Growth of Meromorphic Solutions of Some -Difference Equations
We estimate the growth of the meromorphic solutions of some complex -difference equations and investigate the convergence exponents of fixed points and zeros of the transcendental solutions of the second order -difference equation. We also obtain a theorem about the -difference equation mixing with difference.
1. Introduction and Main Results
In this paper, we mainly use the basic notation of Nevanlinna Theory, such as , , and , and the notation is defined to be any quantity satisfying as possibly outside a set of of finite linear measure (see [1–3]). In addition, we use the notation to denote the order of growth of the meromorphic function and to denote the exponent of convergence of the zeros. We also use the notation to denote the exponent of convergence of fixed points of . We give the definition of as following.
Definition 1. Let be a nonconstant meromorphic function. The exponent of convergence of fixed points of is defined by
Recently, a number of papers focused on complex difference equations, such as [4–6] and on difference analogues of Nevanlinna’s theory, such as [7, 8]. Correspondingly, there are many papers focused on the -difference (or -difference) equations, such as [9–14].
Because of the intimate relations between iteration theory and the functional equations of Schröder, Böttcher and Abel and Bergweiler et al.  studied the following functional equation where is a complex number and , and are rational functions, and . They obtained the following two theorems.
Theorem A. All meromorphic solutions of (2) satisfy .
Theorem B. All transcendental meromorphic solutions of (2) satisfy .
What will happen if the right-hand side of (2) is a rational function in ? That is, for the functional equation where is a complex number, , , , , , and are coefficients, and is irreducible in . Gundersen et al.  studied the case that on the left-hand side of (3). Following the results in , we continue to study the properties of the solutions of (3) in the case of on the left-hand side. In fact, we obtain the following theorem.
Theorem 2. Suppose that is a nonconstant meromorphic solution of equation of (3) and the coefficients are small functions of . Then, and .
In particular, we concern the second-order -difference equation with rational coefficients, that is, in the case of . From Theorem 2, we know that if is a nonconstant meromorphic solution, then . Thus, the second-order -difference equation is the following form:
First of all, we give some remarks.
Remark 3. If and are not zero at the same time, by Theorem 2, we derive that the solution of (4) is of order zero.
Remark 4. If , by Theorem A, the solutions of (4) is also of order zero.
Remark 5. If , by Theorem 2, the order of the solutions is less than . Thus, a question arises: does the equation have a solution which is of order nonzero under this situation? This question is still open.
In , Chen and Shon proved some theorems about the properties of solutions of the difference Painlevé I and II equations, such as the exponents of convergence of fixed points and the zeros of transcendental solutions. A natural question arises: how about the exponents of convergence of the fixed points and the zeros of transcendental solutions of the -difference equation (4)? Do the transcendental solutions have infinitely many fixed points and zeros? The following theorem, in which the coefficients are constants, answers the above questions partly.
Theorem 6. Suppose that is a transcendental solution of the equation where , the coefficients , , , , , , and are constants, and at least one of is nonzero. Then, and (i) has infinitely many fixed points, and (ii) has infinitely many zeros, whenever .
In the rest of the paper, we consider (3) when . In , Heittokangas et al. considered the essential growth problem for transcendental meromorphic solutions of complex difference equations, which is to find lower bounds for their characteristic functions. Following this idea, Zheng and Chen  obtained the following theorem for -difference equations.
Theorem C. Suppose that is a transcendental solution of equation where , the coefficients are rational functions, and , are relatively prime polynomials in over the field of rational functions satisfying , , and . If has infinitely many poles, then for sufficiently large , holds for some constant . Thus, the lower order of , which has infinitely many poles, satisfies .
Regarding Theorem C, they obtained the lower bound of the order of solutions. Then, how about the upper bound of the order of the solutions? Can the conditions of Theorem C become a little more simple? In fact, we have the following theorem.
Theorem 7. Suppose that is a transcendental solution of (3), where , and the coefficients are rational functions. Then, .
We know that the difference analogues and -difference analogues of Nevanlinna’s theory have been investigated. Consequently, many results on the complex difference equations and -difference equations have been obtained respectively. Thus, mixing the difference and -difference equations together is a natural idea. The following Theorem 8 is just a simple application of the above idea, and further investigation is required.
In what follows, we will consider difference products and difference polynomials. By a difference product, we mean a difference monomial, that is, an expression of type where are complex numbers and are natural numbers. A difference polynomial is a finite sum of difference products, that is, an expression of the form where is a set of distinct complex numbers and the coefficients of difference polynomials are small functions as understood in the usual the Nevanlinna theory; that is, their characteristic is of type .
Theorem 8. Suppose that is a nonconstant meromorphic solution of the equation where and the index set consists of elements and the coefficients and are small functions of . If is of finite order, then .
2. Some Lemmas
The following important result by Valiron and Mohon’ko will be used frequently, one can find the proof in Laine’s book [16, page 29].
Lemma 9. Let be a meromorphic function. Then, for all irreducible rational function in , with meromorphic coefficients , , the characteristic function of satisfies where and In the particular case when One has .
The next lemma on the relationship between and is due to Bergweiler et al. [10, page 2].
Lemma 10. One case see that holds for any meromorphic function and any constant .
Lemma 11 (see ). Let be a monotone increasing function, and let be a nonconstant meromorphic function. If for some real constant , there exist real constants and such that then
Lemma 12 (see [9, Theorem 2.2]). Let be a nonconstant zero-order meromorphic solution of where is a c-difference (or -difference) equation in . If , where is a zero-order meromorphic function such that on a set of logarithmic density 1, and in particular, is a constant, then on a set of logarithmic density 1.
Lemma 13 (see ). Let be a meromorphic function of finite order, and let be a nonzero complex constant. Then one has
Lemma 14 (see ). Let be a meromorphic function of finite order , and let is a nonzero complex constant. Then, for each , one has
It is evident that from Lemma 14.
By (7), (8), and Lemmas 13 and 14, Laine and Yang obtained the following lemma in .
Lemma 15. The characteristic function of a difference polynomial in (8) satisfies provided that f is a meromorphic function of finite order and the index set consists of n elements.
3. Proof of Theorems
Proof of Theorem 2. Set . From (3) and Lemmas 9 and 10 and noting , we immediately obtain and so we have . From (22), we have Since , we have for each . From (24) and Lemma 11, we have .
Proof of Theorem 6. Assume that is a transcendental solution of (5). Since at least one of is non-zero, by Remark 3, we obtain that .
(I) Set . Substituting into (5), we obtain that By (25), we may define By (26), we see that Suppose that , and we split into two cases. If , then we obtain . Thus, the right-hand side of (5) is vanishing. This contradicts to our assumption. If , we obtain and . Then, the right-hand side of (5) becomes ; this also contradicts to our assumption. Thus, we have . By Lemma 12, we obtain that on a set of logarithmic density 1. Thus, on a set of logarithmic density 1. Hence, by (29), has infinitely many fixed points and
(II) By (5), we derive that By (31) and the assumption , we obtain that By Lemma 12 and (32), we have on a set of logarithmic density 1. Thus, on a set of logarithmic density 1. Hence, by (34), has infinitely many zeros and
Proof of Theorem 7. From (3), we have By the properties of the Nevanlinna characteristic function and Lemma 9, we have By and Lemma 10, we obtain Setting , we have Applying Lemma 11 to (39) yields We now prove the lower bound of the order of the solutions. From (3), by the properties of the Nevanlinna characteristic function and Lemma 9, we have By Lemma 10 and noting , we obtain where and for some and all greater than some . Hence, for , where For sufficiently large and , we note that since , the two series converge, and hence where is a positive constant. Since is a transcendental meromorphic function, we can choose sufficiently large such that for all , by the increasing property of , we have for some constant . Hence, we get for some constant . By the definition of the order of , we have So we have . Thus, we complete the proof.
Proof of Theorem 8. Suppose that the order of is . We rewrite (9) as By the property of the Nevanlinna characteristic function, we have By Lemmas 10 and 15, we obtain for each . Since , we derive for each . Setting , , and applying Lemma 11 to (51), yield for each . Thus, we obtain
The author wishes to express his thanks to the referee for his/her valuable suggestions and comments. The present investigation was supported by the National Natural Science Foundation under Grant no. 11226088 and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant no. 12A110002 of the People’s Republic of China.
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