Abstract
We use the theory of normal families to prove the following. Let and be two polynomials such that (where is a nonnegative integer) and are two distinct complex numbers. Let be a transcendental entire function. If and share the polynomial and if whenever , then . This result improves a result due to Li and Yi.
1. Introduction and Main Results
Let and be two nonconstant meromorphic functions in the complex plane ,and let be a polynomial or a finite complex number. denotes the degree of the polynomial . To simplify the statement of our results in this paper, deviating from the common definition, we consider the zero polynomial as a polynomial of degree . If whenever , we write If and , we write and say that and share (IM ignoring multiplicity). If and have the same zeros with the same multiplicities, we write and say that and share (CM counting multiplicity) (see, [1]). In addition, we use notations , to denote the order and the hyperorder of , respectively, where It is assumed that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory, as found in [1, 2].
In 1977, Rubel and Yang [3] proved the well-known theorem.
Theorem A. Let and be two complex numbers such that, and letbe a nonconstant entire function. If and, then .
This result has undergone various extensions and improvements (see, [1]).
In 1979, Mues and Steinmetz [4] proved the following result.
Theorem B. Let and be two complex numbers such that , and let be a nonconstant entire function. If and , then .
In 2006, Li and Yi [5] proved the following related result.
Theorem C. Let andbe two complex numbers such that, and letbe a nonconstant entire function. If and , then.
Remark 1.1. Meanwhile, Li and Yi [5] give an example to show that cannot be omitted in Theorem C.
In recent years, there have been several papers dealing with entire functions that share a polynomial with their derivatives.
In 2006, Wang [6] proved the following result.
Theorem D. Letbe a nonconstant entire function, and letbe a polynomial of degreeLetbe an integer. Ifand iffor everywiththen
In 2007, Li and Yi [7] proved the following result.
Theorem E. Letbe a nonconstant entire function of hyperorderand letbe a nonconstant polynomial. If then for some constant .
In 2008, Grahl and Meng [8] proved the following result.
Theorem F. Let be a nonconstant entire function, and letbe a nonconstant polynomial. Letbe an integer. Ifand if for some positivewe havefor everywiththen is constant.
From the ideas of Theorem D to Theorem F, it is natural to ask whether the values , in Theorem C can be replaced by two polynomials , . The main purpose of this paper is to investigate this problem. We prove the following result.
Theorem 1.2. Let and be two polynomials such that (where is a nonnegative integer) and , are two distinct complex numbers. Let be a transcendental entire function. If and , then .
Remark 1.3. The following example shows the hypothesis that is transcendental cannot be omitted in Theorem 1.2.
Example 1.4. Let , and . Then While does not satisfy the result of Theorem 1.2.
Remark 1.5. The case of Theorem 1.2 yields Theorem C.
It seems that we cannot get the result by the methods used in [4, 5]. In order to prove our theorem, we need the following result which is interesting in its own right.
Theorem 1.6. Letand be two polynomials such that (where is a nonnegative integer) and , are two distinct complex numbers. Let be a nonconstant entire function, and and , then is of finite order.
2. Some Lemmas
In order to prove our theorems, we need the following lemmas.
Let be a meromorphic function in . is called a normal function if there exists a positive such that for all , where denotes the spherical derivative of .
Let be a family of meromorphic functions in a domain . We say that is normal in if every sequence contains a subsequence which converges spherically and uniformly on compact subsets of ; see [9].
Normal families, in particular, of holomorphic functions often appear in operator theory on spaces of analytic functions; for example, see in [10, Lemma 3] and in [11, Lemma 4].
Lemma 2.1 (see [12]). Let be a family of analytic functions in the unit disc with the property that for each , all zeros of have multiplicity at least . Suppose that there exists a number such that whenever and . If is not normal in , then for , there exist(1)a number (2)a sequence of complex numbers , ,(3)a sequence of functions , and(4)a sequence of positive numbers such that converges locally and uniformly (with respect to the spherical metric) to a nonconstant analytic function on , and moreover, the zeros of are of multiplicity at least , .
Lemma 2.2 (see [13]). A normal meromorphic function has order at most two. A normal entire function is of exponential type and thus has order at most one.
Lemma 2.3 (see [9, Marty's criterion]). A family of meromorphic functions on a domain is normal if and only if, for each compact subset , there exists a constant such that for each and .
Lemma 2.4 (see [2]). Let be a meromorphic function, and let , , be three distinct meromorphic functions satisfying . Then
Lemma 2.5 (see [5]). Let be a family of functions holomorphic on a domain D, and let and be two finite complex numbers such that . If for each , and , then is normal in D.
3. Proof of Theorem 1.6
If , by , we obtain , , . From the conditions of Theorem 1.6, we obtain and . By Lemmas 2.5 and 2.3 we obtain that is a normal function in D. By Lemma 2.2 we obtain that is a finite order function.
If , by and , we obtain . Now we consider the function and we distinguish two cases.
Case 1. If there exists a constant such that , by Lemmas 2.3 and 2.2, then is of finite order. Hence is of finite order as well.
Case 2. If there does not exist a constant such that , then there exists a sequence such that and for .
Since is a polynomial, there exists an such that Obviously, if , then . Let , and , then is analytic in . Without loss of generality, we may assume for all . We define and Let be fixed; from the above equality, if , then . Noting that , then we obtain the following: if , then
Obviously, are analytic in and as . It follows from Lemma 2.3 that is not normal at .
Therefore, we can apply Lemma 2.1, with . Choosing an appropriate subsequence of if necessary, we may assume that there exist sequences and , such that and and such that the sequence defined by converges locally and uniformly in where is a nonconstant analytic function and . By lemma 2.2, the order of is at most 1.
First, we will prove that on . Suppose that there exists a point such that . Then by Hurwitz's theorem, there exist , as such that for sufficiently large This implies . From the above, we obtain Let , by (3.1), (3.3) and (3.4), it is easy to obtain . Noting that , we have Thus This shows that .
Next we will prove that on . Suppose that there exists a point such that . If , then , where is a constant, together with the fact that gives , which contradicts to the assumptions. Thus . Since as and , by Hurwitz's theorem, there exist as such that for sufficiently large Noting that , from (3.4) and (3.9) (for sufficiently large), we have Since , and , by (3.10), we get which contradicts . This shows that on .
Since is of order at most one, so is , it follows that where are two finite constants. We divide this case into two subcases.Subcase 1. If , from (3.12), we have where is a constant. Since , from (3.13) we have . By a simple calculation, we have , which contradicts .
Subcase 2. If , by we obtain where is a constant. Obviously, has infinitely many solutions. Suppose that there exists a point such that . By (3.14), (3.15), and we get a unique . Which is a contradiction.
Thus is of finite order. This completes the proof of the theorem.
4. Proof of Theorem 1.2
Now we distinguish two cases.
Case 1. If , by , we deduce and . By Theorem C, we obtain .
Case 2. If , by and , we deduce . So is a nonconstant polynomial. By Theorem 1.6, we know that is of finite order. Thus, the hyperorder . Then, by Theorem E, we have where is a nonzero constant. We rewrite it as If , we obtain .
Now, we assume that . Solving (4.2), we obtain where is a nonzero constant, and is a polynomial. Thus, we have Substituting (4.3) and (4.4) into (4.2), we get Next, we will prove that . Suppose that , by (4.4) we obtain
Since is a transcendental entire function and is a polynomial, we deduce . It is well known that and are the Picard values of . By Lemma 2.4, we obtain
By the Nevanlinna First Fundamental Theorem, we immediately obtain If we combine (4.7) and (4.8), we obtain Since , we suppose is a zero of . By the assumption , we have . Substituting into (4.3) and (4.4), we have If , noting that is a polynomial, we have which contradicts with (4.9). Hence, Comparing the above equality to (4.5), we have , a contradiction. Thus, we prove . It is easy to see . By (4.5) we obtain . Finally we deduce . This is a contradiction. So is impossible. This completes the proof of Theorem 1.2.
Acknowledgment
The authors are grateful to the referee for his(her) valuable suggestions and comments. The authors would like to express their hearty thanks to Professor Hongxun Yi for his valuable advice and helpful information. The work was supported by the NNSF of China (no. 10371065), the NSF of Shang-dong Province, China (no. Z2002A01), and the NSFC-RFBR.