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ρ = ρ > . Let d j ( )( z j = 1, 2 ) be meromorphic functions that are not **all** vanishing
identically such that max { ρ p ( ) ( ) d 1 , ρ p d 2 } < ρ p ( ) B . If f and 1 f are two nontrivial linearly 2
independent meromorphic **solutions** whose poles are **of** uniformly bounded multiplicity **of** (1.2), then the polynomial **of** **solutions** w = d f 1 1 + d f 2 2 satisfies i w ( ) = + , p 1

Theorem 1.4 Let P (z), Q(z), A 1 (z), A 0 (z), F satisfy the hypotheses **of** Theorem 1.2. Let d 0 (z), d 1 (z), d 2 (z) be entire functions that are not **all** equal to zero with ρ(d j ) < n (j = 0, 1, 2), ϕ(z) is an entire function with
finite order such that ψ(z) is not a solution **of** equation (1.3). If f (z) is a solution **of** (1.3), then the **differential** polynomial g f (z) = d 2 f 00 + d 1 f 0 + d 0 f satisfies λ(g f − ϕ) = ∞ with at most one finite order solution f 0 .

(ii) Let f (z) be an entire function with ρ (f ) = α ≤ ∞, and let L (f) = a k f (k) +
a k −1 f (k −1) + · · · + a 0 f, where a 0 , a 1 , · · · , a k are entire functions which are not **all**
equal zero and satisfy b = max {ρ (a j ) : j = 0, · · · , k} < α. Then ρ (L (f)) = α.
Lemma 2.2. [3] For any given equation **of** the form (3), there must exists a solution

satisfies σ 2 (f ) 6 σ.
Lemma 2.5 ([7]) Let f (z) be a transcendental meromorphic function, and let α > 1 be a given constant. Then there exist a set E 8 ⊂ (1, ∞) with finite
logarithmic measure and a constant B > 0 that depends only on α and i, j (0 6 i < j 6 k), such that for **all** z satisfying |z| = r / ∈ [0, 1] ∪ E 8 , we have

(ii) If ρ [p+1,q] (F ) < α, then **all** meromorphic **solutions** f whose poles are **of** uniformly
bounded multiplicities **of** equation (1.2) satisfy λ [p+1,q] (f ) = λ [p+1,q] (f ) = ρ [p+1,q] (f ) =
α with at most one exceptional solution f 0 satisfying ρ [p+1,q] (f 0 ) < α.
The main purpose **of** this paper is to consider the growth **of** meromorphic **solutions** **of** **equations** (1.1) and (1.2) with meromorphic coefficients **of** finite [p, q]-order in the complex plane. We obtain the following results which generalize and improve Theorem 9 and Theorem 10.

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exp {( 1 − ε ) ( 1 − ρ ) δ ( P d , θ ) r n } ≤ 2k. (3.25) This is absurd. Hence σ ( f ) ≥ n.
Assume f is a transcendental meromorphic solution whose poles are **of** uni- formly bounded multiplicity **of** equation ( 1.1 ) . By Lemma 2.1, there exist a con- stant B > 0 and a set E 1 ⊂ ( 1, + ∞ ) having finite logarithmic measure such that for **all** z satisfying | z | = r / ∈ [ 0, 1 ] ∪ E 1 , we have

order.
Let n ≥ 2 be an integer and consider the **linear** **differential** equation
A n (z) f (n) + A n−1 (z) f (n−1) + ... + A 1 (z) f ′ + A 0 (z) f = 0. (1.12) It is well-known that if A n ≡ 1, then **all** **solutions** **of** this equation are entire functions but when A n is a nonconstant entire function, equation (1.12) can possess meromorphic **solutions**. For instance the equation

λ (d) = σ(d) = λ(1/f) ≤ σ < σ(f) = σ(g) = +∞.
For each sufficiently large |z| = r, let z r = re iθ r be a point satisfying |g(z r )| = M(r, g). By Lemma 2.6, there exist a constant δ r (> 0), a sequence {r m } m∈N , r m → +∞ and a set E 5 **of** finite logarithmic measure such that the estimation (3.3) holds for **all** z satisfying |z| = r m ∈ E / 5 , r m → +∞ and arg z = θ ∈ [θ r − δ r , θ r + δ r ]. Since |g(z)| is continuous in |z| = r, then there exists a constant λ r (> 0) such that for **all** z satisfying |z| = r sufficiently large and arg z = θ ∈ [θ r − λ r , θ r + λ r ], we have

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A k (z) f (k) + A k−1 (z) f (k−1) + · · · + A 1 (z) f 0 + A 0 (z) f = 0 (1.4) and
A k (z) f (k) + A k−1 (z) f (k−1) + · · · + A 1 (z) f 0 + A 0 (z) f = F(z), (1.5) where k ≥ 2, A j (z) ( j = 0, 1, · · · , k) , F(z) are entire functions with A 0 A k F 6≡ 0, many authors investigated the properties **of** their **solutions** and obtained some interesting results, (see e.g. [3], [4], [7], [19]). It well-known that if A k (z) ≡ 1, then **all** **solutions** **of** (1.4) and (1.5) are entire functions, but when A k (z) is a nonconstant entire function, then equation (1.4) or (1.5) can possess meromorphic **solutions**. For instance the equation

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+ a 0 (z) f = 0 (1.1)
is well understood when the coefficients a j are polynomials [16, 17, 22, 27] or **of**
finite (iterated) order **of** growth in the complex plane [7, 14, 15, 21]. In particular, Wittich [27] showed that **all** **solutions** **of** (1.1) are entire functions **of** finite order if and only if **all** coefficients are polynomials, and Gundersen, Steinbart and Wang [17] listed **all** possible orders **of** **solutions** **of** (1.1) in terms **of** the degrees **of** the polynomial coefficients.

Theorem 1.4 Let P j (z) = P n
i=0 a i,j z i (j = 0, 1, . . . , k − 1) be noncon- stant polynomials, where a 0,j , a 1,j , . . . , a n,j (j = 0, 1, . . . , k − 1) are com- plex numbers such that a n,j a n,0 6= 0 (j = 1, 2, . . . , k − 1), let h j (z), d j (z) (j = 0, 1, . . . , k − 1) be meromorphic functions with h 0 6≡ 0. Suppose that arg an,j 6= arg a n,0 (j = 1, 2, . . . , k − 1), arg(an,1 + an,j ) 6= arg an,0 (j = 2, 3) or a n,j = cj a n,0 (0 < cj < 1) (j = 1, 2, . . . , k − 1) and ρ = max{ρ(h j ), ρ(d j ) : j = 0, 1, . . . , k − 1} < n. Then for any meromorphic solution f 6≡ 0 **of** equation (1.4), f , f 0 , f 00 **all** have infinitely many fixed points and satisfy

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where (d 5 > 0, d 6 > 0) are some constants. This is a contradiction. Therefore,
when max {c 1 , ..., c s−1 } < c 0 , every solution f ≡ 0 **of** (1.7) has inﬁnite order. Now we prove that ρ 2 (f ) = n. Put c = max {cj : j = s} , then 0 < c <
1. Since deg Ps > deg (P j − c j P s) (j = s) , by Lemma 2.5, there exist real numbers b > 0, λ, R 2 and θ 1 < θ 2 such that for **all** r ≥ R 2 and θ 1 ≤ θ ≤ θ 2 ,

(i) max{ρ [p,q] (A i ) : i = 1, 2, · · · , k − 1} < ρ [p,q] (A 0 ) = ρ (0 < ρ < +∞) or that (ii) max{ρ [p,q] (A i ) : i = 1, 2, · · · , k − 1} ≤ ρ [p,q] (A 0 ) = ρ (0 < ρ < +∞) and max{τ M,[p,q] (A i ) : ρ [p,q] (A i ) = ρ [p,q] (A 0 )} <
τ M,[p,q] (A 0 ) = τ (0 < τ < +∞) . Let d j (z) (j = 0, 1, · · · , k) be finite [p, q] −order entire functions that
are not **all** vanishing identically such that h 6≡ 0. If f 6≡ 0 is a solution **of** (2.1), then the **differential** polynomial (1.2) satisfies

(ii) There exists a set E 4 ⊂ [0, 2π) which has **linear** measure zero, such that
if θ ∈ [0, 2π) \ E 4 , then there is a constant R = R (θ) > 0 such that (2.1)
holds for **all** z satisfying arg z = θ and |z| ≥ R.
Lemma 2.4 ([12]). Let f (z) be an entire function and suppose that G(z) := log

f (k) + A (z) f = 0 (k ≥ 2) , where A is a meromorphic function in the complex plane.
1. Introduction and main results
In this paper, we shall assume that the reader is familiar with the fundamental results and the standard notations **of** the Nevanlinna value distribution theory **of** meromorphic functions see [10, 17]. For the definition **of** the iterated order **of** a meromorphic function, we use the same definition as in [11] , ([2], p. 317), ([12], p. 129). For **all** r ∈ R, we define exp 1 r := e r and exp p+1 r := exp exp p r , p ∈ N. We also define for **all** r sufficiently large log 1 r := log r and log p+1 r := log log p r , p ∈ N. Moreover, we denote by exp 0 r := r, log 0 r := r, log −1 r := exp 1 r and exp −1 r := log 1 r. Definition 1.1. [11, 12] Let f be a meromorphic function. Then the iterated p−order ρ p (f ) **of** f is defined as

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Then i λ (f ) = i λ (f ) = i (f ) = p + 1 and λ p+1 (f ) = λ p+1 (f ) = σ p+1 (f ).
To avoid some problems caused by the exceptional set, we recall the following lemma. Lemma 2.9 [9] Let g : [0, +∞) → R and h : [0, +∞) → R be monotone non- decreasing functions such that g (r) 6 h (r) for **all** r / ∈ E 5 ∪ [0, 1], where E 5 ⊂ (1, +∞) is a set **of** finite logarithmic measure. Let α > 1 be a given constant. Then there exists an r 0 = r 0 (α) > 0 such that g (r) 6 h (αr) for **all** r > r 0 .

Theorem 3 [11] Let k > 2 and A (z) be a transcendental meromorphic function **of** finite iterated order ρ p (A) = ρ > 0 such that δ (∞, A) = δ > 0. Suppose, moreover, that either:
(i) **all** poles **of** f are **of** uniformly bounded multiplicity or that
(ii) δ (∞, f) > 0. If ϕ (z) 6≡ 0 is a meromorphic function with finite iterated p−order ρ p (ϕ) < +∞, then every meromorphic solution f (z) 6≡ 0 **of** (1) satisfies

Abstract. In this paper we investigate the growth **of** **solutions** **of** the **differential** equation f .k/ C A k 1 .´/ f .k 1/ C C A 1 .´/ f 0 C A 0 .´/ f D 0; where A 0 .´/ ; : : : ; A k 1 .´/ are entire func- tions with 0 < .A 0 / 1=2: We will show that if there exists a real constant ˇ < .A 0 / and a set E ˇ .1; C1/ with log dens E ˇ D 1; such that for **all** r 2 E ˇ ; we have min j´jDr jA j .´/ j exp .r ˇ / .j D 1; 2; : : : ; k 1/ ; then every solution f 6 0 **of** the above **differential** equation is **of** infinite order with hyper-order 2 .f / .A 0 /. The paper extends previous results by the author and Hamani.

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