Further Extending Results of Some Classes of Complex Difference and Functional Equations

Abstract

The main purpose of this paper is to present some properties of the meromorphic solutions of complex difference equation of the form , where and are two finite index sets, are distinct, nonzero complex numbers, and are small functions relative to is a rational function in with coefficients which are small functions of . We also consider related complex functional equations in the paper.

1. Introduction and Main Results

Let be a meromorphic function in the complex plane. We assume that the reader is familiar with the standard notations and results in Nevanlinna's value distribution theory of meromorphic functions such as the characteristic function , proximity function , counting function , the first and second main theorems (see, e.g., [1–4]). We also use to denote the counting function of the poles of whose every pole is counted only once. The notation denotes any quantity that satisfies the condition: as possibly outside an exceptional set of of finite linear measure. A meromorphic function is called a small function of if and only if

Recently, a number of papers (see, e.g., [5–9]) focusing on Malmquist type theorem of the complex difference equations emerged. In 2000, Ablowitz et al. [5] proved some results on the classical Malmquist theorem of the complex difference equations in the complex differential equation by utilizing Nevanlinna theory. They obtained the following two results.

Theorem A.

If the second-order difference equation

(1.1)

with polynomial coefficients () and (), admits a transcendental meromorphic solution of finite order, then

Theorem B.

If the second-order difference equation

(1.2)

with polynomial coefficients () and (), admits a transcendental meromorphic solution of finite order, then

One year later, Heittokangas et al. [7] extended the above two results to the case of higher-order difference equations of more general type. They got the following.

Theorem C.

Let . If the difference equation

(1.3)

with the coefficients of rational functions () and () admits a transcendental meromorphic solution of finite order, then

Theorem D.

Let . If the difference equation

(1.4)

with the coefficients of rational functions () and () admits a transcendental meromorphic solution of finite order, then

Laine et al. [9] and Huang and Chen [8], respectively, generalized the above results. They obtained the following theorem.

Theorem E.

Let be distinct, nonzero complex numbers, and suppose that is a transcendental meromorphic solution of the difference equation

(1.5)

with coefficients () and (), which are small functions relative to where is a collection of all subsets of . If the order is finite, then .

In the same paper, Laine et al. also obtained Tumura-Clunie theorem about difference equation.

Theorem F.

Suppose that are distinct, nonzero complex numbers and that is a transcendental meromorphic solution of

(1.6)

where the coefficients are nonvanishing small functions relative to and where and are relatively prime polynomials in over the field of small functions relative to . Moreover, we assume that ,

(1.7)

and that, without restricting generality, is a monic polynomial. If there exists such that for all sufficiently large,

(1.8)

where , then either the order , or

(1.9)

where is a small meromorphic function relative to .

Remark 1.1.

Huang and Chen [8] proved that the Theorem F remains true when the left hand side of (1.6) is replaced by the left hand side of (1.5), meanwhile, the condition (1.8) would be replaced by a corresponding form.

Moreover, Laine et al. [9] also gave the following result.

Theorem G.

Suppose that is a transcendental meromorphic solution of

(1.10)

where is a polynomial of degree is a collection of all subsets of . Moreover, we assume that the coefficients are small functions relative to and that Then

(1.11)

where

In this paper, we consider a more general class of complex difference equations. We prove the following results, which generalize the above related results.

Theorem 1.2.

Let be distinct, nonzero complex numbers and suppose that is a transcendental meromorphic solution of the difference equation

(1.12)

with coefficients , and are small functions relative to where and are two finite index sets, denote

(1.13)

If the order is finite, then

Corollary 1.3.

Let be distinct, nonzero complex numbers and suppose that is a transcendental meromorphic solution of the difference equation

(1.14)

with coefficients and , which are small functions relative to where is a finite index set, denote

(1.15)

If the order is finite, then

Remark 1.4.

In Corollary 1.3, if we take

(1.16)

then Corollary 1.3 becomes Theorem E. Therefore, Theorem 1.2 is a generalization of Theorem E.

Example 1.5.

Let Then it is easy to check that solves the following difference equation:

(1.17)

Example 1.6.

Let It is easy to check that satisfies the difference equation

(1.18)

In above two examples, we both have and Therefore, the estimations in Theorem 1.2 and Corollary 1.3 are sharp.

Theorem 1.7.

Suppose that are distinct, nonzero complex numbers and that is a transcendental meromorphic solution of

(1.19)

where the coefficients are nonvanishing small functions relative to and and are relatively prime polynomials in over the field of small functions relative to , and are two finite index sets, denote

(1.20)

Moreover, we assume that ,

(1.21)

and that, without restricting generality, is a monic polynomial. If there exists such that for all sufficiently large,

(1.22)

(1.23)

where , then either the order , or

(1.24)

where is a small meromorphic function relative to .

If the left hand side of (1.19) in Theorem 1.7 is replaced by the left hand side of (1.14) in Corollary 1.3, then (1.23) implies (1.22). Since we have

(1.25)

by the fundamental property of counting function. Therefore, we get the following result easily.

Corollary 1.8.

Suppose that are distinct, nonzero complex numbers and that is a transcendental meromorphic solution of

(1.26)

where the coefficients are nonvanishing small functions relative to and and are relatively prime polynomials in over the field of small functions relative to , is a finite index set, denote

(1.27)

Moreover, we assume that ,

(1.28)

and that, without restricting generality, is a monic polynomial. If there exists such that for all sufficiently large,

(1.29)

where , then either the order , or

(1.30)

where is a small meromorphic function relative to .

Finally, we give a result corresponding to Theorem G.

Theorem 1.9.

Let be distinct, nonzero complex numbers and suppose that is a transcendental meromorphic solution of

(1.31)

where is a polynomial of degree , and are two finite index sets. Denote

(1.32)

Moreover, we assume that the coefficients and are small functions relative to and that Then

(1.33)

where

2. Main Lemmas

In order to prove our results, we need the following lemmas.

Lemma 2.1 (see [10]).

Let be a meromorphic function. Then for all irreducible rational functions in ,

(2.1)

such that the meromorphic coefficients satisfy

(2.2)

one has

(2.3)

Lemma 2.2 (see [11]).

Let be distinct meromorphic functions and

(2.4)

Then

(2.5)

where and are two finite index sets, and ().

Remark 2.3.

If we suppose that and hold for all , and denote and then we have the following estimation by the proof of Lemma 2.2

(2.6)

Lemma 2.4 (see [6]).

Let be a meromorphic function with order and let be a fixed nonzero complex number, then for each one has

(2.7)

Lemma 2.5 (see [12]).

Let be a meromorphic function and let be given by

(2.8)

where are small meromorphic functions relative to . Then either

(2.9)

or

(2.10)

Lemma 2.6 (see [9, 13]).

Let be a nonconstant meromorphic function and let , be two polynomials in with meromorphic coefficients small relative to . If and have no common factors of positive degree in over the field of small functions relative to , then

(2.11)

Lemma 2.7 (see [14]).

Let be a transcendental meromorphic function, and be a nonconstant polynomial of degree . Given denote and . Then given and one has

(2.12)

for all large enough.

Lemma 2.8 (see [15]).

Let be positive and bounded in every finite interval, and suppose that holds for all large enough, where and are real constants. Then

(2.13)

where .

3. Proof of Theorems

Proof of Theorem 1.2.

We assume that is a meromorphic solution of finite order of (1.12). It follows from Lemmas 2.1, 2.2, and 2.4 that for each

(3.1)

This yields the asserted result.

Proof of Theorem 1.7.

Suppose is a transcendental meromorphic solution of (1.19) and the second alternative of the conclusion is not true. Then according to Lemmas 2.5 and 2.6, we get

(3.2)

Thus, we have

(3.3)

Now assuming the order , then we have and

(3.4)

for all . By using Lemmas 2.1 and 2.2, we conclude that

(3.5)

It follows from this that

(3.6)

We prove the following inequality by induction:

(3.7)

The case has been proved. We assume that above inequality holds when . Next, we prove that inequality (3.7) holds for We have

(3.8)

Noting that thus we have

(3.9)

and so

(3.10)

This implies that

(3.11)

It follows from (3.7) that

(3.12)

Let be large enough such that

(3.13)

Since

(3.14)

we have for any

(3.15)

thus for each

(3.16)

for large enough holds. We now fix and let , thus

(3.17)

Finally, let and we conclude that the order Therefore, we get a contradiction and the assertion follows.

Proof of Theorem 1.9.

We assume is a transcendental meromorphic solution of (1.31). Denoting again According to the last assertion of Lemmas 2.7 and 2.2, we get that

(3.18)

Since holds for large enough for we may assume to be large enough to satisfy

(3.19)

outside a possible exceptional set of finite linear measure. By the standard idea of removing the exceptional set (see [4, page 5]), we know that whenever

(3.20)

holds for all large enough. Denote , thus inequality (3.20) may be written in the form

(3.21)

By Lemma 2.8, we have

(3.22)

where

(3.23)

Denoting now , thus we obtain the required form. Theorem 1.9 is proved.