Some Oscillation Results of Higher-Order Linear Differential Equations with Meromorphic Coefficients

Abstract

We investigate the growth of solutions of higher-order nonhomogeneous linear differential equations with meromorphic coefficients. We also discuss the relationship between small functions and solutions of such equations.

1. Introduction and Main Results

Let 𝑓(𝑧) be a meromorphic function in the whole complex plane. Throughout this paper, we assume that the reader is familiar with the fundamental results of the Nevanlinna’s value distribution theory of meromorphic functions and the standard notations such as the order 𝜎(𝑓), the exponent of convergence of zero-sequence 𝜆(𝑓), and the exponent of convergence of the sequence of distinct zeros 𝜆(𝑓). Moreover, a meromorphic function 𝜓(𝑧) is called a small function with respect to 𝑓(𝑧) if 𝑇(𝑟,𝜓)=𝑜(𝑇(𝑟,𝑓)) as 𝑟→∞, possible outside of a set of 𝑟 with finite measure, where 𝑇(𝑟,𝑓) is the Nevanlinna characteristic function of 𝑓(𝑧). The study of oscillation of solutions of linear differential equations has attracted many interests since the work of Bank and Laine; for more details see [1]. One of the main subject of this research is the growth and zero distribution of solutions of linear differential equations. In this paper, we first discuss the growth of solutions of higher-order linear differential equation𝑓^(𝑘)+𝐻_𝑘−1𝑓^(𝑘−1)+⋯+𝐻₀𝑓=𝐹(𝑧),𝑘≥2, (1.1) where 𝐻_𝑗(𝑗=1,…,𝑘−1), 𝐻₀≢0 and 𝐹≢0 are entire functions of finite order. Some results on the growth of solutions of (1.1) have been obtained by several researchers, see [2–4]. In the case that the coefficients 𝐻_𝑗(𝑗=0,…,𝑘−1) are polynomials, the growth of solutions of (1.1) has been extensively studied; see [2]. In 1992, Hellerstein et al. [3] proved that every transcendental solution of (1.1) is of infinite order, if there exists some 𝑑∈{0,1,…,𝑘−1} such that max_𝑗≠𝑑{𝜎(𝐻_𝑗),𝜎(𝐹)}<𝜎(𝐻_𝑑)≤1/2. As for sectorial growth conditions on the coefficients of (1.1) that imply that all solutions are of infinite order, see, for example, [5]. In addition, a special case for this was investigated by Wang and Laine [6]. Recently, Wang and Laine [4] obtain the following.

Theorem A. Suppose that 𝐻_𝑗(𝑧)=𝐷_𝑗(𝑧)𝑒^{𝑃𝑗(𝑧)}(𝑗=0,…,𝑘−1) where 𝑃_𝑗(𝑧)=𝑎_𝑗𝑛𝑧^𝑛+⋯+𝑎_𝑗0(𝑗=0,…,𝑘−1) are polynomials with degree 𝑛≥1, 𝐷_𝑗(𝑧) are entire functions of order less than 𝑛, not all vanishing, and 𝐹(𝑧)≢0 is an entire function of order less than 𝑛. If 𝑎_𝑗𝑛(𝑗=0,…,𝑘−1) are distinct complex numbers, then every solution of (1.1) is of infinite order.

Thus a natural question is whether every meromorphic solution of (1.1) has infinite order, if the coefficients of (1.1) are meromorphic? The following theorem partially answers this question.

Theorem 1.1. Let 𝐴_𝑗(𝑧)(𝑗=1,…,𝑘−1), 𝐴₀≢0 and 𝐹≢0 be meromorphic functions with max{𝜎(𝐴_𝑗),𝑗=0,1,…,k−1),𝜎(𝐹)}<1, and let 𝑎₀,𝑎₁,…,𝑎_𝑘−1 be complex constants such that (i) arg𝑎_𝑗=arg𝑎₀ and 𝑎_𝑗=𝑐_𝑗𝑎₀(0<𝑐_𝑗<1) or (ii) arg𝑎_𝑗≠arg𝑎₀(𝑗=1,…,𝑘−1). Then every meromorphic solution 𝑓≢0 of the equation 𝑓^(𝑘)+𝐴_𝑘−1𝑒^{𝑎𝑘−1𝑧}𝑓^(𝑘−1)+⋯+𝐴₀𝑒^𝑎0𝑧𝑓=𝐹,𝑘≥2, (1.2) satisfies 𝜆(𝑓)=𝜆(𝑓)=𝜎(𝑓)=∞.

Remark 1.2. In Theorem 1.1, if 𝑎_𝑗𝑧(𝑗=0,…,𝑘−1) are replaced by 𝑃_𝑗(𝑧)=𝑎_𝑗𝑛𝑧^𝑛+⋯+𝑎_𝑗0(𝑗=0,…,𝑘−1), with additional hypothesis on 𝑎_𝑗𝑛 similar to 𝑎_𝑗, and furthermore 𝜎(𝐹) is replaced by 𝜎(𝐹)<𝑛, then we have the same conclusion with Theorem 1.1.

In 2000, Chen [7] first studied the fixed points of solutions of second-order linear differential equations and obtained some precise estimation on the number of fixed points. After that, a number of results on fixed points of solutions of differential equations with entire coefficients were obtained; see [8–10]. In 2006, Chen and Shon [11] further studied the relation between small functions and solutions of differential equations and obtain the following.

Theorem B. Let 𝐴_𝑗(𝑧)≢0(𝑗=0,1) be entire functions with 𝜎(𝐴_𝑗)<1, and let 𝑎,𝑏 be complex constants such that 𝑎𝑏≠0 and arg𝑎≠arg𝑏 or 𝑎=𝑐𝑏(0<𝑐<1). If 𝜓(𝑧)≢0 is an entire function with finite order, then every solution 𝑓≢0 of equation 𝑓^′′+𝐴₁(𝑧)𝑒^𝑎𝑧𝑓′+𝐴₀(𝑧)𝑒^𝑏𝑧𝑓=0 (1.3) satisfies 𝜆(𝑓−𝜓)=𝜆(𝑓′−𝜓)=𝜆(𝑓^′′−𝜓)=∞.

Motivated by Theorem B, we try to consider the relation between small functions with meromorphic solutions of (1.2). Indeed, such relationship on higher order differential equations is more difficult than that of second order differential equations. Moreover, the method used in the proof of Theorem B cannot deal with the case of higher-order linear differential equations.

Theorem 1.3. Under the assumption of Theorem 1.1, if 𝜙≢0 is a meromorphic function of finite order and satisfies max{𝜆(𝜙),𝜆(1/𝜙)}<1, then every non-trivial meromorphic solution of (1.2) satisfies 𝜆(𝑓−𝜙)=𝜆(𝑓′−𝜙)=∞.

Remark 1.4. As the remark on Theorem 1.1 in Remark 1.2, the conclusion of Theorem 1.3 can also be extended to the case that 𝑎_𝑗𝑧(𝑗=0,…,𝑘−1) are replaced by 𝑃_𝑗(𝑧)=𝑎_𝑗𝑛𝑧^𝑛+⋯+𝑎_𝑗0(𝑗=0,…,𝑘−1), with some similar additional hypothesis on 𝑎_𝑗𝑛, 𝐹 and 𝜙 in Theorem 1.3.

2. Preliminary Lemmas

To prove our theorems, we need some lemmas.

Lemma 2.1 (see [12]). Let 𝑤(𝑧) be a transcendental meromorphic function with 𝜎(𝑓)=𝜎<∞. Let Γ={(𝑘₁,𝑗₁),…,(𝑘_𝑚,𝑗_𝑚)} be a finite set of distinct pairs of integers satisfying 𝑘_𝑖>𝑗_𝑖≥0 for 𝑖=1,2,…,𝑚. Also let 𝜖>0 be a given constant. Then, there exists a set 𝐸₁⊂(1,∞) that has finite logarithmic measure, such that for all 𝑧 satisfying 𝑧∉𝐸∪[0,1] and for all (𝑘,𝑗)∈Γ, One has 𝑤^(𝑘)(𝑧)𝑤^(𝑗)(𝑧≤|𝑧|)^{(𝑘−𝑗)(𝜎−1+𝜀)}. (2.1)

Let 𝑃(𝑧)=(𝛼+𝛽𝑖)𝑧^𝑛+⋯ is a nonconstant polynomial and 𝛼,𝛽 are real constants. For 𝜃∈[0,2𝜋), set 𝛿(𝑃(𝑧),𝜃)=𝛼cos𝑛𝜃−𝛽sin𝑛𝜃.

Lemma 2.2 (see [13]). Let 𝑃(𝑧) be a non-constant polynomial of degree 𝑛. Let 𝑤(𝑧) be a meromorphic function, not identically zero, of order less than 𝑛, and set 𝑔(𝑧)=𝑤(𝑧)𝑒^𝑃(𝑧). Then for any given 𝜀>0 there exists a zero measure set 𝐻₁⊂[0,2𝜋) such that if 𝜃∈𝜃∈[0,2𝜋)⧵(𝐻₁∪𝐻₂), then for |𝑧|>𝑟(𝜃).(1)If 𝛿(𝑃,𝜃)<0, then exp((1+𝜖)𝛿(𝑃,𝜃)𝑟^𝑛)≤|𝑔(𝑟𝑒^𝑖𝜃)|≤exp((1−𝜖)𝛿(𝑃,𝜃)𝑟^𝑛);(2)If 𝛿(𝑃,𝜃)>0, then exp((1−𝜖)𝛿(𝑃,𝜃)𝑟^𝑛)≤|𝑔(𝑟𝑒^𝑖𝜃)|≤exp((1+𝜖)𝛿(𝑃,𝜃)𝑟^𝑛), where 𝐻₂={𝜃∶𝛿(𝑃,𝜃)=0,0≤𝜃<2𝜋} is a finite set.

Lemma 2.3 (see [14]). Let 𝐴₀,…,𝐴_𝑘−1,𝐹≢0 be finite-order meromorphic functions. If 𝑓 is an infinite-order meromorphic solution of the equation f^(𝑘)+𝐴_𝑘−1𝑓^(𝑘−1)+⋯+𝐴₀𝑓=𝐹, (2.2) then 𝑓 satisfies 𝜆(𝑓)=𝜆(𝑓)=𝜎(𝑓)=∞.

Lemma 2.4 (see [15], page 79). Suppose that 𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧)(𝑛≥2) are meromorphic functions and 𝑔₁(𝑧),𝑔₂(𝑧),…,𝑔_𝑛(𝑧) are entire functions satisfying the following conditions:(1)∑^𝑛_𝑗=1𝑓_𝑗(𝑧)𝑒^{𝑔𝑗(𝑧)}≡0;(2)𝑔_𝑗(𝑧)−𝑔_𝑘(𝑧) are not constants for 1≤𝑗<𝑘≤𝑛;(3)For 1≤𝑗≤𝑛,1≤ℎ<𝑘≤𝑛, 𝑇(𝑟,𝑓_𝑗)=𝑜{𝑇(𝑟,𝑒^{𝑔ℎ−𝑔𝑘})}(𝑟→∞,𝑟∉𝐸), where 𝐸 has a finite measure.

Then 𝑓_𝑗(𝑧)≡0(𝑗=1,2,…,𝑛).

Lemma 2.5 (see [4], Lemma 2.6). Let 𝑓(𝑧) be a an entire function of order 𝜎(𝑓)=𝜎<∞. Suppose that there exists a set 𝐸⊂[0,2𝜋) which has linear measure zero, such that log⁺|𝑓(𝑟𝑒^𝑖𝜃)|≤𝑀𝑟^𝜌 for any ray arg𝑧=𝜃∈[0,2𝜋)⧵𝐸, where 𝑀 is a positive constant depending on 𝜃, while 𝜌 is a positive constant independent of 𝜃. Then 𝜎(𝑓)≤𝜌.

Lemma 2.6 (see [16]). Let 𝑓(𝑧) be a transcendental meromorphic function of order 𝜎(𝑓)=𝜎<∞. Then for any given 𝜀>0, there exists a set 𝐸₃⊂[0,2𝜋) which has linear measure zero, such that if 𝜓₁∈[0,2𝜋)⧵𝐸₃, then there is a constant 𝑅₁=𝑅₁(𝜓₁)>1 such that for all 𝑧 satisfying arg𝑧=𝜓₁ and |𝑧|=𝑟≥𝑅₂, One has exp−𝑟^𝜎+𝜀≤𝑓(𝑧)≤exp𝑟^𝜎+𝜀. (2.3)

3. Proof of Theorem 1.1

Assume that 𝑓(𝑧) is a meromorphic solution of (1.2) of finite order. We will deduce a contradiction later. First, we have 𝜎(𝑓)≥1. Otherwise, it follows from Lemma 2.4 that 𝐴₀≡0 which is a contradiction.

Since 𝑓 is a meromorphic solution of (1.2), we know that the poles of 𝑓 can occur only at the poles of 𝐴_𝑗(𝑗=0,…,𝑘−1) and 𝐹. Let 𝑓=𝑔(𝑧)/𝑑(𝑧), where 𝑑(𝑧) is the canonical product formed with the nonzero poles of 𝑓(𝑧), with 𝜎(𝑑)≤max{𝜎(𝐹),𝜎(𝐴_𝑗),𝑗=0,…,𝑘−1}<1, and 𝑔 is an entire function with 1≤𝜎(𝑔)=𝜎(𝑓)=𝜎<∞. Substituting 𝑓=𝑔/𝑑 into (1.2), by some calculation we can get𝑑𝐹=𝑔^(𝑘)+𝑔^(𝑘−1)𝐴_𝑘−1𝑒^{𝑎𝑘−1𝑧}+𝐵_{𝑘,𝑘−1}𝐴+⋯+𝑔′₁𝑒^𝑎1𝑧+^𝑘−1_𝑖=2𝐴_𝑖𝑒^𝑎𝑖𝑧𝐵_𝑖,1+𝐵_𝑘,1+𝑔𝐴₀𝑒^𝑎0𝑧+^𝑘−1_𝑖=1𝐴_𝑖𝑒^𝑎𝑖𝑧𝐵_𝑖,1+𝐵_𝑘,0, (3.1) where 𝐵_𝑖,𝑗 are defined as a sum of a finite number of terms of the type_{𝑗1⋯𝑗𝑖}𝐶_{𝑗𝑗1⋯𝑗𝑖}𝑑^′𝑑^𝑗1⋯𝑑^(𝑖)𝑑^𝑗𝑖. (3.2)𝐶_{𝑗𝑗1⋯𝑗𝑖} are constants, and 𝑗+𝑗₁+2𝑗₂+⋯+𝑖𝑗_𝑖=𝑘. Now, we rewrite (3.1) into𝑑𝐹𝑔=𝑔^(𝑘)𝑔+⋯+𝐴₁𝑒^𝑎1𝑧+^𝑘−1_𝑖=2𝐴_𝑖𝑒^𝑎𝑖𝑧𝐵_𝑖,1+𝐵_𝑘,1𝑔^′𝑔+𝐴₀𝑒^𝑎0𝑧+^𝑘−1_𝑖=1𝐴_𝑖𝑒^𝑎𝑖𝑧𝐵_𝑖,1+𝐵_𝑘,0. (3.3)

Set max{𝜎(𝐴_𝑗),𝜎(𝐹),𝜎(𝑑),𝑗=0,…,𝑘−1}=𝛽. By Lemma 2.1, for any given 𝜀>0 there exists a set 𝐸₁∈[0,2𝜋) which has linear measure zero, such that if 𝜃∈[0,2𝜋)⧵𝐸₁, then there is a constant 𝑅₁=𝑟₁(𝜃)>1 such that for all 𝑧 satisfying arg𝑧=𝜃 and |𝑧|≥𝑅₁, we have𝑔^(𝑖)(𝑧)𝑔(𝑧≤|𝑧|)^{𝑘(𝜎−1+𝜀)},𝑑^(𝑖)(𝑧)|𝑑(𝑧≤|𝑧|)|^{𝑘(𝛽−1+𝜀)},𝑖=1,…,𝑘. (3.4) Meanwhile, by Lemma 2.6, for the above 𝜀 there exists a set 𝐸₂ of zero linear measure, such that if 𝜃∈[0,2𝜋)⧵𝐸₂, we have for sufficiently large 𝑟=|𝑧|,𝐴_𝑗(𝑧)≤exp𝑟^𝛽+𝜀,𝑗=0,…,𝑘−1. (3.5) Next we divide our proof into two cases.

Case 1. Suppose that arg𝑎_𝑗=arg𝑎₀ and 𝑎_𝑗=𝑐_𝑗𝑎₀(0<𝑐_𝑗<1),𝑗=1,…,𝑘−1. Then for any ray arg𝑧=𝜃, we have 𝛿𝑎_𝑗𝑧,𝜃=𝑐_𝑗𝛿𝑎₀𝑧,𝜃. (3.6)
By Lemma 2.2, for any given 𝜀 we can find a ray arg𝑧=𝜃∈[0,2𝜋)⧵𝐸₁∪𝐸₂∪𝐻₁∪𝐻₂, where 𝐻₁ and 𝐻₂ are defined in Lemma 2.2, 𝐸₁∪𝐸₂∪𝐻₁∪𝐻₂ has zero linear measure, such that 𝛿(𝑎_𝑗𝑧,𝜃)=𝑐_𝑗𝛿(𝑎₀𝑧,𝜃)>0, and for sufficiently large 𝑟, 𝐴₀𝑟𝑒^𝑖𝜃𝑒^{𝑎0𝑟𝑒𝑖𝜃}≥exp(1−𝜀)𝛿𝑎₀𝑟,𝐴𝑧,𝜃_𝑗𝑟𝑒^𝑖𝜃𝑒^{𝑎𝑗𝑟𝑒𝑖𝜃}≤exp(1+𝜀)𝑐_𝑗𝛿𝑎₀𝑧,𝜃𝑟,𝑗=1,…,𝑘−1. (3.7)
Thus, by (3.4) and (3.7), we have for sufficiently large 𝑟, 𝐴₀𝑒^𝑎0𝑧+^𝑘−1_𝑖=1𝐴_𝑖𝑒^𝑎𝑖𝑧𝐵_𝑖,1+𝐵_𝑘,0≥𝑀₀exp(1−𝜀)𝛿𝑎₀𝑟,𝐴𝑧,𝜃_𝑘−1𝑒^{𝑎𝑘−1𝑧}+𝐵_{𝑘,𝑘−1}≤𝑀_𝑘−1exp(1−𝜀)𝛿𝑎_𝑗𝑧,𝜃𝑟=𝑀_𝑘−1exp(1−𝜀)𝑐_𝑘−1𝛿𝑎₀𝑟,⋯𝐴𝑧,𝜃₁𝑒^𝑎1𝑧+^𝑘−1_𝑖=2𝐴_𝑖𝑒^𝑎𝑖𝑧𝐵_𝑖,1+𝐵_𝑘,1≤𝑀₁exp(1+𝜀)𝛿𝑎₁𝑧,𝜃𝑟=𝑀₁exp(1+𝜀)𝑐₁𝛿𝑎₀𝑧,𝜃𝑟, (3.8) where 𝑀_𝑗(𝑗=0,…,𝑘−1) are positive constants.
For 𝜃∈[0,2𝜋)⧵𝐸₁∪𝐸₂∪𝐻₁∪𝐻₂, we claim that log⁺𝑔(𝑧)|𝑧|^𝛽+𝜀 (3.9) is bounded on the ray arg𝑧=𝜃. Otherwise, there exists a sequence of points 𝑧_𝑚=𝑟_𝑚𝑒^𝑖𝜃, such that 𝑟_𝑚→∞, and that log⁺𝑔𝑧_𝑚𝑟_𝑚^𝛽+𝜀⟶∞. (3.10) From (3.10) and the definition of order, we see that 𝑑𝑧_𝑚𝐹𝑧_𝑚𝑔𝑧_𝑚⟶0 (3.11) for 𝑚 is large enough.
By (3.3),(3.4), and (3.8)–(3.11), we get for sufficiently large 𝑚, 𝑀₀exp(1−𝜀)𝛿𝑎₀𝑧,𝜃𝑟_𝑚≤𝐴₀𝑒^{𝑎0𝑟𝑚𝑒𝑖𝜃}+^𝑘−1_𝑖=1𝐴_𝑖𝑒^{𝑎𝑖𝑟𝑚𝑒𝑖𝜃}𝐵_𝑖,1+𝐵_𝑘,0≤−𝑑𝑟_𝑚𝑒^𝑖𝜃𝐹𝑟_𝑚𝑒^𝑖𝜃𝑔+𝑔^(𝑘)𝑔+⋯+𝐴₁𝑒^{𝑎1𝑟𝑚𝑒𝑖𝜃}+^𝑘−1_𝑖=2𝐴_𝑖𝑒^{𝑎𝑖𝑟𝑚𝑒𝑖𝜃}𝐵_𝑖,1+𝐵_𝑘,1𝑔^′𝑔≤^𝑘−1_𝑗=1𝑟^{𝑘(𝜎+𝜀)}𝑀_𝑗exp(1+𝜀)𝑐_𝑗𝛿𝑎₀𝑧,𝜃𝑟_𝑚𝑒^𝑖𝜃+𝑟^{𝑘(𝜎−1+𝜀)}. (3.12) Clearly, we can choose sufficiently small 𝜀 such that 0<𝜀<min{(1−𝑐_𝑗)/(1+𝑐_𝑗)∶𝑗=1,…,𝑘−1}. Then by (3.12), we can obtain a contradiction provided that 𝑚 is sufficiently large.
Therefore, log⁺𝑔(𝑧)|𝑧|^𝛽+𝜀 (3.13) is bounded, and we have |𝑔(𝑧)|≤𝑀exp{𝑟^𝛽+𝜀} on the ray arg𝑧=𝜃.

Case 2. Suppose that arg𝑎_𝑗≠arg𝑎₀(𝑗=1,…,𝑘−1). By Lemma 2.2, there exists a ray arg𝑧=𝜃∈[0,2𝜋)⧵𝐸₁∪𝐸₂∪𝐻₁∪𝐻₂, where 𝐸₁,𝐸₂,𝐻₁, and 𝐻₂ are defined, respectively, as in Case 1, such that 𝛿𝑎₀𝑧,𝜃>0,𝛿𝑎_𝑗𝑧,𝜃<0,𝑗=1,…,𝑘−1. (3.14) Further, for any given 𝜀>0, we have for sufficiently large 𝑟, 𝐴₀𝑟𝑒^𝑖𝜃𝑒^{𝑎0𝑟𝑒𝑖𝜃}≥exp(1−𝜀)𝛿𝑎₀𝑟,𝐴𝑧,𝜃_𝑗𝑟𝑒^𝑖𝜃𝑒^{𝑎𝑗𝑟𝑒𝑖𝜃}≤exp(1−𝜀)𝛿𝑎_𝑗𝑧,𝜃𝑟,𝑗=1,…,𝑘−1. (3.15)
Thus, by (3.4) and (3.15), we have for sufficiently large 𝑟, 𝐴₀𝑒^𝑎0𝑧+^𝑘−1_𝑖=1𝐴_𝑖𝑒^𝑎𝑖𝑧𝐵_𝑖,1+𝐵_𝑘,0≥𝑀₀exp(1−𝜀)𝛿𝑎₀𝑟,𝐴𝑧,𝜃_𝑘−1𝑒^{𝑎𝑘−1𝑧}+𝐵_{𝑘,𝑘−1}≤𝑀_𝑘−1𝑟^{𝑘(𝜎−1+𝜀)},⋯𝐴₁𝑒^𝑎1𝑧+^𝑘−1_𝑖=2𝐴_𝑖𝑒^𝑎𝑖𝑧𝐵_𝑖,1+𝐵_𝑘,1≤𝑀₁𝑟^{𝑘(𝜎−1+𝜀)}, (3.16) where 𝑀_𝑗(𝑗=0,…,𝑘−1) are positive constants.
For 𝜃∈[0,2𝜋)⧵𝐸₁∪𝐸₂∪𝐻₁∪𝐻₂, we claim that log⁺𝑔(𝑧)|𝑧|^𝛽+𝜀 (3.17) is bounded on the ray arg𝑧=𝜃. Otherwise, there exists a sequence of points 𝑧_𝑚=𝑟_𝑚𝑒^𝑖𝜃, such that 𝑟_𝑚→∞, and that log⁺𝑔𝑧_𝑚𝑟_𝑚^𝛽+𝜀⟶∞. (3.18) From (3.18) and the definition of order, we see that 𝑑𝑧_𝑚𝐹𝑧_𝑚𝑔𝑧_𝑚⟶0 (3.19) for 𝑚 is large enough.
Then by similar reasoning as in Case 1, we can also obtain a contradiction. So log⁺𝑔(𝑧)|𝑧|^𝛽+𝜀 (3.20) is bounded, and we have |𝑔(𝑧)|≤𝑀exp{𝑟^𝛽+𝜀} on the ray arg𝑧=𝜃.
Combining Case 1 and Case 2, for any given ray arg𝑧=𝜃∈[0,2𝜋)⧵𝐸, 𝐸 of linear measure zero, we have |𝑔(𝑧)|≤𝑀exp{𝑟^𝛽+𝜀} on the ray arg𝑧=𝜃, provided that 𝑟 is sufficiently large. Thus by Lemma 2.5, we get 𝜎(𝑔)≤𝛽+𝜀<1, which is a contradiction. Again by Lemma 2.3, we complete the proof of Theorem 1.1.

4. Proof of Theorem 1.3

Let 𝑓(𝑧)≢0 be a meromorphic solution of (1.2). Set 𝐵_𝑗(𝑧)=𝐴_𝑗(𝑧)𝑒^𝑎𝑗𝑧. First, we prove that 𝜆(𝑓−𝜙)=∞.

Set 𝑓=𝑔₀+𝜙. By Theorem 1.1, we have 𝜎(𝑔₀)=𝜎(𝑓)=∞. Substituting 𝑓=𝑔₀+𝜙 into (1.2), we get𝑔₀^(𝑘)+𝐵_𝑘−1𝑔₀^(𝑘−1)+⋯+𝐵₀𝑔₀=𝐹−𝜙^(𝑘)+𝐵_𝑘−1𝜙^(𝑘−1)+⋯+𝐵₀𝜙. (4.1) We remark that (4.1) may have finite-order solutions, but we only need to discuss the solutions whose order are ∞.

If 𝜙^(𝑘)+𝐵_𝑘−1𝜙^(𝑘−1)+⋯+𝐵₀𝜙=𝐹, then by Theorem 1.1, we have 𝜎(𝜙)=∞. It is a contradiction. Hence we have 𝐹−(𝜙^(𝑘)+𝐵_𝑘−1𝜙^(𝑘−1)+⋯+𝐵₀𝜙)≢0. It follows from Lemmas 3 and (4.1) that 𝜆(𝑔₀)=𝜆(𝑔₀)=𝜎(𝑔₀)=∞.

Now we prove that 𝜆(𝑓′−𝜙)=∞. Set 𝑔₁=𝑓′−𝜙. Differentiating both side of (1.2), we get𝑓^(𝑘+1)+𝐵_𝑘−1𝑓^(𝑘)+𝐵^′_𝑘−1+𝐵_𝑘−2𝑓^(𝑘−1)+⋯+𝐵^′₁+𝐵₀𝑓′+𝐵^′₀𝑓=𝐹′. (4.2) Rewriting (1.2), we have1𝑓=𝐹−𝐵₀𝑓^(𝑘)+𝐵_𝑘−1𝑓^(𝑘−1)+⋯+𝐵₁𝑓^′. (4.3) Substituting (4.3) into (4.2), we get𝑓^(𝑘+1)+𝐵_𝑘−1−𝐵^′₀𝐵₀𝑓^(𝑘)+𝐵^′_𝑘−1+𝐵_𝑘−2−𝐵^′₀𝐵₀𝐵_𝑘−1𝑓^(𝑘−1)+⋯+𝐵^′₁+𝐵₀−𝐵^′₀𝐵₀𝐵₁𝑓′=𝐹′−𝐵^′₀𝐹. (4.4) We rewrite (4.4) into the following form:𝑓^(𝑘+1)+𝐶_𝑘−1𝑓^(𝑘)+𝐶_𝑘−2𝑓^(𝑘−1)+⋯+𝐶₀𝑓′=𝐹′−𝐵^′₀𝐹, (4.5) where 𝐶_𝑘−1=𝐵_𝑘−1−𝐵^′₀/𝐵₀,𝐶_𝑗=𝐵^′_𝑗+1+𝐵_𝑗−(𝐵^′₀/𝐵₀)𝐵_𝑗+1,𝑗=0,…,𝑘−2. Substituting 𝑓^′=𝑔₁+𝜙,…,𝑓^(𝑘+1)=𝑔₁^(𝑘)+𝜙^(𝑘) into (4.5), we have𝑔₁^(𝑘)+𝐶_𝑘−1𝑔₁^(𝑘−1)+𝐶_𝑘−2𝑔₁^(𝑘−2)+⋯+𝐶₀𝑔₁=𝐹′−𝐵^′₀𝐹−𝜙^(𝑘)+𝐶_𝑘−1𝜙^(𝑘−1)+𝐶_𝑘−2𝜙^(𝑘−2)+⋯+𝐶₀𝜙. (4.6)

Next we prove𝑄₁(𝑧)=𝜙^(𝑘)+𝐶_𝑘−1𝜙^(𝑘−1)+𝐶_𝑘−2𝜙^(𝑘−2)+⋯+𝐶₀𝜙−𝐹′+𝐵^′₀𝐹≢0. (4.7) Conversely, assume that 𝑄₁(𝑧)≡0.

Dividing 𝜙 into both sides of 𝑄₁(𝑧)≡0, we have𝐷_𝑘−1𝑒^{𝑎𝑘−1𝑧}+⋯+𝐷₁𝑒^𝑎1𝑧+𝐷₀𝑒^𝑎0𝑧+𝐷(𝑧)≡0, (4.8) where𝐷₀=𝐴₀+𝐴^′₀𝐹+𝑎₀𝐴₀𝐷𝐹,₁=𝐴₁𝜙^′𝜙+𝐴^′₁+𝐴₁𝑎₁−𝐴^′₀𝐴₀𝐴₁,𝐷₂=𝐴₂𝜙^′′𝜙+𝐴^′₂+𝐴₂𝑎₂−𝐴^′₀𝐴₀𝐴₂𝜙^′𝜙,⋯𝐷_𝑗=𝐴_𝑗𝜙^(𝑗)𝜙+𝐴^′_𝑗+𝐴_𝑗𝑎_𝑗−𝐴^′₀𝐴₀𝐴_𝑗𝜙^(𝑗−1)𝜙,⋯𝐷_𝑘−1=𝐴_𝑘−1𝜙^(𝑘−1)𝜙+𝐴^′_𝑘−1+𝐴_𝑘−1𝑎_𝑘−1−𝐴^′₀𝐴₀𝐴_𝑘−1𝜙^(𝑘−2)𝜙,𝜙𝐷=^(𝑘)𝜙−𝐴^′₀𝐴₀𝜙^(𝑘−1)𝜙−𝐹^′. (4.9) Set max{𝜎(𝐴_𝑗),𝜎(𝐹),𝑗=0,…,𝑘−1}=𝛽. Clearly from (4.9),𝜎𝐷₀(𝑧)≤𝛽. (4.10) Meanwhile, by (4.9), we have𝑚𝑟,𝐷_𝑗≤3𝑚𝑟,𝐴_𝑗𝜙+𝑚𝑟,^(𝑗)𝜙𝜙+𝑚𝑟,^(𝑗−1)𝜙+𝑚𝑟,𝐴^′_𝑗𝐴+𝑚𝑟,^′0𝐴₀+𝑂(1)≤4𝑚𝑟,𝐴_𝑗𝑚(),+𝑂log𝑟,𝑗=1,…,𝑘−1,𝑟,𝐷≤𝑚𝑟,𝐹′+𝑂log𝑟 (4.11) outside a possible exceptional set 𝐸 of finite linear measure. On the other hand, again by (4.9), we have𝑁𝑟,𝐷_𝑗1≤𝑁𝑟,𝜙1+𝑁𝑟,𝐴₀+𝑘𝑁𝑟,𝐴₀+𝑁𝑟,𝜙+𝑁𝑟,𝐴_𝑗,𝑁()1𝑗=1,…,𝑘−1,𝑟,𝐷≤𝑁𝑟,𝜙1+𝑁𝑟,𝐴₀+2𝑁𝑟,𝐴₀+𝑘𝑁𝑟,𝜙+𝑁𝑟,𝐹^′. (4.12) Combining (4.11) and (4.12), we get𝜎𝐷_𝑗𝜆𝜙1≤max,𝜆𝜙,𝜎𝐴_𝑗𝜎(𝐷)𝜆𝜙1,𝑗=0,…,𝑘−1<1,𝑗=1,…,𝑘−1,≤max,𝜆𝜙,𝜎𝐹^′,𝜎𝐴₀<1. (4.13) By Lemma 2.6, for any given positive constants 𝜀, there exists a ray arg𝑧=𝜃∈[0,2𝜋)⧵𝐸₁, where 𝐸₁ has zero linear measure, such that for sufficiently large 𝑟,|𝐷(𝑧)|≤exp𝑟^{𝜎(𝐷)+𝜀}. (4.14) Therefore by (4.13), and Lemma 2.2, we can find a ray arg𝑧=𝜃∈[0,2𝜋)⧵𝐸₁∪𝐻₁∪𝐻₂, where 𝐻₁ and 𝐻₂ are defined in Lemma 2.2, 𝐻₁∪𝐻₂ has zero linear measure, such that 𝛿(𝑎_𝑗𝑧,𝜃)=𝑐_𝑗𝛿(𝑎₀𝑧,𝜃)>0, and for sufficiently large 𝑟,𝐷₀𝑟𝑒^𝑖𝜃𝑒^{𝑎0𝑟𝑒𝑖𝜃}≥exp(1−𝜀)𝛿𝑎₀𝑟,𝐷𝑧,𝜃_𝑗𝑟𝑒^𝑖𝜃𝑒^{𝑎𝑗𝑟𝑒𝑖𝜃}≤exp(1+𝜀)𝑐_𝑗𝛿𝑎₀𝑧,𝜃𝑟,𝑗=1,…,𝑘−1. (4.15) Thus, by (4.8) and (4.14) and (4.15), we haveexp(1−𝜀)𝛿𝑎₀𝑧,𝜃𝑟≤^𝑘−1_𝑗=1exp(1+𝜀)𝑐_𝑗𝛿𝑎₀𝑧,𝜃𝑟+exp𝑟^{𝜎(𝐷)+𝜀}. (4.16) Since 𝜀 can be chosen arbitrarily small, we can obtain a contradiction. Thus we have 𝑄₁(𝑧)≢0. By Lemma 2.3 and (4.6), we get 𝜆(𝑓′−𝜙)=∞.

Acknowledgments

The author would like to thank the referee for his/her valuable suggestions and comments. This research is supported by NNSF of China (no. 11001057), NSF of Jiangsu Province (BK2010234), Project of Qinglan of Jiangsu Province.