Value Distribution and Arbitrary-Order Derivatives of Meromorphic Solutions of Complex Linear Differential Equations in the Unit Disc

In this paper, we investigate the value distribution of meromorphic solutions and their arbitrary-order derivatives of the complex linear differential equation f ′′ + A(z) f ′ + B(z) f = F(z) in ∆ with analytic or meromorphic coefficients of finite iterated p-order, and obtain some results on the estimates of the iterated exponent of convergence of meromorphic solutions and their arbitrary-order derivatives taking small function values.


Introduction and Main Results
Throughout this paper, we use the standard notations of the classic Nevanlinna theory (see, e.g., [1,2]), such as m(r, f ), n(r, f ), N(r, f ), T(r, f ), and M(r, f ).
Firstly, we introduce some definitions on the growth and the value distribution of fast-growing meromorphic functions in ∆ (see, e.g., [4][5][6][7][8][9]).Definition 1 ([6]).Let f (z) be a meromorphic function in ∆.Then, we define the iterated p-order of f (z) as If f (z) is analytic in ∆, we also define Remark 1. From Tsuji [10] and Laine [2], respectively, we can see that if f (z) is analytic in ∆, then we have Definition 2 ([6,7]).Let f (z) be a meromorphic function in ∆.Then, we define the growth index of the iterated order of f (z) as If f (z) is analytic in ∆, we also define Definition 3 ([4,5]).Let f (z) be a meromorphic function in ∆.Then, we define the iterated p-exponent of convergence of the sequence of zeros of f (z) and the iterated p-exponent of convergence of the sequence of distinct zeros of f (z), respectively, as

Definition 4 ([9]
).Let f (z) be a meromorphic function in ∆ with the iterated p-order σ p ( f )(0 Then, we define the iterated p-type of f (z) as Definition 5 ([1]).Let f (z) be a meromorphic function in ∆.Then, for a ∈ C = C {∞}, we define the deficiency of the value a with respect to f (z) as .
Next, we introduce some background relative to our main results.It is well-known that Bank and Laine started the original complex oscillation theory of solutions of linear differential equations in C in 1982 (see [2]).Following that, many scholars in the field of complex analysis have investigated the growth and the value distribution of meromorphic solutions of complex linear differential equations as the theory of complex linear differential equations in C has matured (see, e.g., [2,3,11,12]).Naturally, the question of whether we can get the corresponding results on complex linear differential equations in ∆ has arisen.This question is interesting and meaningful.On the one hand, complex linear differential equations in ∆ have many similar properties to those in C. On the other hand, it is much more difficult to study complex linear differential equations in ∆ than in C, due to the lack of corresponding effective tools.Some results on this topic can be seen in, for example, [4][5][6][7][8][9][13][14][15][16][17][18][19][20].
In particular, Latreuch and Belaïdi [18] investigated the distribution of zeros of meromorphic solutions and their arbitrary-order derivatives for a second-order non-homogeneous complex linear differential equation in ∆ with meromorphic coefficients of finite iterated p-order, and got the following result: Theorem 1 ([18]).Let A(z), B(z)( ≡ 0), and F(z)( ≡ 0) be meromorphic functions in ∆ with finite iterated p-order, such that B j (z) ≡ 0 and F j (z) They also noted that some special conditions on the coefficients in (1) can guarantee that the assumptions B j (z) ≡ 0 and F j (z) ≡ 0 (j = 1, 2, • • • ) in Theorem 1 hold, which makes Theorem 1 more concrete.More details can be seen in Theorems 2 and 3.
Theorem 5 ([17]).Let A(z), B(z)( ≡ 0), F(z)( ≡ 0), and ϕ(z) be analytic functions in ∆ with finite iterated p-order, such that σ p (B) , and ϕ(z) is not a solution of (1).Then, all non-trivial solutions of (1) in ∆ satisfy with at most one possible exceptional solution, f 0 (z), such that Theorem 6 ([17]).Let A(z), B(z)( ≡ 0), F(z)( ≡ 0), and ϕ(z) be meromorphic functions in ∆ with finite iterated p-order, such that σ p (B) Inspired by Theorems 1-6, we proceed further in this direction.Note that there exists a dominant coefficient whose iterated p-order is strictly larger than those of the other coefficients in Theorems 2, 3, 5, and 6.A question arises naturally: What can we say if there exists more than one coefficient having the maximal iterated p-order?In the following, we introduce a condition on the iterated p-type to deal with coefficients having the maximal iterated p-order to obtain Theorems 7 and 8. Theorem 7. Let p ∈ N + \{1}, A(z), B(z)( ≡ 0), F(z)( ≡ 0), and ϕ(z)( ≡ 0) be analytic functions in ∆ with finite iterated p-order, such that max{σ p (A), B), and ϕ(z) is not a solution of (1).Then, all non-trivial solutions f (z) of (1) in ∆ satisfy Remark 2. The partial result of Theorem 7 for the case p = 1 will be shared in Lemma 7.

Lemmas for Proofs of Main Results
Lemma 1 ([14]).Let f (z) be a meromorphic function in ∆ with i( f ) = p and σ p ( f ) < ∞.Then, for any ε(> 0), there exists a subset E ⊂ [0, 1) with E dr , and F(z)( ≡ 0) be meromorphic functions in ∆ with finite iterated p-order.If f (z) is a meromorphic solution of complex linear differential equation and ϕ(z) be meromorphic functions in ∆ with finite iterated p-order, such that , where 0 < R ≤ ∞, and f (z) be a solution of complex linear differential equation where C = C(k) > 0 is a constant depending on p and on the initial values of f (z) at the point z θ , where A j (z θ ) = 0 for some j = 0, 1 • • • , k − 1.
Therefore, the proof of Lemma 7 is complete.

Proofs of Theorems 7 and 8
In this section, we denote by E a subset of [0, 1) with E dr 1−r < ∞ and by H a subset of [0, 1) with H dr 1−r = ∞, and assume that E and H appear not necessarily to be the same on each occasion.