Growth Analysis of Meromorphic Solutions of Linear Difference Equations with Entire or Meromorphic Coefﬁcients of Finite ϕ -Order

: Many researchers’ attentions have been attracted to various growth properties of meromorphic solution f (of ﬁnite ϕ -order) of the following higher order linear difference equation A n ( z ) f ( z + n ) + ... + A 1 ( z ) f ( z + 1 ) + A 0 ( z ) f ( z ) = 0, where A n ( z ) , . . . , A 0 ( z ) are entire or meromorphic coefﬁcients (of ﬁnite ϕ -order) in the complex plane ( ϕ : [ 0, ∞ ) → ( 0, ∞ ) is a non-decreasing unbounded function). In this paper, by introducing a constant b (depending on ϕ ) deﬁned by lim r → ∞ log r log ϕ ( r ) = b < ∞ , and we show how nicely diverse known results for the meromorphic solution f of ﬁnite ϕ -order of the above difference equation can be modiﬁed.


Introduction and Preliminaries
Throughout this paper, a meromorphic function is meant to be analytic in the whole complex plane C except possibly for poles. In the following, let C := C ∪ {∞} and N denote the extended complex plane and the set of positive integers, respectively. The readers are assumed to be familiar with the basic results and standard notations of Nevanlinna's value distribution theory of meromorphic functions (see, e.g., Reference [1][2][3][4]). Yet, here, some fundamental notations for Nevanlinna  is called the (Nevanlinna) characteristic of f . Consider the following higher order linear difference (discrete) equation where A n (z), . . . , A 0 (z) are meromorphic (or entire) functions with A n (z) · A 0 (z) ≡ 0. A lot of interests in such a difference equation as the Equation (1) have recently been renewed, in particular, together with Nevanlinna theory [2,4] (see, e.g., Reference [5][6][7][8][9][10][11][12][13][14] and the references cited therein). For a later use, from the Equation (1), we find that, for f ( ≡ 0), Then, some easily-derivable implications among measure, logarithmic measure, upper density, and upper logarithmic density are given in the following remark.
For a more refined growth of meromorphic solutions of the Equation (1), the following (modified) definitions are recalled. Here, and in the following, let ϕ : [0, ∞) → (0, ∞) be a non-decreasing unbounded function.

Definition 2 (Reference
and In addition, the ϕ-lower type of an entire function f with It is noted that Definitions 1 and 2, where ϕ(r) = r may become the standard definitions of order, lower order, type, and lower type, respectively.
Several interesting and important results about (1) are recalled in the following theorems.
Theorem 1 (Reference [8], Theorem 9.2). Assume that there exists an integer p (0 where A j (z) (j = 0, . . . , n) are entire functions. If f (z) is a meromorphic solution of the Equa- Instead of the restriction (3), assuming that among the maximal order σ, exactly one has its type strictly greater than the others, Laine and Yang (Reference [11], Theorem 5.2) obtained the following conclusion for any meromorphic solution of the Equation (1): In Theorem 1, the Equation (1) has only one dominating coefficient A p . The following two theorems are concerned with the case when there are at least two coefficients which have the maximal order.
Here, the following natural question is occurred: When the coefficients of the Equation (1) are entire or meromorphic functions of finite ϕ-order, what would the growth properties of solutions of the linear difference Equation (1) be like? In this paper, for an answer to this question, by introducing a constant b, which depends on ϕ, defined by we show how nicely diverse known results for the meromorphic solution f of finite ϕ-order of the difference Equation (1) can be amended.

Main Results
In this section, main theorems are provided.

Preliminary Lemmas
For proof of the main results in Section 2, here, diverse estimations regarding meromorphic functions are recalled and established in the following lemmas. We begin with an elementary fact for the upper and lower limits.
Then, there exist strictly increasing sequences {x n } and {y n } in (1, ∞) such that u n x n < x n+1 and u n y n < y n+1 for each n ∈ N, and x n → ∞ and y n → ∞ as n → ∞, and Proof. We prove only the upper limit case. Let ε > 0 be given.
For the lower limit case, we consider the following fact: Let ε > 0 be given. Then, Lemma 2 (Reference [8], Theorem 8.2). Let f be a meromorphic function, η a non-zero complex number, and let γ > 1 be a given real constant. Then, there exist a subset E 1 ⊂ (1, ∞) of finite logarithmic measure and a constant A depending only on γ and η, such that, where n(t) = n(t, ∞, f ) + n t, ∞, 1 f denotes the sum of zeros and poles, respectively, of f , counting multiplicities, which lie in the disk |z| ≤ t.
By using Lemma 6, as well as Lemmas 2 and 3, we may give an analogue of Lemma 7, and hence Lemma 8, for finite ϕ-lower order, which is stated in the following lemma without proof. Lemma 9. Let η 1 , η 2 be two arbitrary complex numbers such that η 1 = η 2 and let f be a transcendental meromorphic function of finite ϕ-lower order µ. Then, there exists a subset E 5 ⊂ (1, ∞) of infinite logarithmic measure such that, for any given ε > 0, for sufficiently large |z| = r ∈ E 5 .
Proof. We first note that only the proof of the assertions in Lemma 6 was given in Reference [17], p = q = 1, Lemma 2.4, as that of this lemma remains to be showed in the same way. It seems meaningful for the authors and the interested reader to copy and modify the proof in Reference [17], p = q = 1, Lemma 2.4, in a little more detailed manner. Indeed, employing Lemma 1 in Definition 1, there exists a sequence {r n } in (1, ∞) such that r n → ∞ as n → ∞, 1 + 1 n r n < r n+1 (n ∈ N), and lim r n →∞ log T(r n , f ) log ϕ(r n ) = σ.
Then, combining the inequalities (33)-(35) gives that for all n ≥ n 0 and r ∈ E 6 , Since ε > 0 is arbitrary, we have Obviously, sets r n , 1 + 1 n r n (n ∈ N) are mutually disjoint. Therefore, we have Let a n := log 1 + 1/n (n ∈ N). Clearly a n > 0 (n ∈ N). We find that lim n→∞ a 1 n n = 1 and, therefore, the root test cannot be employed whether the series ∞ ∑ n=n 0 a n is convergent or By the integral test, the last series in (36) diverges to ∞. Hence, m (E 6 ) = ∞. This completes the proof.

Proof of Main Results
Proof of Theorem 5. The proof here would proceeded in line with that of (Reference [8], Theorem 9.2) which is modified in a little detailed manner (see, in particular, (39) and (41)). Let f ( ≡ 0) be a transcendental meromorphic solution of the Equation (1). If σ( f , ϕ) = ∞, then the result is obvious. So we assume that σ( f , ϕ) = σ < ∞. Suppose to the contrary that From (6), a positive real number η can be chosen such that From (37) and (38), we may choose ε > 0 so small that From (2), we find For such an ε > 0 in (39), using in (29) in Lemma 8, and (39), we find from (40) that, for sufficiently large |z| = r, Finally, taking logarithm on both sides of the inequality composed by the first and last terms in (41), and dividing each side of the resulting inequality by log ϕ(r), and taking the upper limit as r → ∞ on both sides of the last resultant inequality, we obtain σ( Proof of Theorem 6. Here, the proof would run parallel to that of (Reference [14], Theorem 1.1) which is modified in a little detailed manner (see Theorem 2) (see, in particular, (47) and (50)).
Proof of Theorem 9. The process of the proof would be flowed as in that of Theorem II in Reference [21], which is modified in a little detailed manner (see, in particular, (66) and (69)). Let f ( ≡ 0) be a transcendental meromorphic solution of the Equation (1). If σ( f , ϕ) = ∞, then the result is clear. So we suppose that σ( f , ϕ) = σ < ∞. From Definition 3, which gives that, for sufficiently large r, Combining (64) and (40), we obtain for sufficiently large r. By using (29) in Lemma 8 and the relation between T(r, f ) and m(r, f ) in (65), we get for sufficiently large r. In view of (14), by Lemma Applying (68) to (66), we get that, for sufficiently large r ∈ E 7 , Taking logarithm on both sides of the inequality (69), and dividing the resulting inequality by log ϕ(r), and taking the upper limit as r(∈ E 7 ) → ∞ on both sides of the last resultant inequality, we finally obtain which, upon ε > 0 being arbitrary, leads to the desired inequality σ( f , ϕ) ≥ σ(A , ϕ) + b.

Concluding Remarks
In this paper, in order to answer the following natural question: When the coefficients of the Equation (1) are entire or meromorphic functions of finite ϕ-order, what would the growth properties of solutions of the linear difference Equation (1) be like?, we introduced the constant b in (5), depending on ϕ. Then we showed how nicely diverse known results for the meromorphic solution f of finite ϕ-order of the difference Equation (1) can be amended.

Posing a Problem
Considering the results presented in this paper, by using the constant b in (5), some known other results for this subject are supposed to be amendable as those in Theorems 5-10, which are left to the interested readers for future investigation.