Finite Logarithmic Order Meromorphic Solutions of Complex Linear Delay-Differential Equations

In this article, we study the growth of meromorphic solutions of linear delay-differential equation of the form \begin{equation*} \sum_{i=0}^{n}\sum_{j=0}^{m}A_{ij}(z)f^{(j)}(z+c_{i})=F(z), \end{equation*}% where $A_{ij}(z)$ $(i=0,1,\ldots ,n,j=0,1,\ldots ,m,n,m\in \mathbb{N})$ and $% F(z)$ are meromorphic of finite logarithmic order, $c_{i}(i=0,\ldots ,n)$ are distinct non-zero complex constants. We extend those results obtained recently by Chen and Zheng, Bellaama and Bela\"{\i}di to the logarithmic lower order.


INTRODUCTION AND MAIN RESULTS
Throughout this article, we assume the readers are familiar with the fundamental results and standard notations of the Nevanlinna distribution theory of meromorphic functions such as m(r, f ), N (r, f ), M (r, f ), T (r, f ), which can be found in [13,15,25].The concepts of logarithmic order and logarithmic type of entire or meromorphic functions were introduced by Chern, [9,10].Since then, many authors used them in order to generalize previous results obtained on the growth of solutions of linear difference equations and linear differential equations in which the coefficients are entire or meromorphic functions in the complex plane C of positive order different to zero, see for example [1,6,11,14,19,21,22], their new results were on the logarithmic order, the logarithmic lower order and the logarithmic exponent of convergence, where they considered the case when the coefficients are of zero order see, for example, [2-4, 7, 12, 17, 18, 23].In this article, we also use these concepts to investigate the lower logarithmic order of solutions to more general homogeneous and non homogeneous linear delay-differential equations, where we generalize those results obtained in [5,8].We start by stating some important definitions.Definition 1.1 ( [3, 10]).The logarithmic order and the logarithmic lower order of a meromorphic function f are defined by It is clear that, the logarithmic order of any non-constant rational function f is one, and thus, any transcendental meromorphic function in the plane has logarithmic order no less than one.Moreover, any meromorphic function with finite logarithmic order in the plane is of order zero.Definition 1.2 ( [3, 7]).The logarithmic type and the logarithmic lower type of a meromorphic function f are defined by It is clear that, the logarithmic type of any non-constant polynomial P equals its degree deg P , that any non-constant rational function is of finite logarithmic type, and that any transcendental meromorphic function whose logarithmic order equals one in the plane must be of infinite logarithmic type.

Definition 1.3 ( [10]
).Let f be a meromorphic function.Then, the logarithmic exponent of convergence of poles of f is defined by where n(r, f ) denotes the number of poles and N (r, f ) is the counting function of poles of f in the disc |z| ≤ r.
Then any meromorphic solution f of (1.2) satisfies ρ(f Theorem 1.8 ( [5]).Consider the delay differential equation (1.2) with meromorphic coefficients.Suppose that one of the coefficients, say A l0 with µ(A l0 ) > 0, is dominate in the sense that: Note that the case when the coefficients are of order zero is not included in the above results and because the logarithmic order is an effective technique to express the growth of solutions of the linear difference equations and the linear differential equations even when the coefficients are zero order entire or meromorphic functions, in this article, our main aim is to investigate the logarithmic lower order of meromorphic solutions of equations (1.1) and (1.2) to extend and improve the above theorems.When the coefficients of (1.1) and (1.2) are meromorphic functions and there is one dominating coefficient by its logarithmic lower order or by its logarithmic lower type, we get the following two theorems.Theorem 1.9.Let A ij (z) (i = 0, 1, . . ., n, j = 0, 1, . . ., m, n, m ∈ N) be meromorphic functions, and a, l ∈ {0, 1, ..., n} , b ∈ {0, 1, ..., m} such that (a, b) = (l, 0).Suppose that one of the coefficients, say A l0 with λ log 1 Then any meromorphic solution f of ( 1 Theorem 1.10.Let A ij (z) (i = 0, 1, . . ., n, j = 0, 1, . . ., m, n, m ∈ N) be meromorphic functions, and a, l ∈ {0, 1, ..., n} , b ∈ {0, 1, ..., m} such that (a, b) = (l, 0).Suppose that one of the coefficients, say A l0 with µ(A l0 ) > 0 and δ(∞, A l0 ) > 0, is dominate in the sens that: , where S := {F, A ij : (i, j) = (a, b), (l, 0)} and ρ log (S) := max{ρ log (g) : g ∈ S}.
Then any meromorphic solution f of (1.2) satisfies

SOME LEMMAS
The following lemmas are important to our proofs.

Lemma 2.1 ( [16]).
Let k and j be integers such that k > j ≥ 0. Let f be a meromorphic function in the plane C such that f (j) does not vanish identically.Then, there exists an r 0 > 1 such that Remark 2.1.It is shown in [13, p. 66], that for an arbitrary complex number c = 0, the following inequalities hold as r → +∞ for a general meromorphic function f (z).Therefore, it is easy to obtain that

Lemma 2.2 ( [3]
).Let f be a meromorphic function with 1 ≤ µ log (f ) < +∞.Then there exists a set E 1 ⊂ (1, +∞) with infinite logarithmic measure such that for any given ε > 0 and r ∈ E 1 ⊂ (1, +∞) , we have Lemma 2.3.Let f be a meromorphic function with 1 ≤ µ log (f ) < +∞.Then there exists a set E 2 ⊂ (1, +∞) with infinite logarithmic measure such that Consequently, for any given ε > 0 and all sufficiently large r ∈ E 2 , we have Proof.By the definition of the logarithmic lower type, there exists a sequence {r n } ∞ n=1 tending to ∞ satisfying 1 + 1 n r n < r n+1 , and Then for any given ε > 0, there exists an integer n 1 such that for n ≥ n 1 and any r ∈ n n+1 r n , r n , we have . Set Then from (2.1), we obtain so for any given ε > 0 and all sufficiently large r ∈ E 2 , we get where lm

Lemma 2.4 ( [3]
).Let η 1 , η 2 be two arbitrary complex numbers such that η 1 = η 2 and let f be a finite logarithmic order meromorphic function.Let ρ be the logarithmic order of f .Then for each ε > 0, we have 3. PROOF OF THEOREM 1.9 Let f be a meromorphic solution of (1.2).If f has infinite logarithmic order, then the result holds.Now, we suppose that ρ log (f ) < ∞.We divide (1.2) by f (z + c l ) to get .
By (3.1) and Remark 2.1, for sufficiently large r, we have From Lemma 2.1, for sufficiently large r, we obtain By Lemma 2.4, for any given ε > 0 and all sufficiently large r, we have From the definition of λ log 1 A l0 , for any given ε > 0 with sufficiently large r, we have By using the assumptions (3.3)-(3.5),we may rewrite (3.2) as This proof is also divided into four cases: Case (i): If max{µ log (A ab ), ρ log (S)} < µ log (A l0 ), then by the definitions of µ log (A l0 ) and ρ log (S) for any given ε > 0 and all sufficiently large r, we have By the definition of µ log (A ab ) and Lemma 2.2, there exists a subset E 1 ⊂ (1, +∞) of infinite logarithmic measure such that for any given ε > 0 and for all sufficiently large r ∈ E 1 , we have (3.9)T (r, A ab ) ≤ (log r) µ log (A ab )+ε .
We may choose sufficiently small ε satisfying for all sufficiently large r ∈ E 1 , by (3.22) we have Case (iv): When µ log (A l0 ) = µ log (A ab ) = ρ log (S) and Then, by substituting (3.11), (3.15), (3.16), (3.19) and (3.20) into (3.6), for all sufficiently large r ∈ E 1 , we have +1+ε .Now, we may choose sufficiently small ε satisfying for all sufficiently large r ∈ E 1 , we deduce from (3.23) that Further, for the homogeneous case, by substituting ( , for all sufficiently large r ∈ E 1 , we get Therefore, for ε satisfying and for all sufficiently large r ∈ E 1 , by (3.24) we have The proof of Theorem 1.9 is complete.

PROOF OF THEOREM 1.10
Let f be a meromorphic solution of (1.2).If f has infinite logarithmic order, then the result holds.Now, we suppose that ρ log (f ) < ∞.By (3.1) and Remark 2.1, for sufficiently large r, we have .

EXAMPLE
The following example is for illustrating the sharpness of some assertions in Theorem 1.10.We see that f satisfies ρ log (f ) = 1 = µ log (A 20 ).

Author's contributions:
The study was carried out in collaboration of all authors.All authors read and approved the final manuscript.

ρ
log (f ) = lim sup r−→+∞ log T (r, f ) log log r , µ log (f ) = lim inf r−→+∞ log T (r, f ) log log r .whereT (r, f ) denotes the Nevanlinna characteristic of the function f .If f is an entire function, thenρ log (f ) = lim sup r−→+∞ log log M (r, f ) log log r = lim sup r−→+∞ log T (r, f ) log log r , µ log (f ) = lim inf r−→+∞ log log M (r, f ) log log r = lim inf r−→+∞ log T (r, f ) log log r ,where M (r, f ) denotes the maximum modulus of f in the circle |z| = r.