On the Growth of Higher Order Complex Linear Differential Equations Solutions with Entire and Meromorphic Coefficients

We revisit the problem of studying the solutions growth order in complex higher order linear differential equations with entire and meromorphic coefficients of [p, q]-order, proving how it is related to the growth of the coefficient of the unknown function under adequate assumptions. Our study improves the previous results due to J. Liu J. Tu L. Z Shi, L. M. Li T. B. Cao , and others.


Introduction, Definitions and Notations
Complex linear differential equations (1) where the coefficients A 0 , A 1 , ... , A k−1 , (k ≥ 2), and F( ≡ 0) are entire or meromorphic functions, are relevant and they have extensively been studied by many authors (cf. [1][2][3][4][5][6][7][8][9][10][11]). In this line, Juneja-Kapoor-Bajpai studied entire functions of [p, q]-order with the aim of accurately discussing the growth of these functions, ( [12,13]). Additionally, more recently, Liu-Tu-Shi [14] modified slightly the aforementioned [p, q]-order definition investigating properties of the solutions of complex linear differential equations, also see [7]. The study of order of an entire or meromorphic function f studies the symmetries or analogies between the growth of the maximum modulus of f and the growth of exponential and logarithmic functions, since the order of growth of a function relates to the rate of growth of the latter ones, ( [7][8][9][12][13][14]). In order to handle this comparison, for each real number r ∈ [0, ∞) belonging to the domain of f ∈ {exp, log}, we consider f 1 (r) = f (r) and f 0 (r) = r. Additionally, for each of such f and p ∈ N, we define f p+1 (r) = f ( f p (r)), this for sufficiently large r when f = log . We will consider exp −1 r = log 1 r and log −1 r = exp 1 r. Moreover, given a set E ⊂ [0, ∞), we denote its linear measure by mE = E dt, and the logarithmic measure for E ⊂ (1, ∞), by m l E = E dt t . Despite the fact that this paper uses standard notions of Nevanlinna theory, we consider it to be convenient to recall some notation that is related to the number of poles of a meromorphic or entire function that are located within a disk centered at the origin in order to facilitate its reading (cf. [15][16][17]). Let n(r, f ) be the number of poles of a function f (counting multiplicities) in |z| ≤ r, and where n(r, f ) is the number of distinct poles of a function f in |z| ≤ r. Subsequently, we define the integrated counting function N(r, f ) by on the other hand, we define the proximity function m(r, f ) by where log + x = max{0, log x}. We should think of m(r, f ) as a measure of how close f is to infinity on |z| = r. Nevertheless, within that context, we recall that T(r, f ) stands for the Nevanlinna characteristic function of the meromorphic function f that is defined on each positive real value r by T(r, f ) = m(r, f ) + N(r, f ).
Additionally, M(r, f ) stands for the so-called maximum modulus function defined for each non-negative real value r by Now, we recall the following definitions, where p, q are positive integers satisfying p ≥ q ≥ 1.
Definition 1 ([7,14]). Let f be a meromorphic function, the [p, q]−order of f is defined by If f is an entire function, then Remark 1. If p = q = 1, above definition reduces to standard order. If, just q = 1, it reduces to p-th order.
Definition 2 ([7,14]). The [p, q]− lower order of a meromorphic function f is defined by If f is an entire function, then Definition 3 ([7,14]). The [p, q]−type of a meromorphic function f of [p, q] -order σ If f is an entire function, then Definition 4 ([7,14]). The [p, q]−convergence exponent of the sequence of zeros of a meromorphic function f is defined by Definition 5 ([7,14]). The [p, q]−convergence exponent of distinct zeros of a meromorphic function f is defined by Liu-Tu-Shi [14] consider the Equation (1) with entire functions as coefficients, and then obtain the following results.
then every nontrivial solution f of (1) satisfies When the coefficients in (2) are meromorphic functions, Li-Cao [7] obtain the following result: The following section contains the main results that deepen the aforementioned theorems regardinf how fast the solutions of linear differential Equations (1) and (2) may grow.

Main Results
In this section, we present our main results.
, where E 1 is a set of r of finite linear measure, then every nontrivial solution of (1) satisfies For the non-homogeneous case (2), we obtain the following result:

Preliminary Lemmas
In this section, we introduce some lemmas and remark that we will use them in the sequel.

Lemma 2 ([17]
). Let f be a transcendental entire function, and z a point with |z| = r, at which | f (z)| = M(r, f ). Subsequently, for all |z| outside a set E 3 of finite logarithmic measure, it holds

Remark 2.
Because the number of zeros of a polynomial P of degree n is finite (at most n) and, indeed, its central index is n for sufficiently large r, the above Lemma 2 holds for any given entire, transcendental or not, function f .

Proof of Main Results
Proof of Theorem 4. From Equation (1), it follows that By Remark 2 and (3), where E is a set of finite linear measures. Assume that then for sufficiently large r, we find that From (4) and (5), it follows that By Lemma 3, there exists a set E 4 ⊂ (1, ∞) of r of infinite logarithmic measure, such that, for all z satisfying |z| = r ∈ E 4 , we have lim r→∞ log p m(r, A 0 ) Subsequently, by the definition of limit, there exists a ε > 0, such that lim r→∞ log p m(r, A 0 ) . By substituting the above inequality in (6), there exists a set E 4 ⊂ (1, ∞) of r of infinite logarithmic measure, such that, for all z satisfying |z| = r ∈ E 4 \E and for any ε > 0, we have From (7) and Lemma 1, we deduce Taking limit r → ∞ after dividing both side by log q r, we obtain that On the other hand, Equation (1) provides Now, Remark 2 provides a set E 3 of finite logarithmic measure, so that, for all z satisfying |z| = r ∈ E 3 and | f (z)| = M(r, f ), we have Subsequently, (5) and the fact σ [p,q] (A 0 ) = σ 1 imply that Hence, having in mind the definition of [p, q]−order, Substituting (10)-(12) into (9), it follows that Because ε > 0 is arbitrary, Lemma 1 and (13) provide By Lemma 5 and (14), we get From (8) and (15), we conclude that This proves the theorem.
Proof of Theorem 5. From Lemma 4, it follows that every nontrivial solution f of Equation (1) satisfies lim r→∞ log T(r, f ) m(r,A 0 ) > 0 (r ∈ E 5 ); hence, there exists a δ > 0 and a sequence {r n } ∞ n=1 tending to infinity, so that, for sufficiently large r n ∈ E 5 and for every nontrivial solution f of Equation (1), we have log T(r n , f ) > δm(r n , A 0 ).
Lemma 3 provides a set E 4 ⊂ (1, ∞) of infinite logarithmic measure, such that, for all r ∈ E 4 \E 5 and for any ε > 0, we have i.e., by (16) and (17), Lemma 1 and Equation (18) imply that On the other hand, from Equation (1), , for sufficiently large r and for any given ε > 0, we have Again, having in mind the definitions of [p, q]−order, we have Now taking Lemma 2 into account, we may assure that there exists some set E 3 of finite logarithmic measure, so that whenever |z| = r ∈ E 3 and | f (z)| = M(r, f ), it holds that (1)), (j = 1, 2, ... , k).
Proof of Theorem 6. Let us rewrite Equation (2) as If f has got a zero at z 0 of order β (β > k), and if A 0 (z), A 1 (z), ... , A k−1 (z) are all of them analytic at z 0 , then F has obtained a zero at z 0 of order β − k. Therefore The classical lemma on logarithmic derivative and (27) bring out that the inequality holds for r / ∈ E, E being a set of finite linear measure. Analogously from (28) and (29), it follows that the inequality T(r, f ) = T r, Subsequently, for sufficiently large r and for any given ε, 0 < ε < c − δ, it holds Substituting (31) into (30), we obtain that First, take logarithm and divide by log q r in both side of (32) and then take limit r → ∞, we can obtain that Definitions make immediate the reverse inequalities This proves the theorem.

Discussion
Keeping the results already established in mind, one may explore, for analogous theorems in which the coefficients of differential equations are bi-complex valued, entire and meromorphic functions of [p, q]-order, with p and q being any two integers with p ≥ q ≥ 1 . Further, the case in which the coefficients of differential equations generated by analytic functions of [p, q]-order in the unit disc may be considered by future researchers in this area. Moreover, the investigation of the problems under the flavor of [p, q] index pair of both complex and bi-complex valued entire and meromorphic functions is still a virgin domain for the new researchers.

Open Problem
The methodologies that were adopted in this paper can be treated algebraically under the flavor of bicomplex numbers, and these may be regarded as an Open Problem to the future workers of this branch.