Abstract
This paper is devoted to the uniqueness problem on meromorphic functions whose differential polynomials share one nonzero finite constant. We improve some previous results and answer two open problems posed by Dyavanal.
Similar content being viewed by others
1 Introduction
We assume that the reader is familiar with the usual notation and basic results of the Nevanlinna theory [4, 10]. Let \(f(z)\) and \(g(z)\) be two nonconstant meromorphic functions, and let a be a complex number. We say that \(f(z)\) and \(g(z)\) share a CM (IM) provided that \(f(z)-a\) and \(g(z)-a\) have the same zeros counting multiplicity (ignoring multiplicity). In addition, f and g sharing ∞ CM (IM) means that f and g have the same poles counting multiplicity (ignoring multiplicity).
The uniqueness theory of meromorphic functions mainly studies conditions under which there is a unique function satisfying the given hypothesis. A great deal of classical results in this field can be seen in [10], where Chap. 9 introduces many works dealing with the relation between two meromorphic functions while their derivatives share values. Over past two decades, the research on the derivatives of polynomials of meromorphic functions sharing values has been ongoing. In 1996, Fang and Hua [3] investigated the relation between two transcendental entire functions f and g when \(f^{n}f'\) and \(g^{n}g'\) share 1 CM. Clearly, \((f^{n+1})'=(n+1)f^{n}f'\). Later, Yang and Hua [9] considered this problem for meromorphic functions f and g, and they proved the following theorem.
Theorem A
Let f and g be two nonconstant meromorphic functions, let \(n\geq11\) be an integer, and let \(a\in\mathbb{C}\setminus\{0\}\). If \(f^{n}f'\) and \(g^{n}g'\) share a CM, then \(f(z)\equiv dg(z)\) for some \((n+1)\)th roots of unity d, or \(g(z)=c_{1}e^{cz}\) and \(f(z)=c_{2}e^{-cz}\), where c, \(c_{1}\), \(c_{2}\) are constants satisfying \((c_{1}c_{2})^{n+1}c^{2}=-a^{2}\).
Without loss of generality, in Theorem A the complex number a can be replaced by 1. Noting that \((\frac{1}{n+2}f^{n+2}-\frac {1}{n+1}f^{n+1})'=f^{n}(f-1)f'\), Fang and Hong [2] obtained the following result.
Theorem B
Let f and g be two transcendental entire functions, let \(n\geq11\) be an integer. If \(f^{n}(f-1)f'\) and \(g^{n}(g-1)g'\) share 1 CM, then \(f(z)\equiv g(z)\).
Three years later, Lin and Yi [6] improved their result to \(n\geq7\) and also studied the case that f and g are meromorphic functions. Moreover, they discussed the other polynomial \(\frac{1}{n+3}f^{n+3}-\frac{2}{n+2}f^{n+2}+\frac{1}{n+1}f^{n+1}\) of f with its derivative as \(f^{n}(f-1)^{2}f'\). In fact, Lin and Yi proved the following two theorems.
Theorem C
Let f and g be two nonconstant meromorphic functions, and let \(n\geq12\) be an integer. If \(f^{n}(f-1)f'\) and \(g^{n}(g-1)g'\) share 1 CM, then
where h is a nonconstant meromorphic function.
Theorem D
Let f and g be two nonconstant meromorphic functions, and let \(n\geq13\) be an integer. If \(f^{n}(f-1)^{2}f'\) and \(g^{n}(g-1)^{2}g'\) share 1 CM, then \(f(z)\equiv g(z)\).
Recently, by introducing the notion of multiplicity, Dyavanal [1] deeply investigated such a uniqueness problem and improved Theorems A, C, and D as follows.
Theorem E
Let f and g be two nonconstant meromorphic functions with zeros and poles of multiplicities at least s, where s is a positive integer. Let \(n\geq2\) be an integer satisfying \((n+1)s\geq12\). If \(f^{n}f'\) and \(g^{n}g'\) share 1 CM, then \(f(z)\equiv dg(z)\) for some \((n+1)\)th roots of unity d, or \(g(z)=c_{1}e^{cz}\) and \(f(z)=c_{2}e^{-cz}\), where c, \(c_{1}\), \(c_{2}\) are constants satisfying \((c_{1}c_{2})^{n+1}c^{2}=-1\).
Theorem F
Let f and g be two nonconstant meromorphic functions with zeros and poles of multiplicities at least s, where s is a positive integer. Let n be an integer satisfying \((n-2)s\geq10\). If \(f^{n}(f-1)f'\) and \(g^{n}(g-1)g'\) share 1 CM, then (1.1) holds.
Theorem G
Let f and g be two nonconstant meromorphic functions with zeros and poles of multiplicities at least s, where s is a positive integer. Let n be an integer satisfying \((n-3)s\geq10\). If \(f^{n}(f-1)^{2}f'\) and \(g^{n}(g-1)^{2}g'\) share 1 CM, then \(f(z)\equiv g(z)\).
In Theorem F, if \(f(z)\not\equiv g(z)\), then f, g must satisfy (1.1), so that
where \(\alpha_{i}\) (≠1) (\(i=1,2,\ldots,n+1\)) and \(\beta_{j}\) (≠1) (\(j=1,2,\ldots,n\)) are distinct roots of \(w^{n+2}=1\) and \(w^{n+1}=1\), respectively. Thus by Valiron–Mokhon’ko theorem (see [10, Thm. 1.13]) \(T(r,f)=(n+1)T(r,h)+S(r,h)\). From (1.2) it follows that the poles of h are not poles of f and
By the second main theorem we have
which leads to \(n\leq(n+1)/s\). From \((n-2)s\geq10\) we have \(n\geq 3\). According to the above argument, we can deduce a contradiction for \(s\geq2\). Therefore, in Theorem F, if \(s\geq2\), then we must have \(f\equiv g\).
In the end of his paper, Dyavanal posed four open problems. Two of them, which we are interested in, are as follows.
Problem 1
Can a CM shared value be replaced by an IM shared value in Theorems E–G?
Problem 2
Are the conditions \((n+1)s\geq12\) in Theorem E, \((n-2)s\geq10\) in Theorem F, and \((n-3)s\geq10\) in Theorem G sharp?
In this paper, we try to answer these two questions. We obtain five theorems, which replace CM by IM in Theorems E–G and reduce n for \(s\geq7\) in Theorems F–G in Sect. 3.
2 Preliminary lemmas
We denote by \(\overline{N}_{(k}(r,\frac{1}{f-a})\) the reduced counting function for zeros of \(f-a\) with multiplicity no less than k. Define
Lemma 2.1
(see [12, Lemma 2.1])
Let \(f(z)\) be a nonconstant meromorphic function, and let p and k be positive integers. Then
This lemma can be proved in the same way as [5, Lemma 2.3] in the particular case \(p=2\).
Lemma 2.2
Let f and g be two nonconstant meromorphic functions sharing 1 CM. Then we have one of the following three cases:
-
(i)
\(T(r,f)\leq N_{2}(r,1/f)+N_{2}(r,1/g)+N_{2}(r,f)+N_{2}(r,g)+S(r,f)+S(r,g)\);
-
(ii)
\(f(z)\equiv g(z)\);
-
(iii)
\(f(z)g(z)\equiv1\).
Lemma 2.3
Let f and g be two nonconstant meromorphic functions. If f and g share 1 IM, then we have one of the following three cases:
-
(i)
\(T(r,f)\leq N_{2}(r,1/f)+N_{2}(r,1/g)+N_{2}(r,f)+N_{2}(r,g)+2\overline {N}(r,f)+\overline{N}(r,g)+2\overline{N}(r,1/f)+ \overline {N}(r,1/g)+S(r,f)+S(r,g)\);
-
(ii)
\(f(z)\equiv g(z)\);
-
(iii)
\(f(z)g(z)\equiv1\).
Proof
We first introduce some new notation. Let \(z_{0}\) be a zero of \(f-1\) with multiplicity p and a zero of \(g-1\) with multiplicity q. We denote by \(N_{E}^{1)}(r,\frac{1}{f-1})\) the counting function of the zeros of \(f-1\) with \(p=q=1\), by \(\overline{N}_{E}^{(2}(r,\frac{1}{f-1})\) the counting function of the zeros of \(f-1\) satisfying \(p=q\geq2\), and by \(\overline{N}_{L}(r,\frac{1}{f-1})\) the counting function of the zeros of \(f-1\) with \(p>q\geq1\), where each point in these counting functions is counted only once.
We set
Suppose that \(H(z)\not\equiv0\). Clearly, \(m(r,H)=S(r,f)+S(r,g)\). If \(z_{0}\) is a common simple zero of \(f-1\) and \(g-1\), then a simple computation on local expansions shows that \(H(z_{0})=0\), and then
The poles of \(H(z)\) only come from the zeros of \(f'\) and \(g'\), the multiple poles of f and g, and the zeros of \(f-1\) and \(g-1\) with different multiplicity. By analysis we can deduce that
where \(N_{0}(r,\frac{1}{f'})\) denotes the counting function of the zeros of \(f'\) but not that of \(f(f-1)\), \(\overline{N}_{0}(r,\frac{1}{f'})\) denotes the corresponding reduced counting function, and \(N_{0}(r,\frac{1}{g'})\) and \(\overline{N}_{0}(r,\frac{1}{g'})\) are defined similarly. At the same time, obviously,
Combining this with (2.3) and (2.4) yields
Since
combining this with (2.5), we have
We apply the second fundamental theorem to f and g and consider the above inequality. Then
Clearly, this leads to
By Lemma 2.1 we have
Then, using this inequality, we get
where \(N_{1)}(r,\frac{1}{f})\) denotes the counting function of simple zeros of f. Similarly, we obtain
Substituting (2.7) and (2.8) into (2.6), this yields Case (i).
It remains to treat the case \(H(z)\equiv 0\). Integrating twice results in
where \(A\neq0\) and B are two constants. If now \(B\neq0,-1\), then we rewrite (2.9) as
and then
By the second fundamental theorem we obtain
which leads to Case (i). A similar reasoning results in Case (i) again, unless either \(A=1\) and \(B=0\) or \(A=-1\) and \(B=-1\). Hence, if \(A=1\) and \(B=0\), then \(f\equiv g\), that is, Case (ii). If \(A=-1\) and \(B=-1\), then \(f\cdot g\equiv1\), which is Case (iii). □
When meromorphic functions \(f_{1}\) and \(f_{2}\) share 1 IM, Sun and Xu [8] once obtained a result, whose proof can be also found in [7]. They proved that \(f_{1}\equiv f_{2}\) or \(f_{1}f_{2}\equiv1\) if
where E is a set of finite linear measure. By Lemma 2.3, when
Case (i) cannot happen, and thus \(f_{1}\equiv f_{2}\) or \(f_{1}f_{2}\equiv1\). Since \(N_{2}(r,f)\leq2\overline{N}(r,f)\) and \(N_{2}(r,1/f)\leq 2\overline{N}(r,1/f)\), Lemma 2.3 is an improvement of Sun and Xu’s result.
3 Main results
Based on Problems 1 and 2 in Sect. 1, we introduce our main results.
Theorem 3.1
Let f and g be two nonconstant meromorphic functions with multiplicities of zeros and poles no less than s, where s is a positive integer. Let \(n\geq2\) be an integer satisfying \((n-4)s\geq19\) for \(s=1,2\) and \(ns\geq28\) for \(s\geq 3\). If \(f^{n}f'\) and \(g^{n}g'\) share 1 IM, then \(f(z)\equiv dg(z)\) for some \((n+1)\)th root d of unity, or \(g(z)=c_{1}e^{cz}\) and \(f(z)=c_{2}e^{-cz}\), where c, \(c_{1}\), \(c_{2}\) are constants satisfying \((c_{1}c_{2})^{n+1}c^{2}=-1\).
Proof
Let \(F=\frac{1}{n+1}f^{n+1}\) and \(G=\frac{1}{n+1}g^{n+1}\). Then \(T(r,F)=(n+1)T(r,f)\), \(T(r,G)=(n+1)T(r,g)\), and \(F'\), \(G'\) share 1 IM. Suppose first that Case (i) of Lemma 2.3 holds. From this we have
At the same time, we have
Then from this inequality and (3.1) it follows that
Using Lemma 2.1, we get
Then substituting (3.3) and (3.4) into (3.2) yields
A similar inequality for G also holds. Therefore we can conclude that
which contradicts the condition \((n-4)s\geq19\) for \(s=1,2\).
Again using Lemma 2.1, we have
Then substituting the two inequalities into (3.2) leads to
Similarly, we can get
which contradicts the condition \(ns\geq28\) for \(s\geq3\).
Thus by Lemma 2.3 there we must have \(F'G'\equiv1\) or \(F'\equiv G'\). Consider case \(F'G'\equiv1\), that is \(f^{n}f'g^{n}g'\equiv1\). Suppose that f has a pole \(z_{0}\) with multiplicity p. Then \(z_{0}\) must be a zero of g of order q satisfying \(nq+q-1=np+p+1\). We rewrite it as \((q-p)(n+1)=2\), which is a contradiction since \(n\geq2\). Similarly to [9], we get \(g=c_{1}e^{cz}\), \(f=c_{2}e^{-cz}\). For the case \(F'\equiv G'\), it is easy to see that \(F\equiv G+c\), where c is a constant, so that \(T(r,f)=T(r,g)+S(r,g)\). If \(c\neq0\), then
Applying the second main theorem to G, we have
which leads to \((n+1)s\leq3\). This contradicts the condition on n and s. Therefore, \(c=0\), and thus \(F\equiv G\), that is, \(f^{n+1}=g^{n+1}\). Hence \(f\equiv dg\) for some \((n+1)\)th root d of unity. □
Theorem 3.2
Let f and g be two nonconstant meromorphic functions with multiplicities of zeros and poles no less than s. Suppose that \(f^{n}(f-1)f'\) and \(g^{n}(g-1)g'\) share 1 IM, where s and n are positive integers. Then we have one of the following two cases:
-
(i)
if \(s=1\) and \(n\geq27\), then \(f(z)\equiv g(z)\), or we have (1.1);
-
(ii)
if \((n-8)s\geq19\) for \(s=2\) and \((n-4)s\geq28\) for \(s\geq5\), then \(f(z)\equiv g(z)\).
Proof
Let \(F=\frac{1}{n+2}f^{n+2}-\frac{1}{n+1}f^{n+1}\) and \(G=\frac {1}{n+2}g^{n+2}-\frac{1}{n+1}g^{n+1} \). Then \(F'\) and \(G'\) share 1 IM, and by the Valiron–Mokhon’ko theorem we have
Suppose now that Case (i) of Lemma 2.3 holds. Then we have
Since \(T(r,F)\leq T(r,F')+N(r,1/F)-N(r,1/F')+S(r,f)\), we get
Combining this inequality with (3.10) leads to
If we use (3.3) and (3.4), then (3.11) means
Then this yields
which contradicts to \((n-8)s\geq19\) for \(s=1,2\). If we use (3.6) and (3.7), then (3.11) implies
Similarly as before, we conclude that
which contradicts with \((n-4)s\geq28\) when \(s\geq3\).
Thus, by Lemma 2.3, \(F'G'\equiv1\) or \(F'\equiv G'\). Consider the case \(F'G'\equiv1\), that is,
Let \(z_{0}\) be a zero of f with multiplicity \(p_{0}\). Then \(z_{0}\) must be a pole of g of order \(q_{0}\) satisfying
We rewrite it as \((n+1)(p_{0}-q_{0})=q_{0}+2\), which implies \(p_{0}\geq q_{0}+1\) and \(q_{0}+2\geq n+1\), so that \(p_{0}\geq t=\max\{n, s+1\}\). Let \(z_{1}\) be a zero of \(f-1\) with multiplicity \(p_{1}\). Then by (3.12) \(z_{1}\) must be a pole of g of order \(q_{1}\) satisfying
Rewrite it as \(p_{1}=1+(n+2)q_{1}/2\), so that \(p_{1}\geq1+(n+2)s/2\). Again from (3.12) we have
By the second main theorem we obtain
and a similar inequality for \(T(r,g)\). Combining the two inequalities, we get
Since \((n-8)s\geq19\) for \(s=1,2\) and \((n-4)s\geq28\) for \(s\geq3\), we have
Thus (3.14) leads to a contradiction. Similarly as in the proof of Theorem 3.1, \(F'\equiv G'\) means that \(F\equiv G\). Let \(h\equiv f/g\). If \(h\not\equiv1\), then \(F\equiv G\) implies (1.1). As we pointed out in Sect. 1, (1.1) leads to a contradiction for \(s\geq2\). Hence, when \(s\geq2\), we must have \(f(z)\equiv g(z)\). For \(s=1\), \(f(z)\equiv g(z)\), or (1.1) holds. □
Theorem 3.3
Let f and g be two nonconstant meromorphic functions with multiplicities of zeros and poles no less than s. Suppose that \(f^{n}(f-1)^{2}f'\) and \(g^{n}(g-1)^{2}g'\) share 1 IM, where s and n are positive integers. If \((n-9)s\geq19\) for \(s=1,2\) and \((n-5)s\geq28\) for \(s\geq3\), then \(f(z)\equiv g(z)\).
Proof
Let \(F=\frac{1}{n+3}f^{n+3}-\frac{2}{n+2}f^{n+2}+\frac{1}{n+1}f^{n+1}\) and \(G=\frac{1}{n+3}g^{n+3}-\frac{2}{n+1}g^{n+2}+\frac{1}{n+1}g^{n+1}\). Then \(F'\) and \(G'\) share 1 IM, and
Suppose now that Case (i) of Lemma 2.3 holds. Then we have
Consider \(T(r,F)\leq T(r,F')+N(r,1/F)-N(r,1/F')+S(r,f)\). Then we obtain
where \(a_{1}\) and \(a_{2}\) are distinct solutions of the equation \(\frac{1}{n+3}w^{2}-\frac{2}{n+2}w+\frac{1}{n+1}=0\). Combining this with (3.16), we get
By (3.3) and (3.4) from (3.17) it follows that
This implies
which is a contradiction unless \((n-9)s\geq19\). For \(s\geq3\), we use (3.6) and (3.7), and (3.17) leads to
Similarly as before, we can conclude that
which contradicts to \((n-5)s\geq28\). Thus, by Lemma 2.3, \(F'G'\equiv1\) or \(F'\equiv G'\). As in the proof Theorem 3.2, the case \(F'G'\equiv1\) leads to a contradiction, so we obtain that \(F\equiv G\). Let \(h\equiv f/g\). Then, similarly as in the proof of Theorem G, we only get \(h\equiv1\). Hence \(f(z)\equiv g(z)\). □
Given specific values of s in Theorems 3.1–3.3, we can compare n in the two conditions of n and s and see that the second condition is always better than the first one for \(s\geq3\). For example, we consider \((n-4)s\geq19\): if \(s=3\), then \(n\geq11\); if \(s=4\), then \(n\geq9\); if \(s=5,6 \), then \(n\geq8\); if \(s=7,8,9\), then \(n\geq7\); if \(10\leq s\leq18\), then \(n\geq6\); and if \(s\geq19\), then \(n\geq5\). For the condition \(ns\geq28\), if \(s=3\), then \(n\geq10\); if \(s=4\), then \(n\geq7\); if \(s=5,6\), then \(n\geq5\); if \(s=7,8\), then \(n\geq4\); if \(s=9,10\), then \(n\geq3\); and if \(11\leq s\leq18\), then \(n\geq2\).
Theorem 3.4
Let f and g be two nonconstant meromorphic functions with multiplicities of zeros and poles no less than s, where s (≥7) is a positive integer. Let n be an integer satisfying \((n-1)s\geq13\). If \(f^{n}(f-1)f'\) and \(g^{n}(g-1)g'\) share 1 CM, then \(f(z)\equiv g(z)\).
Proof
Let \(F=\frac{1}{n+2}f^{n+2}-\frac{1}{n+1}f^{n+1}\) and \(G=\frac {1}{n+2}g^{n+2}-\frac{1}{n+1}g^{n+1} \). Then \(F'\) and \(G'\) share 1 CM, and (3.9) holds. Suppose now that Case (i) of Lemma 2.2 holds. Then
From (3.18) we get
where by Lemma 2.1 for \(N_{2}(r,1/g')\), we use
There also exists a similar inequality for \(T(r,G)\). Therefore we have
which contradicts to \((n-1)s\geq13\). Thus, by Lemma 2.2, \(F'G'\equiv1\) or \(F'\equiv G'\). Then, as in the proof of Theorem 3.2, we can deduce \(f(z)\equiv g(z)\). □
Theorem 3.5
Let f and g be two nonconstant meromorphic functions with multiplicities of zeros and poles no less than s, where s (≥7) is a positive integer. Let n be an integer satisfying \((n-2)s\geq13\). If \(f^{n}(f-1)^{2}f'\) and \(g^{n}(g-1)^{2}g'\) share 1 CM, then \(f(z)\equiv g(z)\).
Proof
Let \(F=\frac{1}{n+3}f^{n+3}-\frac{2}{n+2}f^{n+2}+\frac{1}{n+1}f^{n+1}\) and \(G=\frac{1}{n+3}g^{n+3}-\frac{2}{n+1}g^{n+2}+\frac{1}{n+1}g^{n+1}\). Then \(F'\) and \(G'\) share 1 CM, and (3.15) holds. Suppose now that Case (i) of Lemma 2.2 holds. Proceeding as in the proof of Theorem 3.4, we have
Then we obtain
where we use the inequality \(N_{2}(r,1/g')\leq(4/s)T(r,g)+S(r,g)\). Similarly as before, we get
which contradicts to \((n-2)s\geq13\). Thus by Lemma 2.2, \(F'G'\equiv1\) or \(F'\equiv G'\). As in the proof Theorem 3.3, we must have \(f(z)\equiv g(z)\). □
By giving specific values for \(s\geq7\) it is easy to see that the condition \((n-1)s\geq13\) in Theorem 3.4 and \((n-2)s\geq13\) in Theorem 3.5 are sharper than the condition \((n-2)s\geq10\) in Theorem F and \((n-3)s\geq10\) in Theorem G, respectively.
For further study of related problems, we would like to pose the following question.
Open question
Let n, k be positive integers, and let m be a nonnegative integer. Suppose that \(f^{n}(f-1)^{m}f^{(k)}\) and \(g^{n}(g-1)^{m}g^{(k)}\) share a CM (or IM), where a (\(\not\equiv0,\infty \)) is a small function of f and g. Under what conditions can we get \(f\equiv g\)?
4 Conclusions
Using the notion of multiplicity, in this paper, we provide five results, which extend the main results that were derived in the paper [1] and answer two open problems posed by Dyavanal in the same paper. Obtaining our results from more general hypotheses without complicated calculations is probably the most interesting feature of this paper. Finally, in this paper, we pose one more general open question for further studies.
References
Dyavanal, R.S.: Uniqueness and value-sharing of differential polynomials of meromorphic functions. J. Math. Anal. Appl. 374, 335–345 (2011)
Fang, M.L., Hong, W.: A unicity theorem for entire functions concerning differential polynomials. Indian J. Pure Appl. Math. 32, 1343–1348 (2001)
Fang, M.L., Hua, X.H.: Entire functions that share one value. J. Nanjing Univ. Math. Biq. 13(1), 44–48 (1996)
Hayman, W.: Meromorphic Functions. Clarendon, Oxford (1964)
Lahiri, I., Sarkar, A.: Uniqueness of a meromorphic function and its derivative. J. Inequal. Pure Appl. Math. 5(1), Art. 20 (2004)
Lin, W.C., Yi, H.X.: Uniqueness theorems for meromorphic functions. Indian J. Pure Appl. Math. 35, 121–132 (2004)
Liu, K.: Meromorphic functions sharing a set with applications to difference equations. J. Math. Anal. Appl. 359, 384–393 (2009)
Sun, F.S., Xu, Y.: Shared values for several meromorphic functions. J. Nanjing Norm. Univ. 23, 18–21 (2000) (Chinese)
Yang, C.C., Hua, X.H.: Uniqueness and value-sharing of meromorphic functions. Ann. Acad. Sci. Fenn., Math. 22(2), 395–406 (1997)
Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht (2003)
Yi, H.X.: Meromorphic functions that share one or two values. Complex Var. Theory Appl. 28, 1–11 (1995)
Zhang, Q.C.: Meromorphic function that shares one small function with its derivative. J. Inequal. Pure Appl. Math. 6(4), Art. 116 (2005)
Acknowledgements
The authors would like to thank Professors Weiran Lü and Jun Wang for giving enthusiastic help.
Funding
This work was supported by the National Natural Science Foundation of China (No. 11602305) and the Fundamental Research Funds for the Central Universities (No. 18CX02045A).
Author information
Authors and Affiliations
Contributions
Both authors contributed in drafting this manuscript. Both authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zhang, F., Wu, L. Notes on the uniqueness of meromorphic functions concerning differential polynomials. J Inequal Appl 2019, 66 (2019). https://doi.org/10.1186/s13660-019-2010-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-019-2010-1