)
If b ≠-1, then from (14), we get
(20)
From (20), we get
Using the argument as in case 1, we get a contradiction.
Case 3. b = 0. From (14), we obtain
where p(z) is a polynomial with its degree ≤ k. If , then by second fundamental theorem for small functions, we have
Using the argument as in Case 1, we get a contradiction. Therefore, p(z) ≡ 0. So from (22) and (23), we obtain a = 1 and so f ≡ g. This proves the lemma.
Lemma 2.6. Let f(z) and g(z) be two non-constant entire functions such that f(k)and g(k)share 1 IM, where k be a positive integer. If
then either f(k)g(k)≡ 1 or f ≡ g.
Proof. Since f and g are entire functions, we have and . Proceeding as in the proof of Lemma 2.5, we obtain conclusion of Lemma 2.6.
Lemma 2.7. (See [11].) Let f(z) be a non-constant entire function, and let k(≥ 2) be a positive integer. If f f(k)≠0, then f = eaz+b,where a ≠0, b are constants.
Lemma 2.8. (See [12].) Let f(z) be a non-constant meromorphic function. Let k be a positive integer, and let c be a non-zero finite complex number. Then,
3 Proof of theorems
3.1 Proof of Theorem 1.1
Let F = fn(f - 1)mand G = gn(g - 1)m.
By Lemma 2.8, we have
Similarly, .
Therefore,
If n > 4m + 9k + 14, we obtain Δ > 4k + 11.
So by Lemma 2.5, we get either F(k)G(k)≡ 1 or F ≡ G.
Case 1. F(k)G(k)≡ 1, that is,
Case 1.1 when m = 0, that is,
Next, we prove f ≠0, ∞ and g ≠0, ∞.
Suppose that f has a zero z0 of order p, then z0 is a pole of g of order q. By (26), we get np - k = nq + k, i.e., n(p - q) = 2k, which is impossible since n > 9k + 14.
Therefore, we conclude that f ≠0 and g ≠0.
Similarly, Suppose that f has a pole of order p', then is a zero of g of order q'. By (26), we get np' + k = nq' - k, i.e., n(q' - p') = 2k, which is impossible since n > 9k + 14.
Therefore, we conclude that f ≠∞ oo and g ≠∞.
From (26), we get
From (26)-(27) and Lemma 2.7, we get that f(z) = c1eczand g(z) = c2e-cz, where c, c1 and c2 are three constants satisfying ( -1)k(c1c2)n(nc)2k= 1.
Case 1.2 when m ≥ 1
Let f has a zero z1 of order p1. From (25), we get z1 is a pole of g. Suppose that z1 is a pole of g of order q1. Again by (25), we obtain np1 - k = nq1 + mq1 + k, i.e., n(p1 - q1) = mq1 + 2k, which implies that p1 ≥ q1 + 1 and mq1 + 2k ≥ n. From n > 4m + 9k + 14, we can deduce p1 ≥ 6.
Let f - 1 has a zero z2 of order p2, then z2 is a zero of [fn(f - 1)m](k)of order mp2 - k. Therefore from (25), we obtain z2 is a pole of g of order q2. Again by (25), we obtain mp2 - k = (n + m)q2 + k, i.e., mp2 = (n + m)q2 + 2k, i.e., .
Let z3 be a zero of f' of order p3 that not a zero of f(f - 1), as above, we obtain from (25), p3 - (k - 1) = (n + m)q3 + k, i.e., p3 ≥ n + m + 2k - 1.
Moreover, in the same manner as above, we have similar results for the zeros of [gn(g-1)m](k).
On the other hand, Suppose z4 is a pole of f, from (25), we get z4 is a zero of [gn(g - 1)m](k).
Thus,
We get
From this and the second fundamental theorem, we obtain
Similarly, we have
We can deduce from above
Since n > 4m + 9k + 14, we obtain
i.e., 0.57[T(r, f) + T(r, g)] ≤ S(r, f) + S(r, g),
which is contradiction.
Case 2. F ≡ G, i.e.,
Now we consider following three cases.
Case 2.1 when m = 0, then from (28), we get f = tg for a constant t such that tn= 1.
Case 2.2 when m = 1, then from (28), we have
Suppose . Let be a constant. Then from (29), it follows that h ≠1, hn≠1, hn+1≠1 and , a contradiction. So we suppose h is not a constant. Since , we have .
From (29), we obtain and . Hence, it follows that T(r, f) = nT(r, h) + S(r, f).
By the second fundamental theorem, we have
where a i (≠1) (i = 1, 2,..., n) are distinct roots of the equation hn+1= 1.
So we obtain
which contradicts the assumption , thus f ≡ g.
Case 2.3 when m ≥ 2, then from (28), we obtain
Let , if h is a constant, then substituting f = gh into (30), we deduce
which implies h = 1. Thus, f(z) ≡ g(z). If h is not a constant, then we know by (30) that f and g satisfies the algebraic equation R(f, g) ≡ 0, where .
This completes the proof of Theorem 1.1.
3.2 Proof of Theorem 1.2
Consider F = fn(fm- a), G = gn(gm- a), then F(k)and G(k)share 1 IM.
By Lemma 2.8, we have
and
Similarly, .
Therefore,
Since n > 4m + 9k + 14, we get Δ > 4k + 11, then by Lemma 2.5, we obtain either F(k)G(k)≡ 1 or F ≡ G.
Let F(k)G(k)≡ 1, i.e.,
We can rewrite (31) as
where a1, a2,..., a m are roots of wm- a = 0.
By the similar argument for (32) of case 1.2 of Theorem 1.1, the case F(k)G(k)≡ 1 does not arise.
Let F ≡ G, i.e.,
Obviously, if m and n are both odd or if m is odd and n is even or if m is even and n is odd, then f ≡ - g contradicts F ≡ G. Let and . We put , then and . So from (33), we get .
Since g is non-constant, we see that h is not a constant. Again since gmhas no simple pole, h - h k has no simple zero, where and k = 1, 2,..., n + m - 1. Hence, for k = 1,2,...,n + m - 1, which is impossible.
Therefore either f ≡ g or f ≡ - g.
This completes the proof of Theorem 1.2.
3.3 Proof of Theorem 1.3
Since f and g are entire functions, we have N(r, f) = N(r, g) = 0. Proceeding as in the proof of Theorem 1.1 and applying Lemma 2.6, we obtain that Theorem 1.3 holds.
3.4 Proof of Theorem 1.4
Since f and g are entire functions, we have N(r, f) = N(r, g) = 0. Proceeding as in the proof of Theorem 1.2 and applying Lemma 2.6, we can easily prove Theorem 1.4.
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Acknowledgements
The author want to thanks the referee for his/her thorough review and valuable suggestions toward improved of the paper. This work is supported in part by NSF of China (11071266), in part by NSF project of CQ CSTC (2010BB9218) and in part by ST project of CQEC (KJ110609).
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CW drafted the manuscript and have made outstanding contributions to this paper. CM and JL made suggestions for revision. All authors read and approved the final manuscript.
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Wu, C., Mu, C. & Li, J. Uniqueness of meromorphic functions concerning differential polynomials share one value. J Inequal Appl 2011, 133 (2011). https://doi.org/10.1186/1029-242X-2011-133
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DOI: https://doi.org/10.1186/1029-242X-2011-133