Some results about functions that share functions with their derivative of higher order
© Qi et al.; licensee Springer 2013
Received: 7 January 2013
Accepted: 29 May 2013
Published: 1 July 2013
In this paper, we investigate the growth of some functions that share functions with their derivative of higher order. The first main theorem is an improvement of the result obtained by Lü (Bull. Korean Math. Soc. 48: 951-957, 2011), two examples are given to show that the conclusion is sharp. The second main theorem is of estimating, more exactly, the order of an entire function sharing polynomial, which extends the related result of Lü, Xu and Chen (Arch. Math. 92: 593-601, 2009).
1 Introduction and main results
In addition, we say that two meromorphic functions and share a finite value a IM (ignoring multiplicities) when and have the same zeros. And we say that and share a finite value a CM (counting multiplicities) when and have the same zeros counting multiplicities. denotes the degree of the polynomial . If is a rational function (where and , are two coprime polynomials), then we indicate to denote the degree of the rational function.
The subject on sharing values between entire functions and their derivatives was first studied by Rubel and Yang . In 1977, they proved the result that if a nonconstant entire function f and its first derivative share two distinct finite numbers a, b CM, then . Since then, shared value problems have been studied by many authors and a number of profound results have been obtained (see, e.g., [3, 4]).
In 1982, Bank and Laine  investigated the complex oscillation theory of differential equations and obtained the following main result.
Theorem A Let be a nonconstant polynomial of degree n, and let and be two linearly independent solutions of the equation . Then at least one of and has the property that the exponent of convergence of its zero-sequence is .
Since then, to study properties of the exponent of convergence, the order and the hyper-order for the solutions of some differential equations becomes a hot topic and is discussed by many experts.
In 2008, Li and Gao  deduced the following result.
then , where, and in the sequel, n denotes the degree of P.
Recently, Lü  obtained the result.
Theorem C Let f be a transcendental meromorphic function with finitely many poles, let be an integer, and let () be an entire function such that the order of α is less than that of f, where P, Q are two polynomials. If and share α CM, then , where A is a nonzero constant.
From Theorem C, we see that and share a function with finite order. So, it is natural to ask what will happen if they share functions with infinite order and also what will happen if is replaced by . In this work, we discuss these problems and derive the following result.
where , then .
Remark 1 The following examples show that the conclusion is sharp.
Remark 2 If γ is a polynomial, then the above condition obviously holds.
Remark 3 In Theorem 1.1, if the order of γ is zero, for example, γ is a polynomial, then .
In 2009, Lü, Xu and Chen  obtained the following result.
then is of finite order.
In this paper, we get the following results which improve Theorem D.
then , where , are rational functions.
Corollary 1.3 Let be a nonconstant entire function and let , () be two polynomials. If and , then , where , are rational functions.
Very recently, Li, Gao and Zhang  proved the following result.
Theorem E Let f be a nonconstant entire function. If f and share the value 1 CM, and if , where , then for some nonzero constant c.
By the same method of Li, Gao and Zhang , we also consider the k th derivative and improve the above result as follows.
Theorem 1.4 Let f be a nonconstant entire function, and let k be a positive integer. If f and share the value 1 CM and if , where , then for some nonzero constant c.
2 Some lemmas
In order to prove our theorems, we need the following lemmas.
Lemma 2.1 
a subsequence of functions (also denoted by );
a sequence of complex numbers , ;
a positive sequence ;
- 4., here g is a nonconstant meromorphic (entire) function satisfying and
here M, n are respective positive numbers.
With a similar method to that in [, Lemma 2], we obtain the following Lemma 2.3, which plays an important part in the proof of Theorem 1.1. For the sake of convenience, the detailed proof will be given after Lemma 2.3.
Lemma 2.2 Let f be a meromorphic function of hyper-order . Then, for any , there exists a sequence such that if n is large enough.
a contradiction. Thus, the proof is completed. □
Lemma 2.3 
Remark 4 There exist some mistakes in Theorem 2.2 and its proof and the proof of Theorem 3.1 in , we will correct them in another paper.
Lemma 2.4 
Lemma 2.5 
Lemma 2.6 
3 The proof of Theorem 1.1
Noting that f and R have at most finitely many poles, there exists a positive number r such that f and R have no poles in . Then f and R are holomorphic in D.
Noting that , we have , and then we just need to prove .
Then all are holomorphic in .
It follows from Marty’s criterion that is not normal at .
for a positive number M.
where A is a positive constant and q is an integer.
Noting that and (3.2), we obtain . It contradicts with the assumption that . So, .
Next, by the famous Hayman inequality for , it is easy to obtain contradiction.
Hence, we complete the proof of Theorem 1.1. □
4 Proof of Theorem 1.2
Let and . Then . We consider the function , obviously,
Then all are holomorphic in and as . It follows from Marty’s criterion that is not normal at .
locally uniformly in ℂ.
Thus , which yields that the zeros of are of multiplicity at least 2. Similarly, we can prove that the zeros of g are of multiplicity at least 2.
so each is a simple zero of , that is, ().
Noting (4.8), (4.10) and that has m zeros () in , we obtain from Hurwitz’s theorem that is a zero of with multiplicity m, and thus . This is a contradiction. Hence .
We have shown that g is a nonvanishing entire function that takes the value 1 always with multiplicity at least 2. But this contradicts Nevanlinna’s second fundamental theorem that the sum of the defects is at most 2.
This completes the proof of Theorem 1.2.
5 The proof of Corollary 1.3
By Lemma 2.6, the reader could give the proof of Corollary 1.3 with almost the same argument as that in the proof of Theorem 1.2. Here we omit it.
6 The proof of Theorem 1.4
By a similar way to that of Li, Gao and Zhang , we prove Theorem 1.4 as follows.
Combining (6.3) with (6.4), we see that .
for large enough r. (6.6) contradicts to (6.5).
This completes the proof of Theorem 1.4.
This work was supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the authors worked as visiting scholars. The authors would like to express their hearty thanks to Chern Institute of Mathematics that provided very comfortable research environment to them. Project supported by the funding scheme for training young teachers in colleges and universities in Shanghai (ZZSDJ12020), also supported by the NNSF of China (No. 11271090, 11171184, 11001057), the NSF of Guangdong Province (S2012010010121) and by project 10XKJ01, 12C401 and 12C104 from the Leading Academic Discipline Project of Shanghai Dianji University. The authors wish to thank the referee and managing editor for their very helpful comments and useful suggestions.
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