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Unicity of meromorphic functions and their linear differential polynomials
Journal of Inequalities and Applications volume 2014, Article number: 388 (2014)
This paper studies the unicity of meromorphic functions that share an arbitrary nonzero small function with their general linear differential polynomials. The result derived here extends one by Brück in 1996 and others.
In 1929, R Nevanlinna [1–3] proved his celebrated five-value and four-value theorems, which gave birth to the study of the unicity of meromorphic functions in the open complex plane ℂ; in 1977, by considering the unicity of an entire function f and its first order derivative , Rubel and Yang  showed that when they share two distinct, finite values CM, which was generalized to meromorphic functions independently by Gundersen  as well as by Mues and Steinmetz  in 1983. Since these results, there have been many papers on several related problems about f and its derivatives, and we refer the reader to [3, 7–13] and the references therein for more details.
Before we proceed, we spare the reader for a moment and assume the familiarity with the basics of Nevanlinna’s value distribution theory of meromorphic functions in ℂ such as the first and second main theorems, and the common notations such as the characteristic function , the proximity function and the counting functions (with multiplicities) and (without multiplicities); denotes any quantity satisfying as except possibly on a set of finite Lebesgue measure, not necessarily the same at each occurrence.
Let a, f, g be some meromorphic functions on ℂ. a is said to be a small function to f, provided . Given a, a small function to both f and g or some value in , one says that f and g share a CM provided the two meromorphic functions and have the same zeros with counting multiplicities.
Now, we continue our discussion and observe a question that arises naturally: When only f and share one value CM, then what can one have accordingly? Towards this direction, in 1996, Brück  first proved the following result.
Theorem A Let f be a nonconstant entire function with . When f and share a finite, nonzero value a CM, then there is a constant such that
This interesting result was extended by Zhang  in 1998, when he proved, among some related results, the following one.
Theorem B Let f be a nonconstant meromorphic function such that
When f and share a finite, nonzero value a CM, then we have (1).
Theorem C Let f be a nonconstant meromorphic function such that
When f and share a small function () CM, then we have , where c is a constant such that and .
Remark D There are reasons why the condition on the zeros of is posed by Brück : To control the multiple values of f for each finite value, inspired by the exponential function; yet, it is not natural to generalize this condition by the zeros of when as there is no need to control the multiple values of .
Next, we define a linear differential polynomial of order , such as
associated with f, where are small functions of f for . Then we can prove the following theorems, which are the main results of this paper.
Theorem 1 Let f be a nonconstant meromorphic function satisfying (3) with . When f and share a small function () CM, then we have for a constant c with and .
Theorem 2 Under the assumptions of Theorem 1, we further suppose that f, () and , , are all entire, then we have either , or for a constant and must also be a constant as well.
As suggested by one of the referees, the reader may also feel interested in a very recent paper by Al-Khaladi , where in fact some results of Lahiri and Sarkar  were generalized that in turn was deeply related to a paper of Yu . One notices the paper  was not suitably cited in the key reference of  (not the paper  itself), despite the conclusions as well as the methods being very similar.
It is worth to mention that the machinery used in this paper is standard Nevanlinna’s value distribution theory while the methods involved are combinations of those already applied in Al-Khaladi [16, 17] as well as in Han and Yi .
2 Proofs of the main results
This section is devoted to the detailed proofs of Theorems 1 and 2. We first define the following auxiliary function α as
Here and hereafter, we use ℒ for for brevity. Our hypotheses imply α is such a meromorphic function that yet .
Case I. a is a finite, nonzero constant.
Rewrite (5) as and differentiate it to yield
where we introduced two meromorphic functions and .
Notice that . We get easily
Substitute from the top line to the bottom line in (7) to derive
That is, for the meromorphic function , one has
Now, when , then we see α is a constant, so that for a constant . As a consequence, we assume in the following that .
Next, if , then ; using , one derives
Here, is a constant. Hence, it follows that , which further yields . On the other hand, combining (6) and (10) provides us with the following identity:
If , then for a constant ; yet, since now , we have via an easy computation, so that . Rewrite it as
This then implies that . Consequently, using the first main theorem, one has and thus from (10). This is a contradiction. If , then from (11) one observes
An easy calculation says , so that and thus by (10). This is a contradiction again.
All these foregoing discussions imply that . So, from (9), we have
Keeping in mind , we have and thus
Reset and note that the zeros of α come from the poles of f. When with multiplicity , then and with multiplicities at least and , respectively; using (6), one observes with multiplicity at least since ; thus, recalling , we know with multiplicity at least . Thereby, with multiplicity at least . As a result, it follows that
When , then either (from in fact) or with multiplicity 2 of , or with multiplicity 1 of . In view of (15) for the case where , we can thus conclude that
Here, , , and , are the counting functions of the zeros of with multiplicities ≤k and , respectively.
Altogether with (14), (15), and (16), we obtain
which further implies that, as and ,
This obviously contradicts our assumption. As a result, when a is a finite, nonzero constant, it follows that with a constant .
Case II. a is a nonconstant, small function.
Like before, rewrite (5) as and differentiate it to yield
When , then we have . Recall that the zeros of α come from the poles of f. As a consequence, one deduces that
If , then we have, noticing that ,
Note when , then we have either (from actually) or - with multiplicity k of , or - with multiplicity of . Thus, we can derive that, similar to (16),
Apply the second main theorem for three small functions to to yield
which together with (20) yields (17) again. This gives a contradiction.
Thus, . That is, . Using (18), one derives
so that ; in other words, for a constant c with . We thus conclude that and .
Finally, let us prove the associated Theorem 2. When f, a, and () are all entire functions, we have for some entire function . If a is constant, then we have the same result as in Case I: with a constant ; if a is nonconstant, then admits many zeros, so that a cannot be entire unless , which provides us with the case .
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Both authors are greatly indebted to the anonymous referees.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Wang, C., Han, Q. Unicity of meromorphic functions and their linear differential polynomials. J Inequal Appl 2014, 388 (2014). https://doi.org/10.1186/1029-242X-2014-388
- Nevanlinna theory
- meromorphic functions
- small functions
- entire functions
- differential polynomials