On the Iterated Exponent of Convergence of Solutions of Linear Differential Equations with Entire and Meromorphic Coefficients

Abstract

We investigate the zeros of the difference of the derivative of solutions of the higher-order linear differential equations 𝑓^(𝑘)+𝐴_𝑘−1(𝑧)𝑓^(𝑘−1)+⋯+𝐴₁(𝑧)𝑓^′+𝐴₀(𝑧)𝑓=0 and small functions, where 𝐴₀(𝑧),…,𝐴_𝑘−1(𝑧) are entire or meromorphic functions of finite iterated 𝑝 order.

1. Introduction and Main Results

In this paper, a meromorphic function will mean meromorphic in the whole complex plane. We shall use the standard notations in Nevanlinna value distribution of meromorphic functions [1, 2] such as 𝑇(𝑟,𝑓), 𝑁(𝑟,𝑓), 𝑚(𝑟,𝑓). For the definition of the iterated order of a meromorphic function, we use the same definition as in [3], [4, p. 317], [2, p. 129]. For all 𝑟∈ℝ, we define exp₁𝑟∶=𝑒^𝑟 and exp_𝑝+1𝑟∶=exp(exp_𝑝𝑟), 𝑝∈ℕ. We also define for all 𝑟 sufficiently large log₁𝑟∶=log𝑟 and log_𝑝+1𝑟∶=log(log_𝑝𝑟),𝑝∈ℕ. Moreover, we denote by exp₀𝑟∶=𝑟, log₀𝑟∶=𝑟, log₋₁𝑟∶=exp₁𝑟 and exp₋₁𝑟∶=log₁𝑟.

Definition 1 (see [2, 3]). Let 𝑓 be a meromorphic function. Then the iterated 𝑝 order 𝜌_𝑝(𝑓) of 𝑓 is defined by𝜌_𝑝𝑓=limsup_𝑟→+∞log_𝑝𝑇𝑟,𝑓,.log𝑟𝑝⩾1isaninteger (1) For 𝑝=1, this notation is called order and for 𝑝=2 hyperorder. If 𝑓 is an entire function, then the iterated 𝑝 order 𝜌_𝑝(𝑓) of 𝑓 is defined by 𝜌_𝑝𝑓=limsup_𝑟→+∞log_𝑝𝑇𝑟,𝑓log𝑟=limsup_𝑟→+∞log_𝑝+1𝑀𝑟,𝑓,,log𝑟𝑝⩾1isaninteger (2) where 𝑀(𝑟,𝑓)=max_|𝑧|=𝑟|𝑓(𝑧)|.

Definition 2 (see [3]). The finiteness degree of the order of a meromorphic function 𝑓 is defined by 𝑖𝑓=0for𝑓rational,min𝑗∈ℕ∶𝜌_𝑗𝑓<+∞,for𝑓transcendentalforwhichsome𝑗∈ℕwith𝜌_𝑗𝑓<+∞exists,+∞,for𝑓with𝜌_𝑗𝑓=+∞∀𝑗∈ℕ. (3)

Definition 3 (see [3]). The iterated convergence exponent of the sequence of zeros of a meromorphic function 𝑓(𝑧) is defined by 𝜆_𝑝𝑓=limsup_𝑟→+∞log_𝑝𝑁𝑟,1/𝑓,,log𝑟𝑝⩾1isaninteger (4) where 𝑁(𝑟,1/𝑓) is the counting function of zeros of 𝑓(𝑧) in {𝑧∶|𝑧|<𝑟}. Similarly, the iterated convergence exponent of the sequence of distinct zeros of 𝑓(𝑧) is defined by 𝜆_𝑝𝑓=limsup_𝑟→+∞log_𝑝𝑁𝑟,1/𝑓,,log𝑟𝑝⩾1isaninteger (5) where 𝑁(𝑟,1/𝑓) is the counting function of distinct zeros of 𝑓(𝑧) in {𝑧∶|𝑧|<𝑟}.

Definition 4 (see [3]). The finiteness degree of the iterated convergence exponent of the sequence of zeros of a meromorphic function 𝑓(𝑧) is defined by 𝑖_𝜆𝑓=10,if𝑛𝑟,𝑓=𝑂log𝑟,min𝑗∈ℕ∶𝜆_𝑗𝑓<∞,if𝜆_𝑗𝑓<∞forsome𝑗∈ℕ,∞,if𝜆_𝑗𝑓=∞∀𝑗∈ℕ. (6)

Remark 5. Similarly, we can define the finiteness degree𝑖_𝜆(𝑓) of 𝜆_𝑝(𝑓).

Definition 6 (see [5]). Let 𝑓 be an entire function. Then the iterated 𝑝 type of 𝑓, with iterated 𝑝 order 0<𝜌_𝑝(𝑓)<∞ is defined by 𝜏_𝑝𝑓=limsup_𝑟→+∞log_𝑝𝑀𝑟,𝑓𝑟^{𝜌𝑝(𝑓)},𝑝⩾1isaninteger, (7) where 𝑀(𝑟,𝑓)=max_|𝑧|=𝑟|𝑓(𝑧)|.
We define the linear measure of a set 𝐸⊂[0,+∞) by 𝑚(𝐸)=∫₀^+∞𝜒_𝐸(𝑡)𝑑𝑡 and the logarithmic measure of a set 𝐹⊂(1,+∞) by 𝑙𝑚(𝐹)=∫₁^+∞(𝜒_𝐹(𝑡)/𝑡)𝑑𝑡, where 𝜒_𝐻 is the characteristic function of a set 𝐻.

Recently, Xu, Tu, and Zheng have investigated the relationship between the small functions and the derivative of solutions of higher-order linear differential equation: 𝑓^(𝑘)+𝐴_𝑘−1(𝑧)𝑓^(𝑘−1)+⋯+𝐴₁(𝑧)𝑓^′+𝐴₀(𝑧)𝑓=0, (8) where 𝐴₀(𝑧),…,𝐴_𝑘−1(𝑧) are entire or meromorphic functions and have obtained the following results.

Theorem A (see [6]). Let 𝐴_𝑗(𝑧)??(𝑗=0,1,…,𝑘−1) be entire functions with finite order and satisfy one of the following conditions:(i)max{𝜌(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}<𝜌(𝐴₀)<∞. (ii)0<𝜌(𝐴_𝑘−1)=⋯=𝜌(𝐴₁)=𝜌(𝐴₀)<∞ and max{𝜏(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}=𝜏₁<𝜏(𝐴₀)=𝜏. Then for every solution 𝑓≢0 of (8) and for any entire function 𝜑(𝑧)≢0 satisfying 𝜌₂(𝜑)<𝜌(𝐴₀), we have 𝜆₂𝑓^(𝑖)−𝜑=𝜌₂𝑓=𝜌𝐴₀,(𝑖=0,1,…). (9)

Theorem B (see [6]). Let 𝐴_𝑗(𝑧)??(𝑗=1,…,𝑘−1) be polynomials and 𝐴₀(𝑧) be a transcendental entire function. Then for every solution 𝑓≢0 of (8) and for any entire function 𝜑(𝑧) of finite order, we have(i)𝜆(𝑓−𝜑)=𝜆(𝑓−𝜑)=𝜌(𝑓)=∞, (ii)𝜆(𝑓^(𝑖)−𝜑)=𝜆(𝑓^(𝑖)−𝜑)=𝜌(𝑓^(𝑖)−𝜑)=∞, (𝑖⩾1,𝑖∈ℕ).

When the coefficients are meromorphic functions, they have proved the following theorem.

Theorem C (see [6]). Let 𝐴_𝑗(𝑧)??(𝑗=0,1,…,𝑘−1) be meromorphic functions satisfying max{𝜌(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}<𝜌(𝐴₀) and 𝛿(∞,𝐴₀)>0. Then for every meromorphic solution 𝑓≢0 of (8) and for any meromorphic function 𝜑(𝑧)≢0 satisfying 𝜌₂(𝜑)<𝜌(𝐴₀), we have 𝜆₂𝑓^(𝑖)−𝜑=𝜆₂𝑓^(𝑖)−𝜑⩾𝜌𝐴₀,(𝑖=0,1,…),where𝑓⁽⁰⁾=𝑓. (10)

In this paper, we improve and extend the above results by considering the iterated order, and we obtain the following theorems. For some closely related to applications of this paper, see the papers of Gupta et al. [7, 8].

Theorem 7. Let 𝐴_𝑗(𝑧)??(𝑗=0,1,…,𝑘−1) be entire functions of finite iterated order with 𝑖(𝐴₀)=𝑝(0<𝑝<∞) and satisfy one of the following conditions:(i)max{𝜌_𝑝(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}<𝜌_𝑝(𝐴₀). (ii)max{𝜌_𝑝(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}⩽𝜌_𝑝(𝐴₀)=𝜌(0<𝜌<∞) and max{𝜏_𝑝(𝐴_𝑗)∶𝜌_𝑝(𝐴_𝑗)=𝜌_𝑝(𝐴₀)}<𝜏_𝑝(𝐴₀)=𝜏(0<𝜏<∞). Then for every solution 𝑓≢0 of (8) and for any entire function 𝜑(𝑧)≢0 satisfying 𝜌_𝑝+1(𝜑)<𝜌_𝑝(𝐴₀), we have 𝜆_𝑝+1𝑓^(𝑖)−𝜑=𝜆_𝑝+1𝑓^(𝑖)−𝜑=𝜌_𝑝+1𝑓=𝜌_𝑝𝐴₀,(𝑖=0,1,…). (11)

Theorem 8. Let 𝐴_𝑗(𝑧)??(𝑗=1,2,…,𝑘−1) be polynomials and 𝐴₀(𝑧) be a transcendental entire function with 0<𝜌_𝑝(𝐴₀)<∞. Then for every solution 𝑓≢0 of (8) and for any entire function 𝜑(𝑧) of finite iterated order 𝜌_𝑝(𝜑)<∞, we have(i)𝜆_𝑝(𝑓−𝜑)=𝜆_𝑝(𝑓−𝜑)=𝜌_𝑝(𝑓)=∞. (ii)𝜆_𝑝(𝑓^(𝑖)−𝜑)=𝜆_𝑝(𝑓^(𝑖)−𝜑)=𝜌_𝑝(𝑓^(𝑖)−𝜑)=∞,𝑖⩾1,𝑖∈ℕ.

Theorem 9. Let 𝐴_𝑗(𝑧)??(𝑗=0,1,…,𝑘−1) be meromorphic functions of finite iterated order with 𝑖(𝐴₀)=𝑝(0<𝑝<∞) satisfying max{𝜌_𝑝(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}<𝜌_𝑝(𝐴₀) and 𝛿(∞,𝐴₀)>0. Then for every meromorphic solution 𝑓≢0 of (8) whose poles are of uniformly bounded multiplicity and for any meromorphic function 𝜑(𝑧)≢0 satisfying 𝜌_𝑝+1(𝜑)<𝜌_𝑝(𝐴₀), we have 𝜆_𝑝+1𝑓^(𝑖)−𝜑=𝜆_𝑝+1𝑓^(𝑖)−𝜑=𝜌_𝑝𝐴₀,(𝑖=0,1,…). (12)

Corollary 10. Under the assumptions of Theorem 7, if 𝜑(𝑧)=𝑧, then, for every solution 𝑓≢0 of (8), we have 𝜆_𝑝+1𝑓^(𝑖)−𝑧=𝜆_𝑝+1𝑓^(𝑖)−𝑧=𝜌_𝑝+1𝑓=𝜌_𝑝𝐴₀,(𝑖=0,1,…). (13)

Corollary 11. Under the assumptions of Theorem 8, if 𝜑(𝑧)=𝑧, then, for every solution 𝑓≢0 of (8), we have(i)𝜆_𝑝(𝑓−𝑧)=𝜆_𝑝(𝑓−𝑧)=𝜌_𝑝(𝑓)=∞. (ii)𝜆_𝑝(𝑓^(𝑖)−𝑧)=𝜆_𝑝(𝑓^(𝑖)−𝑧)=𝜌_𝑝(𝑓^(𝑖)−𝑧)=∞, 𝑖⩾1, 𝑖∈ℕ.

Corollary 12. Under the assumptions of Theorem 9, if 𝜑(𝑧)=𝑧, then for every meromorphic solution 𝑓≢0 of (8) whose poles are of uniformly bounded multiplicity, we have 𝜆_𝑝+1𝑓^(𝑖)−𝑧=𝜆_𝑝+1𝑓^(𝑖)−𝑧=𝜌_𝑝𝐴₀,(𝑖=0,1,…). (14)

2. Auxiliary Lemmas

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

Lemma 13 (see [6]). Assume that 𝑓≢0 is a solution of (8), set 𝑔_𝑖=𝑓^(𝑖)−𝜑, and then 𝑔_𝑖 satisfies the equation 𝑔_𝑖^(𝑘)+𝑈^𝑖_𝑘−1𝑔_𝑖^(𝑘−1)+⋯+𝑈^𝑖₀𝑔_𝑖=−𝜑^(𝑘)+𝑈^𝑖_𝑘−1𝜑^(𝑘−1)+⋯+𝑈^𝑖₀𝜑, (15) where 𝑈^𝑖_𝑗=𝑈^𝑖−1′_𝑗+1+𝑈_𝑗^𝑖−1−𝑈^𝑖−1′₀𝑈₀^𝑖−1𝑈^𝑖−1_𝑗+1,𝑈⁰_𝑗=𝐴_𝑗,𝑈𝑗=0,1,2,…,𝑘−1,_𝑘𝑖−1≡1,(𝑖⩾1,𝑖∈ℕ). (16)

Lemma 14 (see [3]). Let 𝑓 be a meromorphic function for which 𝑖(𝑓)=𝑝⩾1 and 𝜌_𝑝(𝑓)=𝜌, and let 𝑘⩾1 be an integer. Then for any 𝜀>0, 𝑚𝑓𝑟,^(𝑘)𝑓=𝑂exp_𝑝−2𝑟^𝜌+𝜀 (17) outside of a possible exceptional set 𝐸₁ of finite linear measure.

Lemma 15 (see [1]). Let 𝑓 be a transcendental meromorphic function and 𝑘⩾1 be an integer. Then 𝑚𝑓𝑟,^(𝑘)𝑓=𝑂log𝑟𝑇𝑟,𝑓 (18) outside of a possible exceptional set 𝐸₂ of finite linear measure, and if 𝑓 is of finite order of growth, then 𝑚𝑓𝑟,^(𝑘)𝑓=𝑂log𝑟. (19)

Lemma 16 (see [9]). Let 𝜑∶[0,+∞)→ℝ and 𝜓∶[0,+∞)→ℝ be monotone nondecreasing functions such that 𝜑(𝑟)⩽𝜓(𝑟) for all 𝑟∉𝐸₃∪[0,1], where 𝐸₃⊂(1,+∞) is a set of finite logarithmic measure. Let 𝛼>1 be a given constant. Then there exists an 𝑟₀=𝑟₀(𝛼)>0 such that 𝜑(𝑟)⩽𝜓(𝛼𝑟) for all 𝑟>𝑟₀.

Lemma 17 (see [10]). Let 𝑓 be a meromorphic solution of (8), assuming 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), we have log𝑚𝑟,𝑓<𝑇(𝑟)log𝑟log𝑇(𝑟)^𝛾,if𝑝=0,log𝑚𝑟,𝑓<𝑟^{2𝑝+𝛾−1}𝑇(𝑟)log𝑇(𝑟)^𝛾,if𝑝>0 (20) outside of an exceptional set 𝐸_𝑝 with ∫_𝐸𝑝𝑡^𝑝−1𝑑𝑡<+∞.

Remark 18. We note that in the above lemma, 𝑝=1 corresponds to Euclidean measure and 𝑝=0 to logarithmic measure.

Lemma 19 (see [11]). Let 𝑓(𝑧) be a meromorphic function of finite iterated order satisfying 𝑖(𝑓)=𝑝. Then there exists a set 𝐸₄⊂(1,+∞) with infinite logarithmic measure such that for all 𝑟∈𝐸₄, we have lim_𝑟→∞log_𝑝𝑇𝑟,𝑓log𝑟=𝜌. (21)

Lemma 20. Let 𝐴₀(𝑧),𝐴₁(𝑧),…,𝐴_𝑘−1(𝑧) be entire functions of finite iterated order with 𝑖(𝐴₀)=𝑝(0<𝑝<∞) and satisfy max{𝜌_𝑝(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}=𝜌₁<𝜌_𝑝(𝐴₀) and set 𝑈¹_𝑗=𝐴^′_𝑗+1+𝐴_𝑗−𝐴^′₀𝐴₀𝐴_𝑗+1,𝑈^𝑖_𝑗=𝑈^𝑖−1′_𝑗+1+𝑈_𝑗^𝑖−1−𝑈^𝑖−1′₀𝑈₀^𝑖−1𝑈^𝑖−1_𝑗+1, (22) where 𝑗=0,1,…,𝑘−1;??𝐴_𝑘≡1,??𝑈_𝑘^𝑖−1≡1 and 𝑖⩾1,??𝑖∈ℕ. Then there exists a set 𝐸₅ with infinite logarithmic measure such that for all 𝑟∈𝐸₅lim_𝑟→∞log_𝑝𝑚𝑟,𝑈^𝑖₀log𝑟=𝜌_𝑝𝐴₀>limsup_𝑟→∞max_{1⩽𝑗⩽𝑘−1}log_𝑝𝑚𝑟,𝑈^𝑖_𝑗log𝑟=𝜌₁. (23)

Proof. We prove this lemma by using mathematical induction. First, when 𝑖=1, we have 𝑈¹_𝑗=𝐴^′_𝑗+1+𝐴_𝑗−(𝐴^′₀/𝐴₀)𝐴_𝑗+1,??𝑗=0,1,…,𝑘−1 and 𝐴_𝑘≡1.
When 𝑗=0, that is, 𝑈¹₀=𝐴^′₁+𝐴₀−(𝐴^′₀/𝐴₀)𝐴₁. Then we have 𝑚𝑟,𝑈¹₀⩽𝑚𝑟,𝐴₁+𝑚𝑟,𝐴₀𝐴+𝑚𝑟,^′1𝐴₁𝐴+𝑚𝑟,^′0𝐴₀+𝑂(1). (24) From −𝐴₀=𝐴^′₁−𝑈¹₀−(𝐴^′₀/𝐴₀)𝐴₁, we have 𝑚𝑟,𝐴₀⩽𝑚𝑟,𝐴₁+𝑚𝑟,𝑈¹₀𝐴+𝑚𝑟,^′1𝐴₁𝐴+𝑚𝑟,^′0𝐴₀+𝑂(1). (25) When 𝑗≠0, from the definitions of 𝑈¹_𝑗, we have 𝑚𝑟,𝑈¹_𝑗⩽𝑚𝑟,𝐴_𝑗+1+𝑚𝑟,𝐴_𝑗𝐴+𝑚𝑟,^′𝑗+1𝐴_𝑗+1𝐴+𝑚𝑟,^′0𝐴₀+𝑂(1), (26)𝑗=1,2,…,𝑘−1. Since 𝐴_𝑗(𝑧) are entire functions with max{𝜌_𝑝(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}<𝜌_𝑝(𝐴₀)<∞, then by Lemma 19, there exists a set 𝐸₄⊂(1,+∞) with infinite logarithmic measure such that for all 𝑟∈𝐸₄𝑚𝑟,𝐴_𝑗+1=𝑜𝑚𝑟,𝐴₀,𝑗=1,2,…,𝑘−1. (27) By this, (26), and Lemma 14, we get for all 𝑟∈𝐸₄max_{1⩽𝑗⩽𝑘−1}𝑚𝑟,𝑈¹_𝑗⩽max_{1⩽𝑗⩽𝑘−1}𝑚𝑟,𝐴_𝑗+𝑜𝑚𝑟,𝐴₀+𝑂exp_𝑝−2𝑟^𝛽+𝑂(1)} (28) for some constant 𝛽<∞ outside of a possible exceptional set 𝐸₁ of finite linear measure. From (24), (25), and (28) we get for all 𝑟∈𝐸₄⧵𝐸₁lim_𝑟→∞log_𝑝𝑚𝑟,𝑈¹₀log𝑟=𝜌_𝑝𝐴₀>𝜌₁=limsup_𝑟→∞max_{1⩽𝑗⩽𝑘−1}log_𝑝𝑚𝑟,𝐴_𝑗log𝑟⩾limsup_𝑟→∞max_{1⩽𝑗⩽𝑘−1}log_𝑝𝑚𝑟,𝑈¹_𝑗.log𝑟 (29) Now, suppose that (23) holds, for 𝑖⩽𝑛,𝑛∈ℕ; that is, there is a set 𝐸₅ with infinite logarithmic measure such that for all 𝑟∈𝐸₅limsup_𝑟→∞log_𝑝𝑚𝑟,𝑈^𝑛₀log𝑟=𝜌_𝑝𝐴₀>limsup_𝑟→∞max_{1⩽𝑗⩽𝑘−1}log_𝑝𝑚𝑟,𝑈^𝑛_𝑗log𝑟=𝜌₁. (30) Next, we prove that (23) holds for 𝑖=𝑛+1. Since 𝑖=𝑛+1, then we have 𝑈_𝑗^𝑛+1=𝑈^𝑛′_𝑗+1+𝑈^𝑛_𝑗−𝑈^𝑛′₀𝑈^𝑛₀𝑈^𝑛_𝑗+1, (31) where 𝑗=0,1,…,𝑘−1;??𝑈^𝑛_𝑘≡1. When 𝑗=0, we have 𝑈₀^𝑛+1=𝑈^𝑛′₁+𝑈^𝑛₀−(𝑈^𝑛′₀/𝑈^𝑛₀)𝑈^𝑛₁. Then we obtain that 𝑚𝑟,𝑈₀^𝑛+1⩽𝑚𝑟,𝑈^𝑛₀+𝑚𝑟,𝑈^𝑛₁𝑈+𝑚𝑟,^𝑛′0𝑈^𝑛₀𝑈+𝑚𝑟,^𝑛′1𝑈^𝑛₁+𝑂(1). (32) Since −𝑈^𝑛₀=𝑈^𝑛′₁−𝑈₀^𝑛+1−(𝑈^𝑛′₀/𝑈^𝑛₀)𝑈^𝑛₁, then we have 𝑚𝑟,𝑈^𝑛₀⩽𝑚𝑟,𝑈₀^𝑛+1+𝑚𝑟,𝑈^𝑛₁𝑈+𝑚𝑟,^𝑛′0𝑈^𝑛₀𝑈+𝑚𝑟,^𝑛′1𝑈^𝑛₁+𝑂(1). (33) When 𝑗≠0, from the definitions of 𝑈_𝑗^𝑛+1,??𝑗=1,2,…,𝑘−1 and 𝑈^𝑛_𝑘≡1, we have 𝑚𝑟,𝑈_𝑗^𝑛+1⩽𝑚𝑟,𝑈^𝑛_𝑗+1+𝑚𝑟,𝑈^𝑛_𝑗𝑈+𝑚𝑟,^{𝑛′𝑗+1}𝑈^𝑛_𝑗+1𝑈+𝑚𝑟,^𝑛′0𝑈^𝑛₀+𝑂(1). (34) From (30)–(34), there exists a set 𝐸₅ with infinite logarithmic measure such that for all 𝑟∈𝐸₅lim_𝑟→∞log_𝑝𝑚𝑟,𝑈₀^𝑛+1log𝑟=lim_𝑟→∞log_𝑝𝑚𝑟,𝑈^𝑛₀log𝑟=𝜌_𝑝𝐴₀>𝜌₁=limsup_𝑟→∞max_{1⩽𝑗⩽𝑘−1}log_𝑝𝑚𝑟,𝑈^𝑛_𝑗log𝑟=limsup_𝑟→∞max_{1⩽𝑗⩽𝑘−1}log_𝑝𝑚𝑟,𝑈_𝑗^𝑛+1.log𝑟 (35) Thus, the proof of Lemma 20 is completed.

Lemma 21. Let 𝐴₀,𝐴₁,…,𝐴_𝑘−1 be meromorphic functions of finite iterated order. If limsup_𝑟→∞max_{1⩽𝑗⩽𝑘−1}log_𝑝𝑚𝑟,𝐴_𝑗log𝑟=𝛽₁ (36) and there exists a set 𝐸₆ with infinite logarithmic measure such that lim_𝑟→∞log_𝑝𝑚𝑟,𝐴₀log𝑟=𝛽₂>𝛽₁ (37) holds for all 𝑟∈𝐸₆, then every meromorphic solution 𝑓≢0 of (8) satisfies 𝜌_𝑝(𝑓)=∞ and 𝜌_𝑝+1(𝑓)⩾𝛽₂.

Proof. Suppose that 𝑓≢0 is a meromorphic solution of (8). If 𝜌_𝑝(𝑓)=𝜌<∞, then from (8) we have 𝑚𝑟,𝐴₀⩽^𝑘_𝑗=1𝑚𝑓𝑟,^(𝑗)𝑓+^𝑘−1_𝑗=1𝑚𝑟,𝐴_𝑗+𝑂(1). (38) By using Lemma 14, we obtain 𝑚𝑟,𝐴₀⩽𝑂exp_𝑝−2𝑟^𝜌+𝜀+^𝑘−1_𝑗=1𝑚𝑟,𝐴_𝑗+𝑂(1) (39) outside of a possible exceptional set 𝐸₁ of finite linear measure. By the hypothesis of Lemma 21, there exists a set 𝐸₆ having infinite logarithmic measure such that for all |𝑧|=𝑟∈𝐸₆⧵𝐸₁, we have for any 𝜀(0<2𝜀<𝛽₂−𝛽₁)exp_𝑝−1𝑟^𝛽2−𝜀⩽𝑂exp_𝑝−2𝑟^𝜌+𝜀+(𝑘−1)exp_𝑝−1𝑟^𝛽1+𝜀 (40) and so by Lemma 16, we obtain 𝛽₂⩽𝛽₁, (41) and from this we obtain a contradiction. Hence 𝜌_𝑝(𝑓)=∞.
Now, assume that 𝑓≢0 is a meromorphic solution of (8) with 𝜌_𝑝(𝑓)=∞. By (8) we have 𝑚𝑟,𝐴₀⩽^𝑘_𝑗=1𝑚𝑓𝑟,^(𝑗)𝑓+^𝑘−1_𝑗=1𝑚𝑟,𝐴_𝑗+𝑂(1). (42) By Lemma 15 and (42), we have 𝑚𝑟,𝐴₀⩽𝑂log𝑟𝑇𝑟,𝑓+^𝑘−1_𝑗=1𝑚𝑟,𝐴_𝑗,𝑟∉𝐸₂, (43) where 𝐸₂⊂[1,+∞) is a set having finite linear measure. By the hypothesis of Lemma 21, there exists a set 𝐸₆ having infinite logarithmic measure such that, for all |𝑧|=𝑟∈𝐸₆⧵𝐸₂, we have by using (43) exp_𝑝−1𝑟^𝛽2−𝜀⩽𝑂log𝑟𝑇𝑟,𝑓+(𝑘−1)exp_𝑝−1𝑟^𝛽1+𝜀, (44) where 0<2𝜀<𝛽₂−𝛽₁. By (44) and Lemma 16, we have 𝜌_𝑝+1𝑓⩾𝛽₂. (45)

Lemma 22 (see [12, Theorem 3]). Let 𝑓(𝑧) be a transcendental meromorphic function, and let 𝛼>1 be a given constant. Then there exists a set 𝐸₇⊂(1,∞) with finite logarithmic measure and a constant 𝐵>0 that depends only on 𝛼 and 𝑖,𝑗(0⩽𝑖<𝑗⩽𝑘), such that for all 𝑧 satisfying |𝑧|=𝑟∉[0,1]∪𝐸₇, we have 𝑓^(𝑗)(𝑧)𝑓^(𝑖)(𝑧)𝑇⩽𝐵𝛼𝑟,𝑓𝑟log^𝛼𝑟log𝑇𝛼𝑟,𝑓^𝑗−𝑖. (46)

From the above lemma, we obtain the following result.

Lemma 23. Let 𝑓 be a transcendental meromorphic function with finite iterated order 𝜌_𝑝(𝑓)=𝜌<+∞,??𝐻={(𝑘₁,𝑗₁),(𝑘₂,𝑗₂),…,(𝑘_𝑞,𝑗_𝑞)} be a finite set of distinct pairs of integers satisfying 𝑘_𝑖>𝑗_𝑖⩾0??(𝑖=1,…,𝑞), and let 𝜀>0 be a given constant. Then, there exists a set 𝐸₈⊂(1,+∞) with finite logarithmic measure, such that, for all 𝑧 satisfying |𝑧|∉𝐸₈∪[0,1] and for all (𝑘,𝑗)∈𝐻, we have 𝑓^(𝑘)(𝑧)𝑓^(𝑗)(𝑧)⩽exp_𝑝−1𝑟^{𝜌−1+𝜀𝑘−𝑗}. (47)

Proof. The definition 𝜌_𝑝𝑓=limsup_𝑟→+∞log_𝑝𝑇𝑟,𝑓log𝑟 (48) implies that for any given 𝜀₁>0 there exists 𝑅>1 such that for all 𝑟>𝑅 we have 𝑇𝑟,𝑓<exp_𝑝−1𝑟^𝜌+𝜀1. (49) Combining (49) with Lemma 22, for 𝛼>0, there exists a set 𝐸₈=[1,𝑅]∪𝐸₇ that has finite logarithmic measure and a constant 𝐵>0, such that if |𝑧|∉𝐸₈∪[0,1], we obtain 𝑓^(𝑘)(𝑧)𝑓^(𝑗)(𝑧)⩽𝐵exp_𝑝−1(𝛼𝑟)^𝜌+𝜀1𝑟log^𝛼𝑟exp_𝑝−2(𝛼𝑟)^{𝜌+𝜀1𝑘−𝑗}. (50) Hence, there exists a constant 𝜀>1+𝜀₁, such that if |𝑧|∉𝐸₈∪[0,1], we have 𝑓^(𝑘)(𝑧)𝑓^(𝑗)(𝑧)⩽exp_𝑝−1𝑟^{𝜌−1+𝜀𝑘−𝑗}. (51)

Lemma 24 (see [5]). Let 𝑓(𝑧) be an entire function of iterated 𝑝 order 0<𝜌_𝑝(𝑓)<+∞ and iterated 𝑝 type 0<𝜏_𝑝(𝑓)<∞. Then, for every given 𝛽<𝜏_𝑝(𝑓), there exists a set 𝐸₉⊂[1,+∞) that has infinite logarithmic measure, such that, for all 𝑟∈𝐸₉, we have log_𝑝𝑀𝑟,𝑓>𝛽𝑟^{𝜌𝑝(𝑓)}. (52)

Lemma 25. Let 𝐴₀(𝑧),𝐴₁(𝑧),…,𝐴_𝑘−1(𝑧) be entire functions with finite iterated order and satisfy max{𝜌_𝑝(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}⩽𝜌_𝑝(𝐴₀)=𝜌₂(0<𝜌₂<∞), and max{𝜏_𝑝(𝐴_𝑗)∶𝜌_𝑝(𝐴_𝑗)=𝜌_𝑝(𝐴₀)}=𝜏₁<𝜏_𝑝(𝐴₀)=𝜏(0<𝜏<∞), and let 𝑈¹_𝑗,??𝑈^𝑖_𝑗 be stated as in Lemma 20. Then, for any given 𝜀(0<2𝜀<𝜏−𝜏₁), there exists a set 𝐸₁₀ with infinite logarithmic measure such that 𝑈^𝑖_𝑗⩽exp_𝑝𝜏₁+𝜀𝑟^𝜌2,𝑈^𝑖₀⩾exp_𝑝(𝜏−𝜀)𝑟^𝜌2, (53) where 𝑖⩾1,𝑖∈ℕ and 𝑗=1,2,…,𝑘−1.

Proof. We prove this lemma by using mathematical induction.(i)We first prove that 𝑈^𝑖_𝑗??(𝑗=0,1,…,𝑘−1) satisfy (53) when 𝑖=1. From the definition of 𝑈¹_𝑗=𝐴^′_𝑗+1+𝐴_𝑗−(𝐴^′₀/𝐴₀)𝐴_𝑗+1(𝑗≠0) and 𝑈¹₀=𝐴^′₁+𝐴₀−(𝐴^′₀/𝐴₀)𝐴₁, we have 𝑈¹₀⩾𝐴₀−𝐴₁𝐴^′₁𝐴₁+𝐴^′₀𝐴₀,𝑈¹_𝑗⩽𝐴_𝑗+𝐴_𝑗+1𝐴^′_𝑗+1𝐴_𝑗+1+𝐴^′₀𝐴₀,𝑗=1,2,…,𝑘−1,𝐴_𝑘≡1. (54) From Lemmas 23, 24, and (54), for any 𝜀(0<4𝜀<𝜏−𝜏₁), there exists a set 𝐸₁₀ with infinite logarithmic measure such that 𝑈¹₀⩾exp_𝑝𝜀𝜏−4𝑟^𝜌2−2exp_𝑝𝜏₁+𝜀4𝑟^𝜌2exp_𝑝−1𝑟^𝑀⩾exp_𝑝𝜀𝜏−2𝑟^𝜌2,𝑈¹_𝑗⩽exp_𝑝𝜏₁+𝜀4𝑟^𝜌2+2exp_𝑝𝜏₁+𝜀4𝑟^𝜌2exp_𝑝−1𝑟^𝑀⩽exp_𝑝𝜏₁+𝜀2𝑟^𝜌2,𝑗≠0, (55) where 𝑀>0 is a constant, not necessarily the same at each occurrence.(ii)Next, we show that 𝑈^𝑖_𝑗??(𝑗=0,1,…,𝑘−1) satisfy (53) when 𝑖=2. From 𝑈²₀=𝑈^1′₁+𝑈¹₀−𝑈^1′₀𝑈¹₀𝑈¹₁,𝑈²_𝑗=𝑈^1′_𝑗+1+𝑈¹_𝑗−𝑈^1′₀𝑈¹₀𝑈¹_𝑗+1,𝑗=1,2,…,𝑘−1,𝑈¹_𝑘≡1, (56) we have 𝑈²₀⩾𝑈¹₀−𝑈¹₁𝑈^1′₁𝑈¹₁+𝑈^1′₀𝑈¹₀,𝑈²_𝑗⩽𝑈¹_𝑗+𝑈¹_𝑗+1𝑈^1′_𝑗+1𝑈¹_𝑗+1+𝑈^1′₀𝑈¹₀,𝑗=1,2,…,𝑘−1. (57)By the conclusions of (i) and Lemma 23, (55)–(57), we obtain for all |𝑧|=𝑟∈𝐸₁₀𝑈²₀⩾exp_𝑝𝜀𝜏−2𝑟^𝜌2−2exp_𝑝𝜏₁+𝜀2𝑟^𝜌2exp_𝑝−1𝑟^𝑀⩾exp_𝑝(𝜏−𝜀)𝑟^𝜌2,𝑈²_𝑗⩽exp_𝑝𝜏₁+𝜀2𝑟^𝜌2+2exp_𝑝𝜏₁+𝜀2𝑟^𝜌2exp_𝑝−1𝑟^𝑀⩽exp_𝑝𝜏₁+𝜀𝑟^𝜌2,𝑗≠0. (58)(iii)Now, suppose that (53) holds for 𝑖⩽𝑛,??𝑛∈ℕ; that is, for any given 𝜀(0<4𝜀<𝜏−𝜏₁), there exists a set 𝐸₁₀ with infinite logarithmic measure such that 𝑈^𝑖_𝑗⩽exp_𝑝𝜏₁+𝜀𝑟^𝜌2,𝑈^𝑖₀⩾exp_𝑝(𝜏−𝜀)𝑟^𝜌2,𝑖⩽𝑛,𝑗=1,2,…,𝑘−1. (59)From 𝑈₀^𝑛+1=𝑈^𝑛′₁+𝑈^𝑛₀−(𝑈^𝑛′₀/𝑈^𝑛₀)𝑈^𝑛₁ and 𝑈_𝑗^𝑛+1=𝑈^𝑛′_𝑗+1+𝑈^𝑛_𝑗−(𝑈^𝑛′₀/𝑈^𝑛₀)𝑈^𝑛_𝑗+1??(𝑗=1,…,𝑘−1),??𝑈^𝑛_𝑘≡1, we have 𝑈₀^𝑛+1⩾𝑈^𝑛₀−𝑈^𝑛₁𝑈^𝑛′₁𝑈^𝑛₁+𝑈^𝑛′₀𝑈^𝑛₀,𝑈_𝑗^𝑛+1⩽𝑈^𝑛_𝑗+𝑈^𝑛_𝑗+1𝑈^𝑛′_𝑗+1𝑈^𝑛_𝑗+1+𝑈^𝑛′₀𝑈^𝑛₀,𝑗=1,2,…,𝑘−1. (60) Then, from Lemma 23 and (59)-(60), for all |𝑧|=𝑟∈𝐸₁₀, 𝑈_𝑗^𝑛+1⩽exp_𝑝𝜏₁+𝜀𝑟^𝜌2+2exp_𝑝𝜏₁+𝜀𝑟^𝜌2exp_𝑝−1𝑟^𝑀⩽exp_𝑝𝜏₁+2𝜀𝑟^𝜌2 (61) for 𝑗≠0, and 𝑈₀^𝑛+1⩾exp_𝑝(𝜏−𝜀)𝑟^𝜌2−2exp_𝑝𝜏₁+𝜀𝑟^𝜌2exp_𝑝−1𝑟^𝑀⩾exp_𝑝(𝜏−2𝜀)𝑟^𝜌2. (62) Thus, the proof of Lemma 25 is complete.

Lemma 26. Let 𝐴₀,𝐴₁,…,𝐴_𝑘−1 be meromorphic functions of finite iterated order with 𝑖(𝐴₀)=𝑝(0<𝑝<∞) such that max{𝜌_𝑝(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}=𝜌₄<𝜌_𝑝(𝐴₀)=𝜌₃ and 𝛿=𝛿(∞,𝐴₀)=liminf_𝑟→∞(𝑚(𝑟,𝐴₀))/(𝑇(𝑟,𝑓))>0. Then every meromorphic solution 𝑓≢0 of (8) satisfies 𝜌_𝑝+1(𝑓)⩾𝜌_𝑝(𝐴₀)=𝜌₃.

Proof. Assume that 𝑓(𝑧)≢0 is a meromorphic solution of (8). By (8) we have 𝑚𝑟,𝐴₀⩽^𝑘_𝑗=1𝑚𝑓𝑟,^(𝑗)𝑓+^𝑘−1_𝑗=1𝑚𝑟,𝐴_𝑗+𝑂(1). (63) By Lemma 15, we obtain 𝑚𝑟,𝐴₀⩽𝑂log𝑟𝑇𝑟,𝑓+^𝑘−1_𝑗=1𝑚𝑟,𝐴_𝑗,𝑟∉𝐸₂, (64) where 𝐸₂⊂[1,+∞) is a set having finite linear measure. By Lemma 19, there exists a set 𝐸₄ having infinite logarithmic measure such that, for all |𝑧|=𝑟∈𝐸₄, we have lim_𝑟→∞log_𝑝𝑇𝑟,𝐴₀log𝑟=𝜌₃. (65) Since 𝛿(∞,𝐴₀)>0, then for any given 𝜀(0<2𝜀<min{𝛿,𝜌₃−𝜌₄}) and for all 𝑟∈𝐸₄, by (65), we have 𝑚𝑟,𝐴₀⩾(𝛿−𝜀)𝑇𝑟,𝐴₀⩾(𝛿−𝜀)exp_𝑝−1𝑟^𝜌3−𝜀. (66) From (64) and (66), we have for any given 𝜀(0<2𝜀<min{𝛿,𝜌₃−𝜌₄}) and for all |𝑧|=𝑟∈𝐸₄⧵𝐸₂(𝛿−𝜀)exp_𝑝−1𝑟^𝜌3−𝜀⩽𝑂log𝑟𝑇𝑟,𝑓+(𝑘−1)exp_𝑝−1𝑟^𝜌4+𝜀. (67) By (67) and Lemma 16, we obtain 𝜌_𝑝+1𝑓⩾𝜌₃. (68)

Lemma 27. Let 𝐴₀,𝐴₁,…,𝐴_𝑘−1 be meromorphic functions of finite iterated order. If there exist positive constants 𝜌₅,𝛼,𝛽(0<𝛼<𝛽) and a set 𝐸₁₁ with infinite logarithmic measure such that 𝐴₀(𝑧)⩾exp_𝑝𝛽𝑟^𝜌5,max𝐴_𝑗(𝑧)∶𝑗=1,2,…,𝑘−1⩽exp_𝑝𝛼𝑟^𝜌5 (69) hold for all |𝑧|=𝑟∈𝐸₁₁, then every meromorphic solution 𝑓≢0 of (8) satisfies 𝜌_𝑝+1(𝑓)⩾𝜌₅.

Proof. Assume that 𝑓≢0 is a meromorphic solution of (8). By (8), we obtain 𝐴₀⩽𝑓^(𝑘)𝑓+^𝑘−1_𝑗=1𝐴_𝑗𝑓^(𝑗)𝑓. (70) By Lemma 22, there exists a set 𝐸₇ having a finite logarithmic measure such that, for all |𝑧|=𝑟∉𝐸₇, we have 𝑓^(𝑗)(𝑧)𝑓(𝑧)⩽𝐵𝑇2𝑟,𝑓^𝑘+1,𝑗=1,2,…,𝑘, (71) where 𝐵>0 is a constant. Substituting (69) and (71) into (70), we obtain for all |𝑧|=𝑟∈𝐸₁₁−𝐸₇exp_𝑝𝛽𝑟^𝜌5⩽𝐵𝑘𝑇2𝑟,𝑓^𝑘+1exp_𝑝𝛼𝑟^𝜌5. (72) Since 0<𝛼<𝛽, then from (72), Definitions 1, 2, and Lemma 16, we can deduce that 𝑖(𝑓)⩾𝑝+1 and 𝜌_𝑝+1𝑓⩾𝜌₅. (73)

Lemma 28 (see [13]). Let 𝑝⩾1 be an integer, and let 𝑓(𝑧) be a meromorphic solution of the differential equation 𝑓^(𝑘)+𝐴_𝑘−1𝑓^(𝑘−1)+⋯+𝐴₀𝑓=𝐹, (74) where 𝐴₀,𝐴₁,…,𝐴_𝑘−1 and 𝐹≢0 are meromorphic functions:(i)if max{𝜌_𝑝(𝐹),𝜌_𝑝(𝐴_𝑗)(𝑗=0,1,2,…,𝑘−1)}<𝜌_𝑝(𝑓)=𝜌⩽∞, then 𝜆_𝑝(𝑓)=𝜆_𝑝(𝑓)=𝜌_𝑝(𝑓), (ii)if max{𝜌_𝑝+1(𝐹),𝜌_𝑝+1(𝐴_𝑗)(𝑗=0,1,2,…,𝑘−1)}<𝜌_𝑝+1(𝑓)=𝜌<∞,??then 𝜆_𝑝+1(𝑓)=𝜆_𝑝+1(𝑓)=𝜌_𝑝+1(𝑓).

Lemma 29 (see [3]). Let 𝐴₀(𝑧),…,𝐴_𝑘−1(𝑧) be entire functions such that 𝑖(𝐴₀)=𝑝(0<𝑝<∞). If either max{𝑖(𝐴_𝑗): 𝑗=1,2,…,𝑘−1}<𝑝 or max{𝜌_𝑝(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}<𝜌_𝑝(𝐴₀), then every solution 𝑓≢0 of (8) satisfies 𝑖(𝑓)=𝑝+1 and 𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀).

Lemma 30 (see [5]). Let 𝐴₀,𝐴₁,…,𝐴_𝑘−1 be entire functions of finite iterated order, and let 𝑖(𝐴₀)=𝑝(0<𝑝<∞). Assume that max{𝜌_𝑝(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}⩽𝜌_𝑝(𝐴₀)=𝜌(0<𝜌<∞) and max{𝜏_𝑝(𝐴_𝑗)∶𝜌_𝑝(𝐴_𝑗)=𝜌_𝑝(𝐴₀)}<𝜏_𝑝(𝐴₀)=𝜏(0<𝜏<∞). Then every solution 𝑓≢0 of (8) satisfies 𝑖(𝑓)=𝑝+1 and 𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀)=𝜌.

3. Proof of the Theorems

Proof of Theorem 7. We consider two cases.
Case??1. Suppose that max{𝜌_𝑝(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}<𝜌_𝑝(𝐴₀)<∞.
(i) First, we prove that 𝜆_𝑝+1(𝑓−𝜑)=𝜆_𝑝+1(𝑓−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). Assume that 𝑓≢0 is a solution of (8). From Lemma 29, we have 𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). Set 𝑔=𝑓−𝜑. Since 𝜌_𝑝+1(𝜑)<𝜌_𝑝(𝐴₀), then 𝜌_𝑝+1(𝑔)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀) and 𝜆_𝑝+1(𝑔)=𝜆_𝑝+1(𝑓−𝜑),𝜆_𝑝+1(𝑔)=𝜆_𝑝+1(𝑓−𝜑). By substituting 𝑓=𝑔+𝜑 into (8), we get that 𝑔 satisfies the following equation: 𝑔^(𝑘)+𝐴_𝑘−1𝑔^(𝑘−1)+⋯+𝐴₀𝑔=−𝜑^(𝑘)+𝐴_𝑘−1𝜑^(𝑘−1)+⋯+𝐴₀𝜑. (75) Set 𝐹=𝜑^(𝑘)+𝐴_𝑘−1𝜑^(𝑘−1)+⋯+𝐴₀𝜑. If 𝐹≡0, then, from Lemma 29, we have 𝜌_𝑝+1(𝜑)=𝜌_𝑝(𝐴₀), which is a contradiction. Hence 𝐹≢0. From the assumptions of Theorem 7, we get max𝜌_𝑝+1(𝐹),𝜌_𝑝+1𝐴_𝑗𝑗=0,1,2,…,𝑘−1<𝜌_𝑝+1𝑔=𝜌_𝑝𝐴₀. (76) From Lemma 28??(ii), we have 𝜆_𝑝+1(𝑓−𝜑)=𝜆_𝑝+1(𝑓−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀).
(ii) Second, we prove that 𝜆_𝑝+1(𝑓^′−𝜑)=𝜆_𝑝+1(𝑓^′−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). Set 𝑔₁=𝑓^′−𝜑, then 𝜌_𝑝+1(𝑔₁)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). By Lemma 13, we get that 𝑔₁ satisfies the (15). Set 𝐹₁=𝜑^(𝑘)+𝑈¹_𝑘−1𝜑^(𝑘−1)+⋯+𝑈¹₀𝜑, where 𝑈¹_𝑗??(𝑗=0,1,…,𝑘−1) are stated as in Lemma 13. If 𝐹₁≡0, then, from Lemmas 20 and 21, we have 𝜌_𝑝+1(𝜑)⩾𝜌_𝑝(𝐴₀), a contradiction with 𝜌_𝑝+1(𝜑)<𝜌_𝑝(𝐴₀). Hence 𝐹₁≢0. By Lemma 28 (ii), we have 𝜆_𝑝+1𝑓^′−𝜑=𝜆_𝑝+1𝑓^′−𝜑=𝜌_𝑝+1𝑓=𝜌_𝑝𝐴₀. (77)
(iii) Now, we prove that 𝜆_𝑝+1(𝑓^′′−𝜑)=𝜆_𝑝+1(𝑓^′′−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). Set 𝑔₂=𝑓^′′−𝜑, then 𝜌_𝑝+1(𝑔₂)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). By Lemma 13, we get that 𝑔₂ satisfies (15). Set 𝐹₂=𝜑^(𝑘)+𝑈²_𝑘−1𝜑^(𝑘−1)+⋯+𝑈²₀𝜑, where 𝑈²_𝑗??(𝑗=0,1,…,𝑘−1) are stated as in Lemma 13. If 𝐹₂≡0, then, from Lemmas 20 and 21, we have 𝜌_𝑝+1(𝜑)⩾𝜌_𝑝(𝐴₀), a contradiction with 𝜌_𝑝+1(𝜑)<𝜌_𝑝(𝐴₀). Hence 𝐹₂≢0. By Lemma 28??(ii), we have 𝜆_𝑝+1𝑓^′′−𝜑=𝜆_𝑝+1𝑓^′′−𝜑=𝜌_𝑝+1𝑓=𝜌_𝑝𝐴₀. (78)
(iv) Set 𝑔₃=𝑓^′′′−𝜑, and then 𝜌_𝑝+1(𝑔₃)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). From Lemmas 13, 20, 21, and 28??(ii), using the same argument as in Case 1??(iii), we can get 𝜆_𝑝+1(𝑓^′′′−𝜑)=𝜆_𝑝+1(𝑓^′′′−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀).
(v) We prove that 𝜆_𝑝+1(𝑓^(𝑖)−𝜑)=𝜆_𝑝+1(𝑓^(𝑖)−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀),??𝑖>3,??𝑖∈ℕ. Set 𝑔_𝑖=𝑓^(𝑖)−𝜑, and then 𝜌_𝑝+1(𝑔_𝑖)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). By Lemma 13, we get that 𝑔_𝑖 satisfies (15). Set 𝐹_𝑖=𝜑^(𝑘)+𝑈^𝑖_𝑘−1𝜑^(𝑘−1)+⋯+𝑈^𝑖₀𝜑, where 𝑈^𝑖_𝑗??(𝑗=0,1,…,𝑘−1,𝑖>3,𝑖∈ℕ) are stated as in Lemma 13. If 𝐹_𝑖≡0, then from Lemmas 20 and 21, we have 𝜌_𝑝+1(𝜑)⩾𝜌_𝑝(𝐴₀), a contradiction with 𝜌_𝑝+1(𝜑)<𝜌_𝑝(𝐴₀). Hence 𝐹_𝑖≢0. By Lemma 28??(ii), we have 𝜆_𝑝+1(𝑓^(𝑖)−𝜑)=𝜆_𝑝+1(𝑓^(𝑖)−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀)??(𝑖>3,𝑖∈ℕ).
Case 2. Suppose that max{𝜌_𝑝(𝐴_𝑗)∶𝑗=1,2,…,𝑘−1}⩽𝜌_𝑝(𝐴₀)=𝜌(0<𝜌<∞) and max{𝜏_𝑝(𝐴_𝑗)∶𝜌_𝑝(𝐴_𝑗)=𝜌_𝑝(𝐴₀)}<𝜏_𝑝(𝐴₀)=𝜏(0<𝜏<∞).
(i) First, we prove that 𝜆_𝑝+1(𝑓−𝜑)=𝜆_𝑝+1(𝑓−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). Assume that 𝑓≢0 is a solution of (8). From Lemma 30, we have 𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). Set 𝑔=𝑓−𝜑. Since 𝜑≢0 is an entire function satisfying 𝜌_𝑝+1(𝜑)<𝜌_𝑝(𝐴₀), then 𝜌_𝑝+1(𝑔)=𝜌_𝑝+1(𝑓−𝜑)=𝜌_𝑝(𝐴₀) and 𝜆_𝑝+1(𝑔)=𝜆_𝑝+1(𝑓−𝜑),??𝜆_𝑝+1(𝑔)=𝜆_𝑝+1(𝑓−𝜑). By substituting 𝑓=𝑔+𝜑 into (8), we get that 𝑔 satisfies (75). Set 𝐹=𝜑^(𝑘)+𝐴_𝑘−1𝜑^(𝑘−1)+⋯+𝐴₀𝜑. If 𝐹≡0, then, from Lemma 30, we have 𝜌_𝑝+1(𝜑)=𝜌_𝑝(𝐴₀), which is a contradiction. Hence 𝐹≢0. From the assumptions of Theorem 7, we get max𝜌_𝑝+1(𝐹),𝜌_𝑝+1𝐴_𝑗𝑗=0,1,2,…,𝑘−1<𝜌_𝑝+1𝑔=𝜌_𝑝𝐴₀. (79) From Lemma 28 (ii), we have 𝜆_𝑝+1(𝑓−𝜑)=𝜆_𝑝+1(𝑓−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀).
(ii) Now, we prove that 𝜆_𝑝+1(𝑓^′−𝜑)=𝜆_𝑝+1(𝑓^′−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). Set 𝑔₁=𝑓^′−𝜑. Since 𝜑≢0 is an entire function satisfying 𝜌_𝑝+1(𝜑)<𝜌_𝑝(𝐴₀), then 𝜌_𝑝+1(𝑔₁)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). By Lemma 13, we get that 𝑔₁ satisfies the (15). If 𝐹₁≡0, then, from Lemmas 25 and 27, we have 𝜌_𝑝+1(𝜑)⩾𝜌_𝑝(𝐴₀), a contradiction with 𝜌_𝑝+1(𝜑)<𝜌_𝑝(𝐴₀). Hence 𝐹₁≢0. By Lemma 28 (ii), we have 𝜆_𝑝+1(𝑓^′−𝜑)=𝜆_𝑝+1(𝑓^′−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀). Similar to the arguments as in Case 1 (iii)–(v) and by using Lemmas 13, 25, and 27, we can get 𝜆_𝑝+1(𝑓^(𝑖)−𝜑)=𝜆_𝑝+1(𝑓^(𝑖)−𝜑)=𝜌_𝑝+1(𝑓)=𝜌_𝑝(𝐴₀),??𝑖>1, 𝑖∈ℕ. Thus, the proof of Theorem 7 is completed.

Proof of Theorem 8. Since 𝐴_𝑗(𝑧)??(𝑗=1,2,…,𝑘−1) are polynomials and 𝐴₀ is a transcendental entire function with 0<𝜌_𝑝(𝐴₀)<∞; that is, 𝐴_𝑗(𝑧)??(𝑗=0,1,2,…,𝑘−1) satisfy the conditions of Theorem 7 (i). By using the same argument as in Theorem 7 and Lemma 28 (i), we can get the conclusions of Theorem 8 easily.

Proof of Theorem 9. Assume that 𝑓≢0 is a meromorphic solution of (8). By (8) we get that the poles of 𝑓(𝑧) can only occur at the poles of 𝐴₀,𝐴₁,…,𝐴_𝑘−1. Note that the multiplicities of poles of 𝑓 are uniformly bounded, and thus we have 𝑁𝑟,𝑓⩽𝑀₁𝑁𝑟,𝑓⩽𝑀₁^𝑘−1_𝑗=0𝑁𝑟,𝐴_𝑗⩽𝑀max𝑁𝑟,𝐴_𝑗∶𝑗=0,1,…,𝑘−1, (80) where 𝑀₁ and 𝑀 are some suitable positive constants. This gives 𝑇𝑁𝑟,𝑓=𝑚𝑟,𝑓+𝑂max𝑟,𝐴_𝑗∶𝑗=0,1,…,𝑘−1. (81) Applying now (81) with Lemma 17, we obtain 𝜌_𝑝+1𝑓⩽𝜌_𝑝𝐴₀. (82) According to the conditions of Theorem 9, we can easily get the conclusions of Theorem 9 by using the similar argument as in Theorem 7, the estimation (82), and Lemmas 26 and 28 (ii).

Acknowledgments

This research is supported by ATRST (Agence Thématique de Recherche en Sciences et Technologies) and University of Mostaganem (UMAB) (PNR Project Code 8/u27/3144).