The Oscillation on Solutions of Some Classes of Linear Differential Equations with Meromorphic Coefficients of Finite [𝑝,𝑞]-Order

Abstract

This paper considers the oscillation on meromorphic solutions of the second-order linear differential equations with the form 𝑓^′′+𝐴(𝑧)𝑓=0, where 𝐴(𝑧) is a meromorphic function with [𝑝,𝑞]-order. We obtain some theorems which are the improvement and generalization of the results given by Bank and Laine, Cao and Li, Kinnunen, and others.

1. Introduction and Main Results

The purpose of this paper is to study the oscillation on solutions of linear differential equations in the complex plane. It is well known that Nevanlinna theory has appeared to be a powerful tool in the field of complex differential equations. We assume that readers are familiar with the standard notations and the fundamental results of the Nevanlinna’s value distribution theory of meromorphic functions (see [1, 3, 4]). Throughout the paper, a meromorphic function 𝑓 means meromorphic in the complex plane ℂ. In addition, we use 𝜎(𝑓) and 𝜆(𝑓) to denote the order and the exponent of convergence of zero sequence of meromorphic function 𝑓(𝑧), respectively. For sufficiently large 𝑟∈[1,∞), we define log_𝑖+1𝑟=log_𝑖(log𝑟)(𝑖∈𝑁) and exp_𝑖+1𝑟=exp(exp_𝑖𝑟)(𝑖∈𝑁) and exp₀𝑟=𝑟=log₀𝑟, exp₋₁𝑟=log𝑟.

For the second-order linear differential equation, 𝑓^′′+𝐴(𝑧)𝑓=0, (1) where 𝐴(𝑧) is an entire function or meromorphic function of finite order. In 1982, Bank and Laine [5] mainly studied the distribution of zeros of solutions of (1) when 𝐴 is an entire function of finite order.

Obviously, all solutions of (1) are entire when 𝐴(𝑧) is entire. However, there are some immediate difficulties when 𝐴 is meromorphic; for example, the solutions of (1) may not be entire, and it is possible that no solution of (1) except the zero solution is single-valued on the plane. In 1983, Bank and Laine [6] investigated the exponent of convergence of zero sequence of nontrivial solutions of (1), when 𝐴 is meromorphic function and obtained the results as follows.

Theorem 1 (see [6, Theorem  5]). Let 𝐴 be a transcendental meromorphic function of order 𝜎(𝐴), where 0<𝜎(𝐴)≤∞. Assume that 𝜆(𝐴)<𝜎(𝐴). Then, if 𝑓≢0 is a meromorphic solution of (1), one has 𝜎(𝐴)≤max𝜆𝑓,𝜆1𝑓. (2)

Theorem 2 (see [6, Theorem  6]). Let 𝐴 be a transcendental meromorphic function. Assume that (1) possesses two linearly independent meromorphic solutions 𝑓₁ and 𝑓₂ satisfying 𝜆(𝑓₁)<∞, 𝜆(𝑓₂)<∞. Then, any solution 𝑓≢0 of (1) which is not a constant multiple of either 𝑓₁ or 𝑓₂ satisfies max𝜆𝑓,𝜆1𝑓=∞, (3) unless all solutions of (1) are of finite order. In the special case where 𝜆(1/𝐴)<∞, one can conclude that 𝜆(𝑓)=∞ unless all solutions of (1) are of finite order.

After their work, many authors have investigated the growth and the exponent of convergence of zero sequence of non-trivial solutions of (1) and obtained many classical results (see [5, 7, 8]).

In 1998, Kinnunen [9] further investigated the oscillation results of entire solutions of (1) when 𝐴(𝑧) is an entire function with finite iterated order and obtained some theorems which improved some theorems given by Bank and Laine [5]. Later, Chen [10] and Wang and Lü [11] studied the oscillation of solutions of (1) when 𝐴(𝑧) is a meromorphic function with finite order by using the Wiman-Valiron theory; they obtained some results which extend some theorems of Kinnunen [9]. In 2007, Liang and Liu [12] considered the complex oscillation on (1) when 𝐴(𝑧) is a meromorphic function with many finite poles. They extended the above oscillation results by using the method of the Wiman-Valiron theory. Although the Wiman-Valiron theory is a powerful tool to investigate entire solutions, it is only useful for the meromorphic function 𝐴(𝑧) with the exponent of the convergence sequence of poles which is less than the order of 𝐴(𝑧) if we consider (1). In 2010, Cao and Li [13] made use of a result due to Chiang and Hayman [14] instead of the Wiman-Valiron theory and obtained four oscillation theorems and three corollaries which extended the above results due to Bank-Laine, Kinnunen, and Liang and Liu.

Theorem 3 (see [13, Theorem  1.6]). Let 𝐴(𝑧) be a meromorphic function with 0<𝑖(𝐴)=𝑛<∞, and assume that 𝜆_𝑛(𝐴)<𝜎_𝑛(𝐴)≠0. Then, if 𝑓 is a nonzero meromorphic solution of (1), one has 𝜎_𝑛(𝐴)≤max{𝜆_𝑛(𝑓),𝜆_𝑛(1/𝑓)}. In the special case where either 𝛿(∞,𝑓)>0 or the poles of 𝑓 are of uniformly bounded multiplicities, one can conclude that max{𝜆_𝑛+1(𝑓),𝜆_𝑛+1(1/𝑓)}≤𝜎_𝑛+1(𝐴)≤max{𝜆_𝑛(𝑓),𝜆_𝑛(1/𝑓)}.

Theorem 4 (see [13, Theorem  1.7]). Let 𝐴 be a meromorphic function with 0<𝑖(𝐴)=𝑛<∞. Assume that (1) possesses two linearly independent meromorphic solutions 𝑓₁ and 𝑓₂. Denote 𝐸∶=𝑓₁𝑓₂. If 𝜆_𝑛(𝐸)<∞, then any nonzero solution 𝑓 of (1) which is not a constant multiple of either 𝑓₁ or 𝑓₂ satisfies 𝜆_𝑛(𝑓)=∞, unless all solutions of (1) are of finite iterated 𝑛-order. In the special case where 𝛿(∞,𝐴)>0  𝑖_𝜆(1/𝐴)<𝑛, or 𝜆_𝑛(1/𝐴)<𝜎_𝑛(𝐴) (e.g., 𝐴 is an entire function), one can conclude that 𝜆_𝑛=∞.

Theorem 5 (see [13, Theorem  1.9]). Let 𝐴 be a meromorphic function with 0<𝑖(𝐴)=𝑛<∞. Assume that 𝑓₁ and 𝑓₂ are two linearly independent meromorphic solutions of (1) such that max{𝜆_𝑛(𝑓₁),𝜆_𝑛(𝑓₂)}<𝜎_𝑛(𝐴). Let Π(𝑧)≢0 be any meromorphic function for which 𝜎_𝑛(Π)<𝜎_𝑛(𝐴). Let 𝑔₁ and 𝑔₂ be two linearly independent solutions of the differential equation 𝑔^′′+(𝐴(𝑧)+Π(𝑧))𝑔=0. Denote 𝐸∶=𝑓₁𝑓₂ and 𝐹∶=𝑔₁𝑔₂. If max𝑖_𝜆1𝐸,𝑖_𝜆1𝐹<𝑛ormax𝜆_𝑛1𝐸,𝜆_𝑛1𝐹<𝜆_𝑛(𝐴), (4) then max{𝜆_𝑛(𝑔₁),𝜆_𝑛(𝑔₂)}≥𝜎_𝑛(𝐴).

In 1976, Juneja and his coauthors [15, 16] firstly introduced the concept of [𝑝,𝑞]-order of entire functions. Recently, Belaïdi [17] and Liu et al. [18] investigated the growth of solutions of complex differential equations 𝑓^(𝑛)+𝐴_𝑛−1(𝑧)𝑓^(𝑛−1)+⋯+𝐴₁(𝑧)𝑓^′+𝐴₀(𝑧)𝑓=0, (5) where 𝐴_𝑗(𝑧), (𝑗=0,1,…,𝑛−1) are entire or meromorphic functions with finite [𝑝,𝑞]-order, by using the idea of [𝑝,𝑞]-order, and obtained some interest results which improved and extended some previous theorems given by [5–7, 9, 19].

Thus, it is interesting to consider the complex oscillation on the meromorphic solutions of (1) for the case when 𝐴 is entire or meromorphic functions in the terms of the idea of [𝑝,𝑞]-order.

In this paper, we further investigated the complex oscillation of meromorphic solutions of (1) when 𝐴(𝑧) is meromorphic by using the idea of [𝑝,𝑞]-order. To state our theorems, we first introduce the concepts of entire functions of [𝑝,𝑞]-order (see [15, 16, 18]). Throughout this paper, we always assume that 𝑝, 𝑞 are positive integers satisfying 𝑝≥𝑞≥1.

Definition 6. If 𝑓(𝑧) is a transcendental entire function, the [𝑝,𝑞]-order of 𝑓(𝑧) is defined by 𝜎_[𝑝,𝑞]𝑓=limsup_𝑟→∞log_𝑝+1𝑀𝑟,𝑓log_𝑞𝑟=limsup_𝑟→∞log_𝑝𝑇𝑟,𝑓log_𝑞𝑟. (6)

Remark 7. If 𝑓(𝑧) is a polynomial, then 𝜎_[𝑝,𝑞](𝑓)=0 for any 𝑝≥𝑞≥1. By Definition 6, we have that 𝜎_[1,1](𝑓)=𝜎₁(𝑓)=𝜎(𝑓), 𝜎_[𝑝+1,1](𝑓)=𝜎_𝑝(𝑓).

Remark 8. If 𝑓(𝑧) is an entire function satisfying 0<𝜎_[𝑝,𝑞](𝑓)<∞, then(i)𝜎_{[𝑝−𝑛,𝑞]}(𝑓)=∞(𝑛<𝑝), 𝜎_{[𝑝,𝑞−𝑛]}(𝑓)=0(𝑛<𝑞), 𝜎_{[𝑝+𝑛,𝑞+𝑛]}(𝑓)=1(𝑛<𝑝) for 𝑛=1,2,….(ii)if [𝑝^′,𝑞^′] is any pair of integers satisfying 𝑞^′=𝑝^′+𝑞−𝑝 and 𝑝^′<𝑝, then 𝜎_{[𝑝′,𝑞′]}(𝑓)=0 if 0<𝜎_[𝑝,𝑞](𝑓)<1 and 𝜎_{[𝑝′,𝑞′]}(𝑓)=∞ if 1<𝜎_[𝑝,𝑞](𝑓)<∞;(iii)𝜌_{[𝑝′,𝑞′]}(𝑓)=∞ for 𝑞^′−𝑝^′>𝑞−𝑝 and 𝜌_{[𝑝′,𝑞′]}(𝑓)=0 for 𝑞^′−𝑝^′<𝑞−𝑝.

Definition 9. A transcendental meromorphic function 𝑓(𝑧) is said to have index-pair [𝑝,𝑞] if 0<𝜎_[𝑝,𝑞](𝑓)<∞ and 𝜎_{[𝑝−1,𝑞−1]}(𝑓) is not a nonzero finite number.

Definition 10. Let 𝑓₁,𝑓₂ be two entire functions such that 𝜎_{[𝑝1,𝑞1]}(𝑓₁)=𝜎₁, 𝜎_{[𝑝2,𝑞2]}(𝑓₂)=𝜎₂ and 𝑝₁≤𝑝₂. Then the following results about their comparative growth can be easily deduced.(i)If 𝑝₂−𝑝₁>𝑞₂−𝑞₁, then the growth of 𝑓₁ is slower than the growth of 𝑓₂;(ii)If 𝑝₂−𝑝₁<𝑞₂−𝑞₁, then 𝑓₁ grows faster than 𝑓₂.(iii)If 𝑝₂−𝑝₁=𝑞₂−𝑞₁>0, then the growth of 𝑓₁ is slower than the growth of 𝑓₂ if 𝜎₂≥1 while the growth of 𝑓₁ is faster than the growth of 𝑓₂ if 𝜌₂<1.(iv)Let 𝑝₂−𝑝₁=𝑞₂−𝑞₁=0; then 𝑓₁, 𝑓₂ are of the same index-pair [𝑝₁,𝑞₁]. If 𝜎₁>𝜎₂, then 𝑓₁ grows faster than 𝑓₂, and if 𝜎₁<𝜎₂, then 𝑓₁ grows slower than 𝑓₂. If 𝜎₁=𝜎₂, Definition 6 does not give any precise estimate about the relative growth of 𝑓₁ and 𝑓₂.

Definition 11 (see [15, 16, 18]). The [𝑝,𝑞] exponent of convergence of the zero sequence and the [𝑝,𝑞] exponent of convergence of the distinct zero sequence of 𝑓(𝑧) are defined respectively, by 𝜆_[𝑝,𝑞]𝑓=lim_𝑟→∞log_𝑝𝑛𝑟,1/𝑓log_𝑞𝑟=lim_𝑟→∞log_𝑝𝑁𝑟,1/𝑓log_𝑞𝑟,𝜆_[𝑝,𝑞]𝑓=lim_𝑟→∞log_𝑝𝑛𝑟,1/𝑓log_𝑞𝑟=lim_𝑟→∞log_𝑝𝑁𝑟,1/𝑓log_𝑞𝑟. (7)

Remark 12. It is easy to know that 𝜆_[𝑝,𝑞]𝑓≤𝜆_[𝑝,𝑞]𝑓≤𝜎_[𝑝,𝑞]𝑓. (8)

Now, we will show our main results on the complex oscillation on meromorphic solutions of (1) when 𝐴(𝑧) is meromorphic with finite [𝑝,𝑞]-order as follows.

Theorem 13. Let 𝐴(𝑧) be a transcendental meromorphic function with 𝜎_[𝑝,𝑞](𝐴)>0. Assume that 𝜆_[𝑝,𝑞](𝐴)<𝜎_[𝑝,𝑞](𝐴). Then, if 𝑓 is a nonzero meromorphic solution of (1), one has 𝜎_[𝑝,𝑞](𝐴)≤max𝜆_[𝑝,𝑞]𝑓,𝜆_[𝑝,𝑞]1𝑓. (9) In the special case where either 𝛿(∞,𝑓)>0 or the poles of 𝑓 are of uniformly bounded multiplicities, one can get that max𝜆_{[𝑝+1,𝑞]}𝑓,𝜆_{[𝑝+1,𝑞]}1𝑓≤𝜎_{[𝑝+1,𝑞]}(𝐴)≤max𝜆_[𝑝,𝑞]𝑓,𝜆_[𝑝,𝑞]1𝑓. (10)

Theorem 14. Let 𝐴(𝑧) be a transcendental meromorphic function with 𝜎_[𝑝,𝑞](𝐴)∶=𝜎(0<𝜎<+∞). Assume that (1) possesses two linearly independent meromorphic solutions 𝑓₁ and 𝑓₂. Denote 𝐸∶=𝑓₁𝑓₂. If 𝜆_[𝑝,𝑞](𝐸)<∞, then any nonzero solution 𝑓 of (1) which is not a constant multiple of either 𝑓₁ or 𝑓₂ satisfies 𝜆_[𝑝,𝑞](𝑓)=∞, unless all solutions of (1) are of finite [𝑝,𝑞]-order. In the special case where 𝛿(∞,𝐴)>0 or 𝜆_[𝑝,𝑞](1/𝐴)<𝜎_[𝑝,𝑞](𝐴), one can conclude that 𝜆_[𝑝,𝑞]=∞.

Theorem 15. Let 𝐴(𝑧) be a transcendental meromorphic function with 𝜎_[𝑝,𝑞](𝐴)>0. Assume that (1) possesses two linearly independent meromorphic solutions 𝑓₁ and 𝑓₂. Denote 𝐸∶=𝑓₁𝑓₂. If 𝛿(∞,𝐴)>0 or 𝜆_[𝑝,𝑞](1/𝐴)<𝜎_[𝑝,𝑞](𝐴), and if either 𝛿(∞,𝑓)>0 or the poles of 𝑓 are of uniformly bounded multiplicities, then one has 𝜆_{[𝑝+1,𝑞]}(𝐸)=𝜆_{[𝑝+1,𝑞]}(𝐸)=𝜎_{[𝑝+1,𝑞]}(𝐸)=max𝜆_{[𝑝+1,𝑞]}𝑓₁,𝜆_{[𝑝+1,𝑞]}𝑓₂≤𝜆_{[𝑝+1,𝑞]}𝑓₁=𝜆_{[𝑝+1,𝑞]}𝑓₂=𝜆_[𝑝,𝑞](𝐴). (11)

Theorem 16. Let 𝐴 be a meromorphic function with 𝜎_[𝑝,𝑞](𝐴)>0. Assume that 𝑓₁ and 𝑓₂ are two linearly independent meromorphic solutions of (1) such that max𝜆_[𝑝,𝑞]𝑓₁,𝜆_[𝑝,𝑞]𝑓₂<𝜎_[𝑝,𝑞](𝐴). (12) Let Π(𝑧)≢0 be any meromorphic function for which 𝜎_[𝑝,𝑞](Π)<𝜎_[𝑝,𝑞](𝐴). Let 𝑔₁ and 𝑔₂ be two linearly independent solutions of the differential equation 𝑔^′′+(𝐴(𝑧)+Π(𝑧))𝑔=0. (13) Denote 𝐸∶=𝑓₁𝑓₂ and 𝐹∶=𝑔₁𝑔₂. If max𝜆_[𝑝,𝑞]1𝐸,𝜆_[𝑝,𝑞]1𝐹<𝜆_[𝑝,𝑞](𝐴), (14) then max{𝜆_[𝑝,𝑞](𝑔₁),𝜆_[𝑝,𝑞](𝑔₂)}≥𝜎_[𝑝,𝑞](𝐴).

Remark 17 (Following Hayman [20]). we will use the abbreviation “n.e.” (nearly everywhere) to mean “everywhere in (0,+∞) except in a set of finite measure” in the proofs of our main results of this paper.

Remark 18. Obviously, Theorems 13–16 are the improvement of Theorems 3–5 given by Cao and Li [13].

2. Some Lemmas

For the proof of our results we need the following lemmas.

Lemma 19 (see [20, Theorem  4]). Let 𝑓(𝑧) be a transcendental meromorphic function not of the form 𝑒^{𝛼𝑧+𝛽}. Then 𝑇𝑓𝑟,𝑓^′≤3𝑁𝑟,𝑓+7𝑁1𝑟,𝑓+4𝑁1𝑟,𝑓^′′𝑓+𝑆𝑟,𝑓^′. (15) Using the same proof of Remark  1.3 in [9], one can easily prove the following lemma.

Lemma 20. Let 𝑓(𝑧) be a meromorphic function of [𝑝,𝑞]-order. Then 𝜎_[𝑝,𝑞]𝑓=𝜎_[𝑝,𝑞]𝑓^′. (16)

Lemma 21. Let 𝑓(𝑧) be a meromorphic function with [𝑝,𝑞]-order and 𝜎_[𝑝,𝑞](𝑓)=𝜎, and let 𝑘≥1 be an integer. Then for any 𝜀>0, 𝑚𝑓𝑟,^(𝑘)𝑓=𝑂exp_𝑝−1(𝜎+𝜀)log_𝑞𝑟 (17) holds outside of an exceptional set 𝐸₁ of finite linear measure.

Proof. Let 𝑘≥1. Since 𝜎=𝜎_[𝑝,𝑞](𝑓)<∞, we have for all sufficiently large 𝑟𝑇𝑟,𝑓<exp_𝑝(𝜎+𝜀)log_𝑞𝑟. (18) By the lemma of the logarithmic derivative, we have 𝑚𝑓𝑟,^(𝑘)𝑓=𝑂log𝑇𝑟,𝑓+log𝑟,𝑟∉𝐸₁, (19) where 𝐸₁⊂(1,∞) is a set of finite linear measure, not necessarily the same at each occurrence. Hence we have 𝑚𝑓𝑟,^′𝑓=𝑂exp_𝑝−1(𝜎+𝜀)log_𝑞𝑟,𝑟∉𝐸₁. (20) Next, assume that we have 𝑚𝑓𝑟,^(𝑘)𝑓=𝑂exp_𝑝−1(𝜎+𝜀)log_𝑞𝑟,𝑟∉𝐸₁, (21) for some 𝑘∈𝑁. Since 𝑁(𝑟,𝑓^(𝑘))≤(𝑘+1)𝑁(𝑟,𝑓), it holds that 𝑇𝑟,𝑓^(𝑘)≤𝑚𝑟,𝑓^(𝑘)+𝑁𝑟,𝑓^(𝑘)𝑓≤𝑚𝑟,^(𝑘)𝑓+()𝑁≤()𝑇+𝑚𝑟,𝑓𝑘+1𝑟,𝑓𝑘+1𝑟,𝑓+𝑂exp_𝑝−1(𝜎+𝜀)log_𝑞𝑟,𝑟∉𝐸₁. (22) By (20), we again obtain 𝑚𝑓𝑟,^(𝑘+1)𝑓^(𝑘)=𝑂exp_𝑝−1(𝜎+𝜀)log_𝑞𝑟,𝑟∉𝐸₁, (23) and hence, for sufficiently large 𝑟∉𝐸₁, 𝑚𝑓𝑟,^(𝑘+1)𝑓𝑓≤𝑚𝑟,^(𝑘+1)𝑓^(𝑘)𝑓+𝑚𝑟,^(𝑘)𝑓=𝑂exp_𝑝−1(𝜎+𝜀)log_𝑞𝑟. (24)

Using the same proof of Lemma  3.6 in [19], we can easily prove the following lemma.

Lemma 22. Let Φ(𝑟) be a continuous and positive increasing function, defined for 𝑟 on (0,+∞), with [𝑝,𝑞]-order 𝜎_[𝑝,𝑞](Φ)=limsup_𝑟→+∞(log_𝑝Φ(𝑟)/log_𝑞𝑟). Then for any subset 𝐸 of [0,+∞) that has finite linear measure, there exists a sequence {𝑟_𝑚}, 𝑟_𝑚∉𝐸 such that 𝜎_[𝑝,𝑞](Φ)=lim_{𝑟𝑚→+∞}log_𝑝Φ𝑟_𝑚log_𝑞𝑟_𝑚. (25)

Lemma 23. Let 𝑔₁(𝑧) and 𝑔₂(𝑧) be two entire functions of [𝑝,𝑞]-order, and denote 𝐹=𝑔₁𝑔₂. Then 𝜆_[𝑝,𝑞](𝐹)=max𝜆_[𝑝,𝑞]𝑔₁,𝜆_[𝑝,𝑞]𝑔₂. (26)

Proof. Let 𝑛(𝑟,𝐹) denote the number of the zeros of 𝐸(𝑧) in disk={𝑧∶|𝑧|≤𝑟} and so on for 𝑔₁ and 𝑔₂. Since, for any given 𝑟>0, we have 𝑛(𝑟,𝐹)≥𝑛(𝑟,𝑔₁) and 𝑛(𝑟,𝐹)≥𝑛(𝑟,𝑔₂), thus by Definition 11, we have 𝜆_[𝑝,𝑞](𝐹)≥max𝜆_[𝑝,𝑞]𝑔₁,𝜆_[𝑝,𝑞]𝑔₂. (27) On the other hand, since the zero of 𝐸(𝑧) must be the zero of 𝑔₁ or 𝑔₂, then for any given 𝑟>0, we have 𝑛(𝑟,𝐹)≤𝑛𝑟,𝑔₁+𝑛𝑟,𝑔₂≤2max𝑛𝑟,𝑔₁,𝑛𝑟,𝑔₂. (28) Therefore, by Definition 11, we have 𝜆_[𝑝,𝑞](𝐹)≤max𝜆_[𝑝,𝑞]𝑔₁,𝜆_[𝑝,𝑞]𝑔₂. (29)
Thus we complete the proof of Lemma 22.

Lemma 24. A meromorphic function 𝑓(𝑧) with [𝑝,𝑞] index can be represented by the form 𝑓(𝑧)=𝑈(𝑧)𝑒^𝑔(𝑧)𝑉(𝑧), (30) where 𝑈(𝑧), 𝑉(𝑧), and 𝑔(𝑧) are entire functions such that 𝜆_[𝑝,𝑞]𝑓=𝜆_[𝑝,𝑞](𝑈)=𝜎_[𝑝,𝑞](𝑈),𝜆_[𝑝,𝑞]1𝑓=𝜆_[𝑝,𝑞](𝑉)=𝜎_[𝑝,𝑞](𝑉),𝜎_[𝑝,𝑞]𝑓=max𝜎_[𝑝,𝑞](𝑈),𝜎_[𝑝,𝑞](𝑉),𝜎_[𝑝,𝑞]𝑒^𝑔. (31)

Proof. By using the essential part of the factorization theorem for meromorphic function of finite [𝑝,𝑞]-order and similar to the proof of Lemma  1.8 in [9], we can get the conclusions of this lemma easily.

Lemma 25 (see [14, Theorem  6.2]). Let 𝑓 be a meromorphic solution of 𝑓^(𝑘)+𝐴_𝑘−1(𝑧)𝑓^(𝑘−1)+⋯+𝐴₁(𝑧)𝑓^′+𝐴₀(𝑧)𝑓=0, (32) where 𝐴₀,…,𝐴_𝑘−1 are meromorphic functions in the plane ℂ. Assume that not all coefficients 𝐴_𝑗 are constants. Given a real constant 𝛾>1, and denoting 𝑇(𝑟)∶=∑^𝑘−1_𝑗−0𝑇(𝑟,𝐴_𝑗), one has log𝑚𝑟,𝑓<𝑇(𝑟)log𝑟log𝑇(𝑟)^𝛾,𝑖𝑓𝑡=0,log𝑚𝑟,𝑓<𝑟^{2𝑡+𝛾−1}𝑇(𝑟)log𝑇(𝑟)^𝛾,𝑖𝑓𝑡=0, (33) outside of an exceptional set 𝐸_𝑡 with ∫_𝐸𝑡𝑙^𝑡−1𝑑𝑙<+∞.

Remark 26. From the above lemma, we can see that 𝑡=1 corresponds to Euclidean measure and 𝑡=0 to logarithmic measure.

Using the above lemma, we can get the following lemma.

Lemma 27. Let 𝐴₀,𝐴₁,…,𝐴_𝑘−1 be meromorphic functions with [𝑝,𝑞] index and 0≤𝜎_[𝑝,𝑞](𝐴_𝑗)<∞(𝑗=0,1,…,𝑘−1). If 𝑓 is a meromorphic solution of (20) whose poles are of uniformly bounded multiplicities or 𝛿(∞,𝑓)>0, then 𝜎_{[𝑝+1,𝑞]}(𝑓)≤max{𝜎_[𝑝,𝑞](𝐴_𝑗)∶𝑗=0,1,…,𝑘−1}.

Proof. Firstly, suppose that 𝜎_[𝑝,𝑞](𝑓)<∞; then we have 𝜎_{[𝑝+1,𝑞]}(𝑓)=0. Since 0≤𝜎_[𝑝,𝑞](𝐴_𝑗)<∞(𝑗=0,1,…,𝑘−1), then we have 𝜎_{[𝑝+1,𝑞]}(𝑓)≤max{𝜎_[𝑝,𝑞](𝐴_𝑗)∶𝑗=0,1,…,𝑘−1}. Second, suppose that 𝜎_[𝑝,𝑞](𝑓)=∞. From (32), we know that the poles of 𝑓(𝑧) can only occur at the poles of 𝐴₀,𝐴₁,…,𝐴_𝑘−1. Since the multiplicities of poles of 𝑓 are uniformly bounded, we have 𝑁𝑟,𝑓≤𝐾₁𝑁𝑟,𝑓≤𝐾₁^𝑘−1_𝑗=0𝑁𝑟,𝐴_𝑗≤𝐾₀max𝑁𝑟,𝐴_𝑗∶𝑗=0,1,…,𝑘−1, (34) where 𝐾₀, 𝐾₁ are some suitable positive constants. Thus, we can get 𝑇𝑁𝑟,𝑓=𝑚𝑟,𝑓+𝑂max𝑟,𝐴_𝑗∶𝑗=0,1,…,𝑘−1. (35) Set 𝛿(∞,𝑓)∶=𝛼₁>0; for sufficiently large 𝑟, we have 𝑚≥𝛼𝑟,𝑓¹2𝑇𝑟,𝑓. (36) From Lemma 24 and (35) or (36), we obtain (𝑟)𝑇(𝑟(𝑟log𝑇𝑟,𝑓≤log𝑚𝑟,𝑓+𝑂log𝑇≤𝑂)log𝑟log𝑇)^𝛾 (37) or ≤2log𝑇𝑟,𝑓𝛼₁𝑚𝑟,𝑓≤𝑂𝑇(𝑟)log𝑟log𝑇(𝑟)^𝛾 (38) outside of an exceptional set 𝐸₀ with finite logarithmic measure.
Using a standard method to deal with the finite logarithmic measure set, one immediately gets from the previous inequalities that 𝜎_{[𝑝+1,𝑞]}(𝑓)≤max{𝜎_[𝑝,𝑞](𝐴_𝑗)∶𝑗=0,1,…,𝑘−1}.
Thus, we complete the proof of this lemma.

Lemma 28. Let 𝐴 be a meromorphic function with [𝑝,𝑞] index, and let 𝑓 be a nonzero meromorphic solution of (1). Then(i)if either 𝛿(∞,𝑓)>0 or the poles of 𝑓 are of uniformly bounded multiplicities, then 𝜎_{[𝑝+1,𝑞]}(𝑓)≤𝜎_[𝑝,𝑞](𝐴);(ii)if 𝛿(∞,𝐴)>0 or 𝜆_[𝑝,𝑞](1/𝐴)<𝜎_[𝑝,𝑞](𝐴), then 𝜎_{[𝑝+1,𝑞]}(𝑓)≥𝜎_[𝑝,𝑞](𝐴).

Proof. Suppose that 𝑓 is a nonzero meromorphic solution of (1). It is easy to see that (i) holds since (i) is just a special case of Lemma 25.
Next, we assume that 𝐴 satisfies 𝛿(∞,𝐴)>0 or 𝜆_[𝑝,𝑞](1/𝐴)<𝜎_[𝑝,𝑞](𝐴). By (1), we have (𝑧)=𝑓−𝐴^′′𝑓. (39) By (39) and Lemma 21, we can get that 𝑚()𝑓𝑟,𝐴≤𝑚𝑟,^′′𝑓=𝑂log𝑟𝑇𝑟,𝑓 (40) holds for all sufficiently large 𝑟∉𝐸, where 𝐸⊂(0,+∞) has finite linear measure. Hence 𝑇()()()()𝑟,𝐴=𝑚𝑟,𝐴+𝑁𝑟,𝐴≤𝑁𝑟,𝐴+𝑂log𝑟𝑇𝑟,𝑓 (41) holds for all sufficiently large |𝑧|=𝑟∉𝐸.
If 𝜎_[𝑝,𝑞](𝐴)=0, since 𝐴 is a meromorphic function with [𝑝,𝑞] index, by Definition 6 and Lemma 22, there exists a sequence {𝑟_𝑚} such that, for all 𝑟_𝑚∉𝐸₃, 𝑇𝑟_𝑚,𝐴≥exp_𝑝−1𝑀log_𝑞𝑟_𝑚 (42) holds for any sufficiently large constant 𝑀>0. If 𝜎_[𝑝,𝑞](𝐴)=𝜎>0, by Lemma 22 there exists a sequence {𝑟_𝑚} such that, for all 𝑟_𝑚∉𝐸₃, 𝑇𝑟_𝑚,𝐴≥exp_𝑝(𝜎−𝜀)log_𝑞𝑟_𝑚 (43) holds for any given 𝜀(0<𝜀<𝜎).
We will consider two cases as follows.
Case  1. Suppose that 𝛿(∞,𝐴)∶=𝛼₂>0. Then for sufficiently large 𝑟, we have 𝛼₂2𝑇(𝑟,𝐴)≤𝑚(𝑟,𝐴)=𝑂log𝑟𝑇𝑟,𝑓. (44) If 𝜎_[𝑝,𝑞](𝐴)=0 and 𝐴 has [𝑝,𝑞] index, from (42) and (44), we have 𝜎_{[𝑝+1,𝑞]}(𝑓)≥0=𝜎_[𝑝,𝑞](𝐴). If 𝜎_[𝑝,𝑞](𝐴)>0, then from (43) and (44), we can get that 𝜎_{[𝑝+1,𝑞]}(𝑓)≥𝜎_[𝑝,𝑞](𝐴).
Case  2. Suppose that 𝜆_[𝑝,𝑞](1/𝐴)<𝜎_[𝑝,𝑞](𝐴)=𝜎. Then the inequality 𝑁(𝑟,𝐴)≤exp_𝑝𝜆_[𝑝,𝑞]1𝐴+𝜀log_𝑞𝑟 (45) holds for any given 𝜀(0<2𝜀<𝜎−𝜆_[𝑝,𝑞](1/𝐴)). Then from (41), (43), and (45), we can get that 𝜎_{[𝑝+1,𝑞]}(𝑓)≥𝜎_[𝑝,𝑞](𝐴).
Thus, we complete the proof of Lemma 28.

3. Proof of Theorem 13

Since 𝜎_[𝑝,𝑞](𝐴)∶=𝜎>0 and 𝑓 is a solution of (1), it is easy to see that 𝑓 can not be rational nor be of the form 𝑒^{𝑎𝑧+𝑏} for constants 𝑎 and 𝑏. Thus, by Lemma 19 we have 𝑇𝑓𝑟,𝑓^′=𝑂𝑁𝑟,𝑓+𝑁1𝑟,𝑓+𝑁1𝑟,𝑓^′′,n.e.as𝑟⟶∞. (46) From (1), we have 𝑁1𝑟,𝑓^′′≤𝑁1𝑟,𝑓+𝑁1𝑟,𝐴. (47) Suppose that (9) fails to hold that is 𝜎_[𝑝,𝑞](𝐴)>max𝜆_[𝑝,𝑞]𝑓,𝜆_[𝑝,𝑞]1𝑓; (48) by the assumption 𝜆_[𝑝,𝑞](𝐴)<𝜎_[𝑝,𝑞](𝐴) and from (46)–(48), we can get 𝜎_[𝑝,𝑞](𝑓/𝑓^′)<𝜎_[𝑝,𝑞](𝐴).

Set 𝜓=𝑓^′/𝑓; by the first main theorem, we can get that 𝜎_[𝑝,𝑞]𝜓<𝜎_[𝑝,𝑞](𝐴). (49) From (1) and 𝜓=𝑓^′/𝑓, we can get −𝐴=𝜓^′+𝜓². Thus, from (49) and Lemma 20, we have 𝜎_[𝑝,𝑞](𝐴)≤𝜎_[𝑝,𝑞](𝜓)<𝜎_[𝑝,𝑞](𝐴), a contradiction. Thus, (9) is true.

In the special case where either 𝛿(∞,𝑓)>0 or the poles of 𝑓 are of uniformly bounded multiplicities, by Lemma 28 we have max𝜆_{[𝑝+1,𝑞]}1𝑓,𝜆_{[𝑝+1,𝑞]}𝑓≤𝜎_{[𝑝+1,𝑞]}𝑓≤𝜎_[𝑝,𝑞](𝐴). (50) Combining the above discussions, we can get (10).

Thus, this completes the proof of Theorem 13.

4. Proof of Theorem 14

Suppose that (1) possesses two linearly independent meromorphic solutions 𝑓₁ and 𝑓₂ such that 𝜆_[𝑝,𝑞](𝐸)<∞, where 𝐸₁∶=𝐸=𝑓₁𝑓₂. Let 𝑓=𝑎𝑓₁+𝑏𝑓₂, where 𝑎 and 𝑏 are nonzero constants, and set 𝐸₂∶=𝑓𝑓₁. From (1), we can see that any pole of 𝑓 is a pole of 𝐴. Since 𝜆_[𝑝,𝑞](1/𝐴)≤𝜎_[𝑝,𝑞](𝐴), and by the assumptions of Theorem 14, we have 𝜆_[𝑝,𝑞](1/𝑓)≤𝜆_[𝑝,𝑞](1/𝐴)<∞. If 𝜆_[𝑝,𝑞](𝑓)<∞, from the above discussion, we can get that 𝜆_[𝑝,𝑞](𝐸₁)<∞ and 𝜆_[𝑝,𝑞](𝐸₂)<∞. By Lemma D(e) in [6], there exists a constant 𝑐>0 such that n.e. as 𝑟→∞, 𝑇𝑟,𝐸_𝑗=𝑂𝑁1𝑟,𝐸_𝑗+𝑇(𝑟,𝐴)+log𝑟=𝑂exp_𝑝𝑐log_𝑞𝑟+𝑇(𝑟,𝐴) (51) for 𝑗=1,2. Since 𝐸₂=𝑎𝑓²₁+𝑏𝐸₁, from (51) we can get that n.e. as 𝑟→∞, 𝑇𝑟,𝑓₁=𝑂exp_𝑝𝑐log_𝑞𝑟+𝑇(𝑟,𝐴). (52) From 𝐴=−(𝑓^′′/𝑓) and Lemma 21, we can get 𝑚(𝑟,𝐴)=𝑂log𝑟𝑇𝑟,𝑓,n.e.𝑟⟶∞. (53) And from (1), we can see that any pole of 𝐴 is at most double and is either a zero or pole of 𝑓. Then we get 𝑁(𝑟,𝐴)≤2𝑁1𝑟,𝑓+𝑁𝑟,𝑓. (54) By assumptions 𝜆_[𝑝,𝑞](𝑓)<∞, 𝜆_[𝑝,𝑞](1/𝑓)<∞ and (54), we can get that 𝑁(𝑟,𝐴)=𝑂(exp_𝑝(𝑑log_𝑞𝑟)) as 𝑟→∞ for some 𝑑>0. Together with (52), (53), and (54), we can get that 𝑇(𝑟,𝑓₁)=𝑂(exp_𝑝(𝑑log_𝑞𝑟)) n.e. as 𝑟→∞. Thus, it follows that 𝑓₁ is of finite [𝑝,𝑞]-order.

By the identity of Abel, we have 𝑓₂𝑓₁^′=𝑊𝑓²₁, (55) where 𝑊 is equal to the Wronskian of 𝑓₁ and 𝑓₂. Hence, by Lemma 20 and (55), we get 𝜎_[𝑝,𝑞]𝑓₂=𝜎_[𝑝,𝑞]𝑓₁𝑓₂𝑓₁≤max𝜎_[𝑝,𝑞]𝑓₂𝑓₁,𝜎_[𝑝,𝑞]𝑓₁=𝜎_[𝑝,𝑞]𝑓₁. (56)

Reversing the roles of 𝑓₁ and 𝑓₂, we can get that 𝜎_[𝑝,𝑞](𝑓₁)=𝜎_[𝑝,𝑞](𝑓₂). Thus, we can get that all solutions of (1) are of finite [𝑝,𝑞]-order if 𝜆_[𝑝,𝑞](𝑓)<∞.

In special case where 𝛿(∞,𝐴)>0 or 𝜆_[𝑝,𝑞](1/𝐴)<𝜎_[𝑝,𝑞](𝐴), by Lemma 28, we can get that all meromorphic solutions 𝑓≢0 of (1) satisfy 𝜎_{[𝑝+1,𝑞]}(𝑓)≥𝜎=𝜎_[𝑝,𝑞](𝐴). Hence, we can obtain that 𝜆_[𝑝,𝑞](𝑓)=∞ holds for any solution 𝑓≢0 of (1) which is not a constant multiple of either 𝑓₁ or 𝑓₂.

5. Proof of Theorem 15

From [6, Page 664], we can get 𝜎_{[𝑝+1,𝑞]}(𝑓₁)=𝜎_{[𝑝+1,𝑞]}(𝑓₂) easily. Suppose that 𝛿(∞,𝐴)∶=𝛼₃>0 or 𝜆_[𝑝,𝑞](1/𝐴)<𝜎_[𝑝,𝑞](𝐴) and that either 𝛿(∞,𝑓)>0 or the poles of 𝑓 are of uniformly bounded multiplicities. Then by Lemma 28 we obtain 𝜎_{[𝑝+1,𝑞]}(𝐸)≤max𝜎_{[𝑝+1,𝑞]}𝑓₁,𝜎_{[𝑝+1,𝑞]}𝑓₂≤𝜎_{[𝑝+1,𝑞]}𝑓₁=𝜎_{[𝑝+1,𝑞]}𝑓₂=𝜎_[𝑝,𝑞](𝐴)<∞. (57) By Lemma D(e) in [6], there a constant 𝑐>0 such that n.e. as 𝑟→∞, 𝑇(𝑟,𝐸)=𝑂𝑁1𝑟,𝐸+𝑇(𝑟,𝐴)+log𝑟=𝑂exp_𝑝𝑐log_𝑞𝑟+𝑇(𝑟,𝐴). (58) By Lemmas 21 and 28, we have 𝑚()𝑓𝑟,𝐴=𝑚𝑟,^′′1𝑓₁=𝑂log𝑟𝑇𝑟,𝑓₁=𝑂exp_𝑝𝛽₁log_𝑞𝑟 (59) for some 𝛽₁<∞ outside of a possible exceptional set 𝐸₄⊂[0,∞) with finite linear measure. If 𝛿(∞,𝐴)∶=𝛼₃>0, then for sufficiently large 𝑟, we have 𝛼₃2𝑇(𝑟,𝐴)≤𝑚(𝑟,𝐴)=𝑂exp_𝑝𝛽₁log_𝑞𝑟,𝑟∉𝐸₄. (60) If 𝜆_[𝑝,𝑞](1/𝐴)<𝜎_[𝑝,𝑞](𝐴)<∞, there exists a constant 𝛽₂<∞ such that 𝑁(𝑟,𝐴)=𝑂exp_𝑝𝛽₂log_𝑞𝑟. (61) From the above equality and (60), we have 𝑇(𝑟,𝐴)=𝑚(𝑟,𝐴)+𝑁(𝑟,𝐴)=𝑂exp_𝑝𝛽log_𝑞𝑟, (62) where 𝛽=max{𝛽₁,𝛽₂}.

Therefore, together with (58) and either (60) or (62), we obtain 𝑇(𝑟,𝐸)=𝑂𝑁1𝑟,𝐸+exp_𝑝𝛽log_𝑞𝑟,𝑟∉𝐸₄. (63)

Suppose that 𝜆_{[𝑝+1,𝑞]}(𝐸)<𝜎_{[𝑝+1,𝑞]}(𝐸)∶=𝜎₁; then from Definitions 6 and 11 we have 𝑁(𝑟,1/𝐸)=𝑂(exp_𝑝+1(𝑏log_𝑞𝑟)) for some 𝑏<𝜎₁. From (63), 𝑇(𝑟,𝐸)=𝑂(exp_𝑝+1(𝑏log_𝑞𝑟)), 𝑟∉𝐸₄, and then by standard reasoning, we obtain 𝜎_{[𝑝+1,𝑞]}(𝐸)≤𝑏<𝜎₁=𝜎_{[𝑝+1,𝑞]}(𝐸). Thus, we get a contradiction. Therefore, we have 𝜆_{[𝑝+1,𝑞]}(𝐸)≥𝜎₁=𝜎_{[𝑝+1,𝑞]}(𝐸). And since 𝜎_{[𝑝+1,𝑞]}(𝐸)≥𝜆_{[𝑝+1,𝑞]}(𝐸)≥𝜆_{[𝑝+1,𝑞]}(𝐸), then we have 𝜎_{[𝑝+1,𝑞]}(𝐸)=𝜆_{[𝑝+1,𝑞]}(𝐸)=𝜆_{[𝑝+1,𝑞]}(𝐸).

By Lemma D(a) in [6], 𝑓₁ and 𝑓₂ have no common zeros. Let 𝑓_𝑗=(𝑔_𝑗/𝑑_𝑗)(𝑗=1,2), where 𝑔_𝑗 and 𝑑_𝑗 have no common zeros. This implies that 𝑔₁ and 𝑔₂ have no common zeros, that 𝜆_[𝑝,𝑞](𝑓_𝑗)=𝜆_[𝑝,𝑞](𝑔_𝑗) for 𝑗=1,2, and that 𝜆_[𝑝,𝑞](𝐸)=𝜆_[𝑝,𝑞](𝑔₁𝑔₂). Then by Lemma 23, we have 𝜆_{[𝑝+1,𝑞]}(𝐸)=max{𝜆_{[𝑝+1,𝑞]}(𝑓₁),𝜆_{[𝑝+1,𝑞]}(𝑓₂)}.

Thus, we can get the following conclusion 𝜆_{[𝑝+1,𝑞]}(𝐸)=𝜆_{[𝑝+1,𝑞]}(𝐸)=𝜎_{[𝑝+1,𝑞]}(𝐸)=max𝜆_{[𝑝+1,𝑞]}𝑓₁,𝜆_{[𝑝+1,𝑞]}𝑓₂≤𝜆_{[𝑝+1,𝑞]}𝑓₁=𝜆_{[𝑝+1,𝑞]}𝑓₂=𝜆_[𝑝,𝑞](𝐴)<∞. (64) Therefore, we complete the proof of Theorem 15.

6. Proof of Theorem 16

From 𝐸∶=𝑓₁𝑓₂ and 𝐹∶=𝑔₁𝑔₂, by using a similar argument as in [9, Lemma  1.7], we can get 𝜆_[𝑝,𝑞](𝐹)=max{𝜆_[𝑝,𝑞](𝑔₁),𝜆_[𝑝,𝑞](𝑔₁)}. Suppose that 𝜆_[𝑝,𝑞](𝐹)<𝜆_[𝑝,𝑞](𝐴)∶=𝜎₁ from (12), we have 𝑁1𝑟,𝐸=𝑂exp_𝑝𝛽₁log_𝑞𝑟 (65) for some 𝛽₁<𝜎_[𝑝,𝑞](𝐴)=𝜎₁. Since 𝜆_[𝑝,𝑞](𝐴)∶=𝜎₁>0 and from Definition 6, for any 𝜀(>0), we have 𝑇(𝑟,𝐴)=𝑂exp_𝑝𝜎₁+𝜀log_𝑞𝑟. (66) By Lemma D(e) in [6], we have 𝑇(𝑟,𝐸)=𝑂(𝑁(𝑟,1/𝐸)+𝑇(𝑟,𝐴)+log𝑟). Thus, we can get that 𝑇(𝑟,𝐸)=𝑂exp_𝑝𝜎₁+𝜀log_𝑞𝑟. (67) Hence, from the above equality, we have 𝜎_[𝑝,𝑞](𝐸)≤𝜎₁. On the other hand, by Lemma B(iv) in [6], we have 𝐸4𝐴=^′𝐸²𝐸−2^′′𝐸−1𝐸², (68) which implies that 𝜎₁≤𝜎_[𝑝,𝑞](𝐸). Since 𝜎_[𝑝,𝑞](Π)<𝜎_[𝑝,𝑞](𝐴), using the same argument as in the above for the function 𝐹, we have 𝜎_[𝑝,𝑞](𝐸)=𝜎_[𝑝,𝑞](𝐹)=𝜎_[𝑝,𝑞](𝐴)=𝜎₁.

From the assumptions of Theorem 16 and Lemma 24, we can write 𝑈𝐸=¹𝑒^𝐺1𝑉₁𝑈,𝐹=²𝑒^𝐺2𝑉₂, (69) where 𝜎_[𝑝,𝑞](𝑈₁)=𝜆_[𝑝,𝑞](𝐸)<𝜎_[𝑝,𝑞](𝐴) and 𝜎_[𝑝,𝑞](𝑈₂)=𝜆_[𝑝,𝑞](𝐹)<𝜎_[𝑝,𝑞](𝐴). And since max{𝜆_[𝑝,𝑞](1/𝐸),𝜆_[𝑝,𝑞](1/𝐹)}<𝜆_[𝑝,𝑞](𝐴), we have 𝜎_[𝑝,𝑞]𝑒^𝐺1=𝜎_[𝑝,𝑞]𝑒^𝐺2=𝜎_[𝑝,𝑞](𝐴). (70)

Substituting (69) into (68), and using the same argument as in the proof of Theorem  3.1 in [7], we can get 𝑐𝑒^{2(𝐺1−𝐺2)}𝑉=−²¹𝑈²₂𝑈²₁𝑉²₂, (71) where 𝑐≠0. From (68), we have 𝐸²𝐹²=𝑈²₁𝑉²₂𝑈²₂𝑉²₁𝑒^{2(𝐺1−𝐺2)}1=−𝑐. (72) For the function 𝐹, similar to (68), we have 4()=𝐹𝐴+Π^′𝐹²𝐹−2^′′𝐹−1𝐹². (73) From (68), (72), and (73), we have 41𝐴+Π+𝑐𝐴=𝐹^′𝐹²𝐹−2^′′𝐹+1𝑐𝐸^′𝐸²−2𝑐𝐸^′′𝐸. (74)

Since 𝜎_[𝑝,𝑞](𝐸)=𝜎_[𝑝,𝑞](𝐹)=𝜎_[𝑝,𝑞](𝐴)=𝜎₁, by Lemma 22, for any 𝜀(>0), we can get 𝑇1𝑟,𝐴1+𝑐1+Π=𝑚𝑟,𝐴1+𝑐1+Π+𝑁𝑟,𝐴1+𝑐+Π=𝑂exp_𝑝−1𝜎₁+𝜀log_𝑞𝑟n.e.as𝑟⟶∞. (75) From (75), we can get 𝜎_[𝑝,𝑞](𝐴(1+(1/𝑐))+Π)=0<𝜎₁=𝜎_[𝑝,𝑞](𝐴) easily. Thus, we can get 𝑐=−1. Since 𝐸²=𝐹², we have 𝐸^′𝐸=𝐹^′𝐹,𝐸^′′𝐸=𝐹^′′𝐹. (76) Then from (68) and (73), we have Π≡0, a contradiction.

Therefore, we complete the proof of Theorem 16.

Acknowledgments

This project is supported by the NSF of China (11301233 and 61202313) and the Natural Science Foundation of Jiangxi Province in China (20132BAB211001 and 20132BAB211002). Zu-Xing Xuan is supported by the Beijing Natural Science Foundation (no. 1132013) and The Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges under Beijing Municipality (CIT and TCD20130513).