Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

Abstract

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomials 𝐿(𝑓)=𝑑₂𝑓′′+𝑑₁𝑓^′+𝑑₀𝑓 generated by solutions of the above equation, where 𝑑₀(𝑧),𝑑₁(𝑧), and 𝑑₂(𝑧) are entire functions that are not all equal to zero.

1. Introduction and Main Results

A function 𝑓(𝑧) is called meromorphic if it is nonconstant and analytic in the complex plane ℂ except at possible isolated poles. If no poles occur, then 𝑓(𝑧) reduces to an entire function. Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna’s value distribution theory of meromorphic functions, for reference see [1]. In addition, we use notations 𝜎(𝑓) and 𝜆(𝑓) to denote the order and the exponent of convergence of zero sequence and 𝜆(𝑓) to denote the sequence of distinct zeros of 𝑓(𝑧), respectively. 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→∞, possibly outside of a set of 𝑟 with finite measure, where 𝑇(𝑟,𝑓) is the Nevanlinna characteristic function of 𝑓(𝑧).

For the second-order linear differential equation 𝑓^′′+𝑒^−𝑧𝑓^′+𝐵(𝑧)𝑓=0, (1.1) where 𝐵(𝑧) is an entire function of finite order, it is well known that each solution 𝑓 of (1.1) is an entire function, and that if 𝑓₁ and 𝑓₂ are any two linearly independent solutions of (1.1), then at least one of 𝑓₁,𝑓₂ must have infinite order, see [2, pages 167-168].

Thus, a natural question is the following: what condition on 𝐵(𝑧) will guarantee that every solution 𝑓≢0 of (1.1) has infinite order? Many researchers have studied the question, for the details see [3, page 291]. For the case that 𝐵(𝑧) is a transcendental entire function, Gundersen [4] proved that if 𝜎(𝐵)≠1, then every solution 𝑓≢0 of (1.1) has infinite order. In 2002, Chen considered the problem and proved the following result which is an improvement of Gundersen’s result.

Theorem A (see [3]). Let 𝑎, 𝑏 be nonzero complex numbers satisfying 𝑎𝑏≠0 and arg𝑎≠arg𝑏 or 𝑎=𝑐𝑏(0<𝑐<1), and let 𝐴_𝑗(𝑧)≢0(𝑗=1,2) be entire functions with 𝜎(𝐴_𝑗)<1, then every solution 𝑓(≢0) of the equation 𝑓^′′+𝐴₁(𝑧)𝑒^𝑎𝑧𝑓^′+𝐴₀(𝑧)𝑒^𝑏𝑧𝑓=0 (1.2) has infinite order.
Some further results on (1.2) were obtained for several cases. Chen [3] got the same conclusion when 𝑎=𝑐𝑏(𝑐>1), and Chen and Shon [5] investigated the more general equations with meromorphic coefficients. Under the same assumption of Theorem A, if 𝐴₁(𝑧) and 𝐴₀(𝑧) are meromorphic functions with 𝜎(𝐴_𝑗)<1(𝑗=0,1), then there is the same conclusion with Theorem A. In 2008, Wang and Laine [6] extended Theorem A to nonhomogeneous second-order linear differential equations.

Theorem B (see [6]). Let 𝐴_𝑗(𝑧)≢0(𝑗=0,1) and 𝐹≢0 be entire functions with max{𝜎(𝐴_𝑗),(𝑗=0,1),𝜎(𝐹)}<1, and let 𝑎,𝑏 be complex constants that satisfy 𝑎𝑏≠0 and 𝑎≠𝑏, then every solution 𝑓 of differential equation 𝑓^′′+𝐴₁(𝑧)𝑒^𝑎𝑧𝑓^′+𝐴₀(𝑧)𝑒^𝑏𝑧𝑓=𝐹 (1.3) is of infinite order.

Remark 1.1. Belaïdi and El Farissi [7] also proved Theorem B and got 𝜆(𝑓)=𝜆(𝑓)=𝜎(𝑓)=∞. We note that (2.21) in [7] cannot be deduced by following their proof. Indeed, as 𝑟→∞, |𝑓(𝑧)|>1 holds just for the points 𝑧 satisfying |𝑓(𝑧)|=𝑀(𝑟,𝑓), not for all 𝑧. However, the difficulty can be got over by using Lemmas 2.5 and  2.6 in [8], and the method can be used in our proof of the following Theorem 1.2.

Since the beginning of the last four decades, a substantial number of research papers have been written to describe the fixed points of general transcendental functions. However, there are few studies on the fixed points of solutions of the general differential equations. In 2000, Chen [9] first studied the problems on the fixed points of solutions of second-order linear differential equations with entire coefficients. Since then, many results on fixed points of solutions of differential equations with entire coefficients were obtained, see [10–12]. In 2006, Chen and Shon [13] further studied the relation between small functions and solutions or differential polynomials of solutions of differential equations and obtained the following.

Theorem C. 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 (1.2) satisfies 𝜆(𝑓−𝜓)=𝜆(𝑓^′−𝜓)=𝜆(𝑓^′′−𝜓)=∞. Furthermore, let 𝑑₀(𝑧),𝑑₁(𝑧),and 𝑑₂(𝑧) be polynomials that are not all equal to zero, and let 𝐿(𝑓)=𝑑₂𝑓^′′+𝑑₁𝑓^′+𝑑₀𝑓. If the order of 𝜓 is less than 1, then 𝜆(𝐿(𝑓)−𝜓)=∞.

Belaïdi and El Farissi [7] also studied the relation between small functions and some differential polynomials generated by solutions of the second-order nonhomogeneous linear differential equation (1.3). They obtained the following.

Theorem D. Let 𝐴_𝑗(𝑧)≢0(𝑗=0,1) and 𝐹≢0 be entire functions with max{𝜎(𝐴_𝑗)(𝑗=0,1),𝜎(𝐹)}<1, and let 𝑎,𝑏 be complex constants that satisfy 𝑎𝑏≠0 and arg𝑎≠arg𝑏 or a=𝑐𝑏(0<𝑐<1). Let 𝑑₀(𝑧),𝑑₁(𝑧),𝑑₂(𝑧) be entire functions that are not all equal to zero with 𝜎(𝑑_𝑗)<1(𝑗=0,1,2), and let 𝜓(𝑧) be an entire function with finite order. If 𝑓 is a solution of (1.3), then the differential polynomial 𝐿(𝑓)=𝑑₂𝑓^′′+𝑑₁𝑓^′+𝑑₀𝑓 satisfies 𝜆(𝐿(𝑓)−𝜓)=∞.

The main purpose of this paper is to study the growth and the oscillation of solutions of second-order linear differential equation with meromorphic coefficients. Also, we will investigate the relation between small functions and differential polynomials generated by solutions of the above equation. Our results can be stated as follows.

Theorem 1.2. Let 𝐴_𝑗(𝑧)≢0 (𝑗=0,1) and 𝐹(𝑧) be meromorphic functions with max{𝜎(𝐹),𝜎(𝐴_𝑗)}<𝑛, and let 𝑃(𝑧)=𝑎_𝑛𝑧^𝑛+⋯+𝑎₀, 𝑄(𝑧)=𝑏_𝑛𝑧^𝑛+⋯+𝑏₀ be polynomials with degree 𝑛 (𝑛≥1), where 𝑎_𝑖, 𝑏_𝑖 (𝑖=0,1,…,𝑛), 𝑎_𝑛𝑏_𝑛≠0 are complex constants such that arg𝑎_𝑛≠arg𝑏_𝑛 or 𝑎_𝑛=𝑐𝑏_𝑛(0<𝑐<1), then every meromorphic solution 𝑓≢0 of the equation 𝑓^′′+𝐴₁𝑒^𝑃(𝑧)𝑓^′+𝐴₀𝑒^𝑄(𝑧)𝑓=𝐹 (1.4) has infinite order and satisfies 𝜆𝑓=𝜆𝑓=𝜎𝑓=∞. (1.5)

Theorem 1.3. Under the assumption of Theorem 1.2, and let 𝑑₀(𝑧),𝑑₁(𝑧),𝑑₂(𝑧) be meromorphic functions that are not all equal to zero with 𝜎(𝑑_𝑗)<1(𝑗=0,1,2), and let 𝜓(𝑧) be a meromorphic function with finite order, if 𝑓≢0 is a meromorphic solution of (1.4), then the differential polynomial 𝐿(𝑓)=𝑑₂𝑓^′′+𝑑₁𝑓^′+𝑑₀𝑓 satisfies 𝜆(𝐿(𝑓)−𝜓)=∞.

Remark 1.4. Clearly, the method used in linear differential equations with entire coefficients cannot deal with the case of meromorphic coefficients. In addition, the proof of the results in [7, 13] relies heavily on the idea of Lemma  5 in [13] or Lemma  2.5 in [7]. However, it seems too complicated to deal with our cases. We will use an important result in uniqueness theory of meromorphic functions, that is Lemma 2.5, to prove our theorems.

2. Preliminary Lemmas

In order to prove our theorems, we need the following lemmas.

Lemma 2.1 (see [14]). 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)

Now we introduced a notation, see [15] and [8, Lemma 2.3]. Let 𝑃(𝑧)=(𝛼+𝛽𝑖)𝑧^𝑛+⋯ is a nonconstant polynomial, and 𝛼,𝛽 is real constants. For 𝜃∈[0,2𝜋), set 𝛿(𝑃(𝑧),𝜃)=𝛼cos𝑛𝜃−𝛽sin𝑛𝜃.

Lemma 2.2 (see [15]). Let 𝑃(𝑧) be a nonconstant 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 [8, Lemma 2.5]). Let 𝑓(𝑧) be an entire function, and suppose that 𝐺(𝑧)∶=log⁺𝑓^(𝑘)|𝑧|^𝜌 (2.2) is unbounded on some ray arg𝑧=𝜃 with constant 𝜌>0, then there exists an infinite sequence of points 𝑧_𝑛=𝑟_𝑛𝑒^𝑖𝜃(𝑛=1,2,…), where 𝑟_𝑛→∞, such that 𝐺(𝑧_𝑛)→∞ and 𝑓^(𝑗)𝑧_𝑛𝑓^(𝑘)𝑧_𝑛≤1!((1𝑟𝑘−𝑗1+𝑜))_𝑛^𝑘−𝑗,𝑗=0,…,𝑘−1, (2.3) as 𝑛→∞.

Lemma 2.4 (see [16]). 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, (2.4) then 𝑓 satisfies 𝜆(𝑓)=𝜆(𝑓)=𝜎(𝑓)=∞.

Lemma 2.5 (see [17, 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)Σ^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 2.6 (see [8, 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 2.7. Under the assumption of Theorem 1.2, and let 𝑓 be a meromorphic solution of (1.4). Set 𝑤=𝑓^′′+𝐴₁𝑒^𝑃(𝑧)𝑓^′+𝐴₀𝑒^𝑄(𝑧)𝑓. If 𝑓≢0 is of finite order, then 𝜎(𝑤)≥𝑛.

Proof. Suppose the contrary that 𝜎(𝑤)<𝑛, we will deduce a contradiction.
First, if 𝑓(𝑧)≡𝐶≠0, then 𝑤=𝐶𝐴₀𝑒^𝑄(𝑧). Clearly, 𝜎(𝑤)=𝑛, this is a contradiction.
Now suppose that 𝑓≢𝐶. If 𝜎(𝑓)<𝑛, then 𝑓^′′+𝐴₁𝑒^𝑃(𝑧)𝑓^′+𝐴₀𝑒^𝑄(𝑧)𝑓−𝑤=0. (2.5) By Lemma 2.5, we have 𝐴₀≡0, and this is a contradiction. Hence, 𝜎(𝑓)≥𝑛.
Since 𝑓 is a meromorphic solution of (1.4), we know that the poles of 𝑓 can occur only at the poles of 𝐴_𝑗(𝑗=0,1) 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=0,1}<𝑛, and 𝑔 is an entire function with 𝑛≤𝜎(𝑔)=𝜎(𝑓)=𝜎≤∞. Substituting 𝑓=𝑔/𝑑 into (2.5), by some calculation we can get 𝑑𝑤=𝑔^′′+𝑔^′𝐴₁𝑒^𝑃(𝑧)𝑑−2^′𝑑+𝑔𝐴₀𝑒^𝑄(𝑧)−𝐴₁𝑒^𝑃(𝑧)𝑑^′𝑑𝑑+2^′𝑑²−𝑑^′′𝑑. (2.6) Now, we rewrite (2.6) into 𝑑𝑤𝑔−𝑔^′′𝑔−𝐴₁𝑒^𝑃(𝑧)𝑑−2^′𝑑𝑔^′𝑔=𝐴₀𝑒^𝑄(𝑧)−𝐴₁𝑒^𝑃(𝑧)𝑑^′𝑑𝑑+2^′𝑑²−𝑑^′′𝑑. (2.7)
Set max{𝜎(𝑤),𝜎(𝐴_𝑗),𝑗=0,1}=𝛽<𝑛. By Lemma 2.1, for any given 𝜀(0<𝜀<1−𝛽), there exists a set 𝐸₂∈[0,2𝜋) which has linear measure zero, such that if 𝜃∈[0,2𝜋)⧵𝐸₂, then there is a constant 𝑅₁=𝑟₁(𝜃)>1 such that for all 𝑧 satisfying arg𝑧=𝜃 and |𝑧|≥𝑅₁, we have 𝑔^(𝑖)(𝑧)𝑔(𝑧≤|𝑧|)^{2(𝜎−1+𝜀)},𝑑^(𝑖)(𝑧)|𝑑(𝑧≤|𝑧|)|^{2(𝛽−1+𝜀)},𝑖=1,2. (2.8)

Case 1. Suppose that 𝑎_𝑛=𝑐𝑏_𝑛(0<𝑐<1), then by Lemmas 2.1 and 2.2, there exists a ray arg𝑧=𝜃∈[0,2𝜋)⧵𝐸₂∪𝐻₁∪𝐻₂, 𝐻₁ and 𝐻₂ being defined in Lemma 2.2, such that 𝛿(𝑃,𝜃)=𝑐𝛿(𝑄,𝜃)>0, and for the above 𝜀 and sufficiently 𝑟, 𝐴₀𝑒^𝑄(𝑧)−𝐴₁𝑒^𝑃(𝑧)𝑑^′𝑑𝑑+2^′𝑑²−𝑑^′′𝑑≥𝐴₀𝑒^𝑄(𝑧)−𝐴₁𝑒^𝑃(𝑧)𝑑^′𝑑−2𝑑^′𝑑²−𝑑^′′𝑑≥12exp(1−𝜀)𝛿(𝑄,𝜃)𝑟^𝑛. (2.9) Also, by Lemmas 2.1 and 2.2, we have 𝐴₁𝑒^𝑃(𝑧)𝑑−2^′𝑑≤𝐴₁𝑒^𝑃(𝑧)+2𝑑^′𝑑≤𝑀exp(1+𝜀)𝑐𝛿(𝑄,𝜃)𝑟^𝑛, (2.10) where 𝑀 is a constant.
Now we claim that log⁺𝑔(𝑧)|𝑧|^𝛽+𝜀 (2.11) is bounded on the ray arg𝑧=𝜃. Otherwise, by Lemma 2.3, there exists a sequence of points 𝑧_𝑚=𝑟_𝑚𝑒^𝑖𝜃, such that 𝑟_𝑚→∞log⁺𝑔𝑧_𝑚𝑟_𝑚^𝛽+𝜀⟶∞. (2.12) From (2.12) and the definition of order, we see that 𝑑𝑧_𝑚𝑤𝑧_𝑚𝑔𝑧_𝑚⟶0, (2.13) for 𝑚 is large enough. By (2.7), (2.8), (2.9), (2.10), and (2.13), we get 12exp(1−𝜀)𝛿(𝑄,𝜃)𝑟^𝑛_𝑚≤𝐴₀𝑒^𝑄(𝑧)−𝐴₁𝑒^𝑃(𝑧)𝑑^′𝑑𝑑+2^′𝑑²−𝑑^′′𝑑≤𝑑𝑤𝑔+𝐴₁𝑒^𝑃(𝑧)𝑑−2^′𝑑𝑔^′𝑔+𝑔^′′𝑔≤𝑀₁exp(1+𝜀)𝑐𝛿(𝑄,𝜃)𝑟^𝑛_𝑚, (2.14) where 𝑀₁ is a constant. Clearly, we can choose 𝜀 such that 0<𝜀<(1−𝑐)/(1+𝑐). Then by (2.14), we can obtain a contradiction. Therefore, log⁺𝑔(𝑧)|𝑧|^𝛽+𝜀 (2.15) is bounded, and we have |𝑔(𝑧)|≤𝑀exp{𝑟^𝛽+𝜀} on the ray arg𝑧=𝜃.

Case 2. Suppose that arg𝑎_𝑛≠arg𝑏_𝑛. By Lemma 2.2, there exists a ray arg𝑧=𝜃∈[0,2𝜋)⧵𝐸₂∪𝐻₁∪𝐻₂, where 𝐸₂, 𝐻₁, and 𝐻₂ are defined, respectively, as in Case 1, such that 𝛿(𝑄,𝜃)>0,𝛿(𝑃,𝜃)<0. (2.16)
Then, for any given 𝜀(0<𝜀<𝑛−𝛽), by Lemma 2.2 and (2.7), we have, for sufficiently large |𝑧|=𝑟, 𝐴₀𝑒^𝑄(𝑧)−𝐴₁𝑒^𝑃(𝑧)𝑑^′𝑑𝑑+2^′𝑑²−𝑑^′′𝑑≥𝐴₀𝑒^𝑄(𝑧)−𝐴₁𝑒^𝑃(𝑧)𝑑^′𝑑−2𝑑^′𝑑²−𝑑^′′𝑑≥12exp(1−𝜀)𝛿(𝑄,𝜃)𝑟^𝑛,𝐴₁𝑒^𝑃(𝑧)𝑑−2^′𝑑≤𝐴₁𝑒^𝑃(𝑧)𝑑+2^′𝑑≤exp𝛽+𝜀exp(1−𝜀)𝛿(𝑃,𝜃)𝑟^𝑛+𝑟^{2(𝛽−1+𝜀)}. (2.17)(2.18) As in Case 1, we prove that log⁺𝑔(𝑧)|𝑧|^𝛽+𝜀 (2.19) is bounded on the ray arg𝑧=𝜃. Otherwise, similarly as in Case 1, there exists a sequence of points 𝑧_𝑚=𝑟_𝑚𝑒^𝑖𝜃, such that 𝑟_𝑚→∞, log⁺𝑔𝑧_𝑚𝑟_𝑚^𝛽+𝜀⟶∞. (2.20) Further, we have 𝑑𝑧_𝑚𝑤𝑧_𝑚𝑔𝑧_𝑚⟶0, (2.21) for 𝑚 is large enough.
By (2.7), (2.8), (2.17), (2.18), and (2.21), we get 12exp(1−𝜀)𝛿(𝑄,𝜃)𝑟^𝑛_𝑚≤𝐴₀𝑒^𝑄(𝑧)−𝐴₁𝑒^𝑃(𝑧)𝑑^′𝑑𝑑+2^′𝑑²−𝑑^′′𝑑≤𝑑𝑤𝑔+𝐴₁𝑒^𝑃(𝑧)𝑑−2^′𝑑𝑔^′𝑔+𝑔^′′𝑔≤exp(1−𝜀)𝛿(𝑃,𝜃)𝑟^𝑛_𝑚+𝑟_𝑚^{2(𝛽−1+𝜀)}. (2.22) Since 𝛿(𝑄,𝜃)>0 and 𝛿(𝑃,𝜃)<0, we obtain a contradiction. So log⁺𝑔(𝑧)|𝑧|^𝛽+𝜀 (2.23) is bounded, and we have 𝑔(𝑧)≤𝑀exp𝑟^𝛽+𝜀 (2.24) on the ray arg𝑧=𝜃.
Combining Cases 1 and 2, for any given ray arg𝑧=𝜃∈[0,2𝜋)⧵𝐸, 𝐸 of linear measure zero, we have (2.24) on the ray arg𝑧=𝜃, provided that 𝑟 is sufficiently large. Thus by Lemma 2.6, we get 𝜎(𝑔)≤𝛽+𝜀<𝑛, which is a contradiction. Then 𝜎(𝑤)≥𝑛.

Lemma 2.8. Under the assumption of Theorem 1.3, let 𝑓(𝑧) be an infinite-order meromorphic solution of (1.4), then 𝜎(𝐿(𝑓))=∞.

Proof. Suppose that 𝑓(𝑧) is a meromorphic solution of (1.4), then by Theorem 1.2, we have 𝜎(𝑓)=∞.
Now suppose that 𝑑₂≢0. Substituting 𝑓^′′=𝐹−𝐴₁𝑒^𝑃(𝑧)𝑓^′−𝐴₀𝑒^𝑄(𝑧) into 𝐿(𝑓), we have 𝐿𝑓−𝑑₂𝐹=𝑑₁−𝑑₂𝐴₁𝑒^𝑃(𝑧)𝑓^′+𝑑₀−𝑑₂𝐴₀𝑒^𝑄(𝑧)𝑓. (2.25)
Differentiating both sides of (2.25), and replacing 𝑓^′′ with 𝑓^′′=𝐹−𝐴₁𝑒^𝑃𝑓^′−𝐴₀𝑒^𝑄𝑓, we obtain 𝐿𝑓^′−𝑑₂𝐹^′−𝑑₁−𝑑₂𝐴₁𝑒^𝑃𝐹=𝑑₂𝐴²₁𝑒^2𝑃−𝑑₂𝐴₁^′+𝑃^′𝑑₂𝐴₁+𝑑₁𝐴₁𝑒^𝑃−𝑑₂𝐴₀𝑒^𝑄+𝑑₀+𝑑^′₁𝑓^′+𝑑₂𝐴₀𝐴₁𝑒^𝑃+𝑄−𝑑₂𝐴₀^′+𝑄^′𝑑₂𝐴₀+𝑑₁𝐴₀𝑒^𝑄+𝑑^′₀𝑓. (2.26) Set 𝛼₁=𝑑₁−𝑑₂𝐴₁𝑒^𝑃,𝛼₀=𝑑₀−𝑑₂𝐴₀𝑒^𝑄,𝛽₁=𝑑₂𝐴²₁𝑒^2𝑃−𝑑₂𝐴₁^′+𝑃^′𝑑₂𝐴₁+𝑑₁𝐴₁𝑒^𝑃−𝑑₂𝐴₀𝑒^𝑄+𝑑₀+𝑑^′₁,𝛽₀=𝑑₂𝐴₀𝐴₁𝑒^𝑃+𝑄−𝑑₂𝐴₀^′+𝑄^′𝑑₂𝐴₀+𝑑₁𝐴₀𝑒^𝑄+𝑑^′₀. (2.27) Then we rewrite (2.25) and (2.26) into 𝛼₁𝑓^′+𝛼₀𝑓=𝐿𝑓−𝑑₂𝛽𝐹,₁𝑓^′+𝛽₀𝑓=𝐿𝑓^′−𝑑₂𝐹^′−𝑑₁−𝑑₂𝐴₁𝑒^𝑃𝐹. (2.28) Set ℎ=𝛼₁𝛽₀−𝛼₀𝛽₁=𝑑₁−𝑑₂𝐴₁𝑒^𝑃𝑑₂𝐴₀𝐴₁𝑒^𝑃+𝑄−𝑑₂𝐴₀^′+𝑄^′𝑑₂𝐴₀+𝑑₁𝐴₀𝑒^𝑄+𝑑^′₀−𝑑₀−𝑑₂𝐴₀𝑒^𝑄𝑑₂𝐴²₁𝑒^2𝑃−𝑑₂𝐴₁^′+𝑃^′𝑑₂𝐴₁+𝑑₁𝐴₁𝑒^𝑃−𝑑₂𝐴₀𝑒^𝑄+𝑑₀+𝑑^′₁=ℎ₀+ℎ_𝑃(𝑧)𝑒^𝑃+ℎ_𝑃+𝑄𝑒^𝑃+𝑄+ℎ_𝑄𝑒^𝑄+ℎ_2𝑃𝑒^2𝑃−𝑑²₂𝐴²₀𝑒^2𝑄, (2.29) where ℎ_𝑖(𝑧)(𝑖∈Λ={0,𝑃,𝑄,𝑃+𝑄,2𝑃}) are meromorphic functions formed by 𝐴₀,𝐴₁,𝑑₀, and 𝑑₁ and their derivatives, with order less than 𝑛, and Λ is a index set. Since any one of {𝑃,𝑄,𝑃+Q,2𝑃} is not equal to 2𝑄, then by Lemma 2.5, we have 𝑑²₂𝐴²₀≡0. This is a contradiction. Thus, ℎ≢0.
By (2.28), we get 1𝑓=ℎ𝛼₁𝐿𝑓^′−𝑑₂𝐹^′−𝛼₁𝐹−𝛽₁𝐿𝑓−𝑑₂𝐹. (2.30) If 𝜎(𝐿(𝑓))<∞, then by (2.30) we have 𝜎(𝑓)<∞. Clearly, it is a contradiction. Hence, 𝜎(𝐿(𝑓))=∞.
Suppose that 𝑑₂≡0,𝑑₁≢0 or 𝑑₂=𝑑₁≡0, and 𝑑₀≢0, then by similar discussion as above, we can get the same conclusion.

3. Proof of Theorem 1.2

Let 𝑓≢0 be a meromorphic solution of (1.4). Conversely, suppose that 𝜎(𝑓)<∞. By Lemma 2.7, we have 𝑛≤𝜎(𝑤)=𝜎(𝐹)<𝑛. This is a contradiction. By Lemma 2.4, 𝑓 satisfies 𝜆(𝑓)=𝜆(𝑓)=𝜎(𝑓)=∞.

4. Proof of Theorem 1.3

Suppose that arg𝑎_𝑛≠arg𝑏_𝑛 or 𝑎_𝑛=𝑐𝑏_𝑛(0<𝑐<1) and that 𝑓 is a meromorphic solution of (1.4). Set 𝑘(𝑧)=𝐿(𝑓)−𝜓. By 𝜎(𝜓)<∞ and Lemma 2.8, we have 𝜎(𝑘)=∞. Without loss of generality, we assume that 𝑑₂≢0. Indeed, the remaining cases can be obtained by similar discussion. Substituting 𝐿(𝑓)=𝑘(𝑧)+𝜓 into (2.30), we have1𝑓=ℎ𝛼₁𝑘^′+𝜓^′−𝑑₂𝐹^′−𝛼₁𝐹−𝛽₁𝑘+𝜓−𝑑₂𝐹=1ℎ𝛼₁𝑘^′−𝛽₁𝑘+𝜙, (4.1) where𝜙(𝑧)=𝛼₁ℎ𝜓^′−𝑑₂𝐹^′−𝛼₁𝐹−𝛽₁ℎ𝜓−𝑑₂𝐹 (4.2) is a meromorphic function of finite order. Then substituting (4.1) into (1.4), we have𝛼₁ℎ𝑘^′′′+𝜙₂𝑘^′′+𝜙₁𝑘^′+𝜙₀𝑘=𝐹−𝜓^′′+𝐴₁𝑒^𝑃𝜓^′+𝐴₀𝑒^𝑄𝜓, (4.3) where 𝜙_𝑗(𝑗=0,1,2) are meromorphic functions formed by 𝛼₁/ℎ,𝛽/ℎ,𝜙, and their derivatives. If 𝐹−(𝜓^′′+𝐴₁𝑒^𝑃𝜓^′+𝐴₀𝑒^𝑄𝜓)≡0, then by Theorem 1.2, we have 𝜎(𝜓)=∞. This is impossible, and hence 𝐹−(𝜓^′′+𝐴₁𝑒^𝑃𝜓^′+𝐴₀𝑒^𝑄𝜓)≢0. Thus, by ℎ≢0, 𝛼₁≢0, (4.3), and Lemma 2.4, we get 𝜆(𝑘)=𝜆(𝑘)=𝜎(𝑘)=∞.

Acknowledgments

The author would like to thank the referees for their valuable suggestions and comments. This research was supported by NNSF of China (no. 11001057), NSF of Jiangsu Province (BK2010234), Project of Qinglan of Jiangsu Province.