Abstract

We will first show that the following differential equation has transcendental entire solutions, where and is an entire function, which improve what we have known and answer an open question raised in the work of Zhang and Yang (2009). And then, the examples are discussed.

1. Introduction and Main Results

In this paper, a meromorphic (resp., entire) function means meromorphic (resp., analytic) in the whole complex plane. We will adopt the standard notations in Nevanlinna's value distribution theory of meromorphic functions, such as the characteristic function , the counting function of the poles , the proximity function , and the reduction counting function (see [1–3]). In addition, denotes any quantity satisfying as , possibly outside a set of of finite linear measure.

Let and be two nonconstant meromorphic functions, . We say that and share the value IM if and have the same zeros. If and have the same zeros with the same multiplicities, we say that they share the value CM. Moreover, if and share 0 CM (resp., IM), we say that and share CM (resp., IM), or we say that and have the same fixed-points CM (resp., IM).

In 2008, Lei et al. [4] proved the follwing.

Theorem A. Suppose that is a nonconstant meromorphic function, , are positive integers, and ,   is a nonzero constant. If and share CM, then , where are nonzero constants such that .

Recently, Zhang and Yang investigated the power of an entire function sharing one value with its derivative and obtained the following results.

Theorem B (see [5]). Let be a nonconstant entire function, and let be an integer. If and share 1 CM, then , and , where is a nonzero constant.

Theorem C (see [6]). Let be a non-constant entire function, and let be a positive integer. If and share 1 CM, and , then , and , where are nonzero constants such that .

According to the above theorems, one may ask an interesting question: what can be said “if and share CM”? Our purpose of this paper is to solve this question by giving the transcendental entire solutions of the equation where , and is an entire function. As an application, we now use a completely different method from that in [4–6] and give the following results.

Theorem 1.1. Let be a positive integer, and be an entire function, then (1.1) has transcendental entire solutions and where c is a nonzero constant.

Obviously, from Theorem 1.1, we have the following result.

Corollary 1.2. Let be a transcendental entire function, and let be a positive integer. If and share CM, then the conclusion of Theorem 1.1 is valid.

Obviously, Corollary 1.2 improves the results in [6] and answers the open question in [7].

In order to illustrate our condition is sharp, we give examples as follows.

Example 1.3. Let , then is a non-constant solution of and , share CM, while .

Example 1.4. Let , then is a non-constant solution of and , share CM, but .

Theorem 1.5. Let be a transcendental entire function, , positive integers, and ; if , share IM, then , where , are nonzero constants and .

2. Preliminaries

In order to prove the theorems above, we need some lemmas.

Lemma 2.1 (see [8, 9]). Suppose that is a meromorphic and transcendental in the whole complex plane and that where and are differential polynomials in with functions of small proximity related to as the coefficients, and the degree of is at most , then

Lemma 2.2 (see [1, 10]). Let be a transcendental meromorphic function, then for every positive integer .

Lemma 2.3 (see [1, 3]). Let be a meromorphic function, and with are three distinct small functions of , then

3. Proof of Theorem 1.1

By differentiation to (1.1), we have Combining (1.1) and (3.1) yields that is, Substituting ,  into (3.3) results in where are differential polynomials in , and the degree of is . Lemma 2.1 gives and We now prove . Suppose that , and , then by (3.6) and Lemma 2.1, we obtain , which is impossible. We may now assume that . So, we get It follows from (3.4) and Lemma 2.2 that , and . These equalities above show that In addition, we can see from the expression of that the multiple zeros of must be the zeros of ; hence, . This together with (3.8) and the first theorem (e.g., see [2, Theorem 1.2]) will result in By (3.7), we find Let be a simple zero of ; it follows from (3.7) and (3.10) that and , which implies that is a zero of . Let Thus, By (3.11), we obtain where , , Substituting (3.13) into (3.7) will yield Also from (3.13), we find where , , It follows from (3.16), (3.13), and (3.10) that From (3.15) and (3.18), we get As noted from and the definitions of and , we can deduce from (3.19) that Suppose that , then , where is a nonzero constant. If , then (1.1) gives , which is the desired result. If , this together with (1.1) will lead to , which contradicts with (3.9). We now assume that , in the same way, (3.20) shows that , which also contradicts with (3.9). Hence, . Then, it follows from (3.4) that and from (3.3) that Obviously, we obtain from (3.21) that By integration, we have , where is a nonzero constant. Substituting this into (1.1) will yield If , by this and (1.1), we have , that is, , where is a nonzero constant.

If , suppose that has infinitely many zeros, and is not a constant, then the zeros of are the zeros of , with multiplicities at least . It follows from the equation above and the second fundamental theorem (or Lemma 2.3) that which with may lead to a contradiction. Now, we suppose that has infinitely many zeros, and is a constant. We suppose that , then we can get a contradiction.

If , suppose that has finitely many zeros, and let where is a polynomial and is an entire function. In this case, (3.23) can be written as

Applying the second fundamental theorem (or Lemma 2.3) to (3.26), we have which is impossible. This also proves Theorem 1.1.

Remark 3.1. By examining the proof of Theorem 1.1 more carefully, here as an extension of Theorem 1.1, we would like to pose the following.

Conjecture 3.2. Let be a transcendental entire function, and let be a positive integer. If and share CM, and , then , where , are nonzero constants such that .

4. Proof of Theorem 1.5

Let , if , Lemma 2.3 implies which contradicts with , and hence , and our desired result follows.

Finally, we conclude the paper with the following.

Question 1. Let be a rational function, and let be an entire function, then for two integers , such that , what can be said about the transcendental entire solution of , can be expressed as for some constants , ?

Question 2. What can be said if the condition in Theorem 1.5 “” is replaced by “”?

Acknowledgment

This works was supported by the Fundamental Research Funds for the Central Universities (no. 10CX04038A).