A Note on the Growth of Meromorphic Functions with a Radially Distributed Value

Abstract

This note is to investigate the growth of transcendental meromorphic functions with radially distributed values. We generalize a more recent result of Chen et al. (2011). The paper is closely related to some previous results due to Fang and Zalcman (2008), and Xu et al. (2009).

1. Introduction and Results

Let 𝑓: ℂ→ℂ be a meromorphic function, where ℂ is the whole complex plane and ℂ=ℂ∪{∞}. We shall use the basic results and notations of Nevanlinna's value distribution theory of meromorphic functions (see [1–3]), such as 𝑇(𝑟,𝑓),  𝑁(𝑟,𝑓), and 𝑚(𝑟,𝑓). Meantime, the Nevanlinna's deficiency 𝛿(𝑎,𝑓) of 𝑓(𝑧) with respect to 𝑎∈ℂ is defined by 𝛿𝑎,𝑓=liminf_𝑟→∞𝑚𝑟,1/𝑓−𝑎𝑇𝑟,𝑓=1−limsup_𝑟→∞𝑁𝑟,1/𝑓−𝑎𝑇𝑟,𝑓 (1.1) and 𝛿(∞,𝑓) is obtained by the above formula with 𝑚(𝑟,𝑓) in place of 𝑚(𝑟,1/(𝑓−𝑎)), and 𝑁(𝑟,𝑓) in place of 𝑁(𝑟,1/(𝑓−𝑎)), respectively. The lower order 𝜇 and order 𝜆 are defined in turn as follows: 𝜇∶=𝜇𝑓=liminf_𝑟→∞log𝑇𝑟,𝑓,𝑓log𝑟𝜆∶=𝜆=limsup_𝑟→∞log𝑇𝑟,𝑓.log𝑟 (1.2)

For an unbounded subset 𝑋 of ℂ and 𝑎∈ℂ, we denote by 𝑛(𝑟,𝑋,𝑓=𝑎) the number of the zeros of 𝑓−𝑎 counting multiplicities in 𝑋∩{𝑧∶|𝑧|≤𝑟}.

In 1992, Yang [4], cf. [5] posed the following interesting conjecture.

Yang's Conjecture
Let 𝑃 be a property (or a set of properties) such that any entire (or meromorphic in ℂ) function satisfying 𝑃 must be a constant. Suppose that 𝑓 is an entire (or meromorphic in ℂ) function of finite lower order 𝜇, and that 𝐿_𝑗∶arg𝑧=𝜃_𝑗(𝑗=1,2,…,𝑞; 0≤𝜃₁<𝜃₂<⋯<𝜃_𝑞<2𝜋; 𝜃_𝑞+1=𝜃₁+2𝜋) are a finite number of rays issuing from the origin. If 𝑓 satisfies 𝑃 in ℂ⧵(⋃^𝑞_𝑗=1𝐿_𝑗), then the order 𝜆 of 𝑓 has the following estimation: 𝜆≤max_{1≤𝑗≤𝑞}𝜋𝜃_𝑗+1−𝜃_𝑗,𝜃_𝑞+1=𝜃₁+2𝜋. (1.3)
Yang's conjecture implies that if meromorphic functions satisfy certain properties in the vicinities of a finite number of rays, then the growth of meromorphic functions will be restricted.

A well-known result of Clunie [6], cf. [7] is that an entire function 𝑓 which satisfies 𝑓(𝑧)𝑓^′(𝑧)≠1 in ℂ must be constant. In [5], L. Yang and C.-C. Yang chose the property 𝑃 as 𝑓(𝑧)𝑓^′(𝑧)≠1 and verified the above conjecture. On the other hand, Fang and Zalcman [8] considered the value distribution of 𝑓+𝑎(𝑓^′)^𝑛 in ℂ, where 𝑎 is a nonzero finite complex number and 𝑛≥2 is a positive integer. Actually, they [8] gave an affirmative answer to a question suggested by Ye [9]. Later on, Xu et al. [10] further generalized 𝑓+𝑎(𝑓^′)^𝑛 to 𝑓+𝑎(𝑓^(𝑘))^𝑛 in ℂ for a positive integer 𝑘 with 𝑛≥𝑘+1 and investigated its value distribution.

More recently, Chen et al. [11] chose another property 𝑃 as 𝑓+𝑎(𝑓^′)^𝑛 to continue to study Yang's Conjecture and proved the following results.

Theorem A. Let 𝑓(𝑧) be a transcendental meromorphic function with 𝛿(∞,𝑓^′)>0 in ℂ and let 𝐿_𝑗∶arg𝑧=𝜃_𝑗(𝑗=1,2,…,𝑞) be a finite number of rays issued from the origin such that −𝜋≤𝜃₁<𝜃₂<⋯<𝜃_𝑞<𝜋,𝜃_𝑞+1=𝜃₁+2𝜋 (1.4) with 𝜔=max{𝜋/(𝜃_𝑗+1−𝜃_𝑗)∶1≤𝑗≤𝑞}. Set 𝑌=ℂ⧵(⋃^𝑞_𝑗=1𝐿_𝑗). If 𝑓(𝑧) satisfies limsup_𝑟→∞log𝑛𝑟,𝑌,𝑓+𝑎𝑓^′𝑛=𝑏log𝑟≤𝜌 (1.5) with a positive number 𝜌, finite complex numbers 𝑎≠0 and 𝑏, for any positive integer 𝑛≥2, then the order 𝜆 of 𝑓(𝑧) has the estimation 𝜆≤max{𝜔,𝜌}.

Theorem B. Let 𝑓(𝑧) be a transcendental meromorphic function of finite lower order 𝜇 with 𝛿(∞,𝑓^′)>0 in ℂ. For 𝑞 pairs of real numbers {𝛼_𝑗,𝛽_𝑗} such that −𝜋≤𝛼₁<𝛽₁≤𝛼₂<𝛽₂≤⋯≤𝛼_𝑞<𝛽_𝑞≤𝜋 (1.6) with 𝜔=max{𝜋/(𝛽_𝑗−𝛼_𝑗)∶1≤𝑗≤𝑞}, suppose that limsup_𝑟→∞log𝑛𝑟,𝑌,𝑓+𝑎𝑓^′𝑛=𝑏log𝑟≤𝜌 (1.7) with a positive number 𝜌, finite complex numbers 𝑎≠0 and 𝑏, and 𝑌=⋃^𝑞_𝑗=1{𝑧∶𝛼_𝑗≤arg𝑧≤𝛽_𝑗}, for any positive integer 𝑛≥2, and that ^𝑞_𝑗=1𝛼_𝑗+1−𝛽_𝑗<4𝜎arcsin𝛿∞,𝑓^′2,𝛼_𝑞+1=𝛼₁+2𝜋 (1.8) with 𝜎=max{𝜔,𝜌,𝜇}. Then the order 𝜆 of 𝑓(𝑧) has the estimation 𝜆≤max{𝜔,𝜌}.

Now there arises a natural question.

Question 1. What can be said if 𝑓^′ in Theorems A and B is replaced by the 𝑘th derivative 𝑓^(𝑘)?

In this paper, we will prove the following results which generalize Theorems A and B.

Theorem 1.1. Let 𝑓(𝑧) be a transcendental meromorphic function with 𝛿(∞,𝑓^(𝑘))>0 for a positive integer 𝑘 in ℂ and let 𝐿_𝑗∶arg𝑧=𝜃_𝑗(𝑗=1,2,…,𝑞) be a finite number of rays issued from the origin such that −𝜋≤𝜃₁<𝜃₂<⋯<𝜃_𝑞<𝜋,𝜃_𝑞+1=𝜃₁+2𝜋 (1.9) with 𝜔=max{𝜋/(𝜃_𝑗+1−𝜃_𝑗)∶1≤𝑗≤𝑞}. Set 𝑌=ℂ⧵(⋃^𝑞_𝑗=1𝐿_𝑗). If 𝑓(𝑧) satisfies limsup_𝑟→∞log𝑛𝑟,𝑌,𝑓+𝑎𝑓^(𝑘)𝑛=𝑏log𝑟≤𝜌 (1.10) with a positive number 𝜌, finite complex numbers 𝑎≠0 and 𝑏, for any positive integer 𝑛≥𝑘+1, then the order 𝜆 of 𝑓(𝑧) has the estimation 𝜆≤max{𝜔,𝜌}.

Remark 1.2. Let 𝑘=1. Then by Theorem 1.1 we get Theorem A.

Corollary 1.3. Let 𝑓(𝑧) be a transcendental entire function, let the notations 𝜃_𝑗 (𝑗=1,2,…,𝑞+1), 𝜔, and 𝑌 be defined as in Theorem 1.1, and suppose that the function 𝑓(𝑧) fulfills the same condition (1.10) as in Theorem 1.1. Then the order 𝜆 of 𝑓(𝑧) has the estimation 𝜆≤max{𝜔,𝜌}.

Theorem 1.4. Let 𝑓(𝑧) be a transcendental meromorphic function of finite lower order 𝜇 with 𝛿(∞,𝑓^(𝑘))>0 for a positive integer 𝑘 in ℂ. For 𝑞 pairs of real numbers {𝛼_𝑗,𝛽_𝑗} such that −𝜋≤𝛼₁<𝛽₁≤𝛼₂<𝛽₂≤⋯≤𝛼_𝑞<𝛽_𝑞≤𝜋 (1.11) with 𝜔=max{𝜋/(𝛽_𝑗−𝛼_𝑗)∶1≤𝑗≤𝑞}, suppose that limsup_𝑟→∞log𝑛𝑟,𝑌,𝑓+𝑎𝑓^(𝑘)𝑛=𝑏log𝑟≤𝜌 (1.12) with a positive number 𝜌, finite complex numbers 𝑎≠0 and 𝑏, and 𝑌=⋃^𝑞_𝑗=1{𝑧∶𝛼_𝑗≤arg𝑧≤𝛽_𝑗}, for any positive integer 𝑛≥𝑘+1, and that ^𝑞_𝑗=1𝛼_𝑗+1−𝛽_𝑗<4𝜎arcsin𝛿∞,𝑓^(𝑘)2,𝛼_𝑞+1=𝛼₁+2𝜋 (1.13) with 𝜎=max{𝜔,𝜌,𝜇}. Then the order 𝜆 of 𝑓(𝑧) has the estimation 𝜆≤max{𝜔,𝜌}.

Remark 1.5. Let 𝑘=1. Then by Theorem 1.4 we get Theorem B.

Corollary 1.6. Let 𝑓(𝑧) be a transcendental entire function of finite lower order 𝜇, let the notations 𝛼_𝑗, 𝛽_𝑗 (𝑗=1,2,…,𝑞), 𝜔, 𝜎, and 𝑌 be defined as in Theorem 1.4, and suppose that the function 𝑓(𝑧) fulfills the same conditions (1.12) and (1.13) as in Theorem 1.4. Then the order 𝜆 of 𝑓(𝑧) has the estimation 𝜆≤max{𝜔,𝜌}.

In order to prove our results, we require the Nevanlinna theory of meromorphic functions in an angular domain. For the sake of convenience, we recall some notations and definitions. Let 𝑓 be a meromorphic function on the angular domain Ω(𝛼,𝛽)={𝑧∶𝛼≤arg𝑧≤𝛽}, where 0<𝛽−𝛼≤2𝜋. Nevanlinna et al. [12, 13] introduced the following notations: 𝐴_𝛼,𝛽=𝜔𝑟,𝑓𝜋^𝑟₁1𝑡^𝜔−𝑡^𝜔𝑟^2𝜔log⁺𝑓𝑡𝑒^𝑖𝛼+log⁺𝑓𝑡𝑒^𝑖𝛽𝑑𝑡𝑡,𝐵_𝛼,𝛽𝑟,𝑓=2𝜔𝜋𝑟^𝜔𝛽_𝛼log⁺𝑓𝑟𝑒^𝑖𝜃()𝐶sin𝜔𝜃−𝛼𝑑𝜃,_𝛼,𝛽𝑟,𝑓=2_{𝑏1<𝑚<𝑟}1𝑏_𝑚^𝜔−𝑏_𝑚^𝜔𝑟^2𝜔sin𝜔𝜃_𝑚−𝛼, (1.14) where 𝜔=𝜋/(𝛽−𝛼) and 𝑏_𝑚=|𝑏_𝑚|𝑒^𝑖𝜃𝑚 are the poles of 𝑓 in Ω(𝛼,𝛽) appearing according to their multiplicities. The function 𝐶_𝛼,𝛽(𝑟,𝑓) is called the angular counting function (counting multiplicities) of the poles of 𝑓 in Ω(𝛼,𝛽), and 𝐶_𝛼,𝛽(𝑟,𝑓) is called the angular reduced counting function (ignoring multiplicities) of the poles of 𝑓 in Ω(𝛼,𝛽). Further, Nevanlinna's angular characteristic function 𝑆_𝛼,𝛽(𝑟,𝑓) is defined as follows: 𝑆_𝛼,𝛽𝑟,𝑓=𝐴_𝛼,𝛽𝑟,𝑓+𝐵_𝛼,𝛽𝑟,𝑓+𝐶_𝛼,𝛽𝑟,𝑓. (1.15)

Throughout the paper, we denote by 𝑅(𝑟,∗) a quantity satisfying 𝑅(𝑟,∗)=𝑂log(𝑟𝑇(𝑟,∗)),∀𝑟∉𝐸, (1.16) where 𝐸 denotes a set of positive real numbers with finite linear measure. It is not necessarily the same for every occurrence in the context.

2. Some Lemmas

In this section we present some lemmas which will be needed in the sequel.

Lemma 2.1 (see [5, 12–14]). Let 𝑓 be meromorphic in ℂ. Then in Ω(𝛼,𝛽) for an arbitrary finite complex number 𝑎, we have 𝑆_𝛼,𝛽1𝑟,𝑓−𝑎=𝑆_𝛼,𝛽𝑟,𝑓+𝑂(1), (2.1) and for each positive integer 𝑘, we have 𝐴_𝛼,𝛽𝑓𝑟,^(𝑘)𝑓+𝐵_𝛼,𝛽𝑓𝑟,^(𝑘)𝑓=𝑅𝑟,𝑓. (2.2)

Lemma 2.2 (see [10]). Let 𝑛≥2 and 𝑘 be positive integers, let 𝑎 be a nonzero finite complex number, and let 𝑝(𝑧) be a polynomial. Then the solution of the differential equation 𝑎[𝜔^(𝑘)(𝑧)]^𝑛+𝜔(𝑧)=𝑝(𝑧) must be polynomial.

Lemma 2.3 (see [1, Theorem 3.1]; [1, page 33, (2.1)]). Let 𝑓 be meromorphic in ℂ, let 𝑎₁,𝑎₂,…,𝑎_𝑞, where 𝑞≥2, be distinct complex numbers, 𝛿>0, and suppose that |𝑎_𝜇−𝑎_𝜈|≥𝛿 for 1≤𝜇<𝜈≤𝑞. Then, for each positive integer 𝑘, we have 𝑇𝑟,𝑓^(𝑘)≤𝑇𝑟,𝑓+𝑘𝑁,𝑚𝑟,𝑓+𝑂log𝑟𝑇𝑟,𝑓𝑟,^𝑞_𝜈=11𝑓−𝑎_𝜈≥^𝑞_𝜈=1𝑚1𝑟,𝑓−𝑎_𝜈−𝑞log⁺3𝑞𝛿−log2,∀𝑟∉𝐸. (2.3) Next we slightly modify the proof of Lemma 2.4 in [10] to give the following key lemma, which is an important generalization of Lemma 3 in [11].

Lemma 2.4. Let 𝑓 be transcendental meromorphic in ℂ, let 𝑎≠0 and 𝑏 be finite complex numbers, and let 𝑛 and 𝑘 be positive integers with 𝑛≥𝑘+1. Then in Ω(𝛼,𝛽), 𝐵_𝛼,𝛽𝑟,𝑓^(𝑘)≤(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛.−𝑏+𝑅𝑟,𝑓 (2.4)

Proof. Put 𝑔=𝑓+𝑎𝑓^(𝑘)𝑛𝑔−𝑏,𝜙=^(𝑘)𝑔. (2.5)(2.6) Then 𝜙≢0, for otherwise 𝑔 would be a polynomial of degree at most 𝑘−1. This and (2.5) together with Lemma 2.2 imply that 𝑓 must be a polynomial, a contradiction. By the Nevanlinna's basic reasoning, Lemmas 2.1, and 2.3, (2.5), and (2.6), we have 𝐴_𝛼,𝛽+𝐵_𝛼,𝛽𝑟,𝜙=𝑅𝑟,𝑔=𝑅𝑟,𝑓. (2.7)
Now a simple calculation for (2.5) shows that 𝑔^(𝑘)=𝑓^(𝑘)+𝑎𝑓^{(𝑘)𝑛(𝑘)}=𝑓^(𝑘)1+𝑄𝑓^(𝑘), (2.8) where 𝑄(𝑓^(𝑘)) is a homogeneous differential polynomial in 𝑓^(𝑘) of degree 𝑛−1 and of the form 𝑄𝑓^(𝑘)=𝑎𝑓^{(𝑘)𝑛−𝑘−1}×𝑛!𝑓^(𝑘+1)𝑘()!+𝑘()𝑛−𝑘𝑘−1𝑛!()!𝑓2!𝑛−𝑘+1^(𝑘)𝑓^{(𝑘+1)𝑘−2}𝑓^(𝑘+2)+⋯+𝑛𝑓^{(𝑘)𝑘−1}𝑓^(2𝑘). (2.9) Then by (2.5), (2.6), and (2.8), we get 𝑓^(𝑘)1+𝑄𝑓^(𝑘)=𝜙𝑓+𝑎𝑓^(𝑘)𝑛−𝑏. (2.10) From (2.5)–(2.10), and Lemma 2.1, it thus follows that 𝐶_𝛼,𝛽1𝑟,𝑓^(𝑘)+𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)+1≤𝐶_𝛼,𝛽1𝑟,𝜙+𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛−𝑏≤𝐶_𝛼,𝛽𝑟,𝜙+𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛−𝑏+𝑅𝑟,𝑓≤𝑘𝐶_𝛼,𝛽𝑟,𝑓+(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛≤𝑘−𝑏+𝑅𝑟,𝑓𝐶𝑘+1^𝛼,𝛽𝑟,𝑓^(𝑘)+(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛.−𝑏+𝑅𝑟,𝑓 (2.11)
On the other hand, by the Nevanlinna's basic reasoning, Lemmas 2.1 and 2.3, we deduce that 𝐴_𝛼,𝛽+𝐵_𝛼,𝛽1𝑟,𝑓^{(𝑘)𝑛−1}+𝐴_𝛼,𝛽+𝐵_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)≤𝐴+1_𝛼,𝛽+𝐵_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)+𝐴_𝛼,𝛽+𝐵_𝛼,𝛽𝑄𝑓𝑟,^(𝑘)𝑓^{(𝑘)𝑛−1}+𝐴_𝛼,𝛽+𝐵_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)≤𝐴+1_𝛼,𝛽+𝐵_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)+1𝑄𝑓^(𝑘)≤𝐴+1+𝑅𝑟,𝑓_𝛼,𝛽+𝐵_𝛼,𝛽𝑄𝑓𝑟,^(𝑘)′𝑄𝑓^(𝑘)+𝑄𝑓^(𝑘)+1^′𝑄𝑓^(𝑘)+𝐴+1_𝛼,𝛽+𝐵_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)′+𝑅𝑟,𝑓≤𝑆_𝛼,𝛽𝑟,𝑄𝑓^(𝑘)′−𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)′+𝑅𝑟,𝑓≤𝑆_𝛼,𝛽𝑟,𝑄𝑓^(𝑘)+𝐶_𝛼,𝛽𝑟,𝑓−𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)′+𝑅𝑟,𝑓≤𝑆_𝛼,𝛽𝑟,𝑄𝑓^(𝑘)+1𝐶𝑘+1^𝛼,𝛽𝑟,𝑓^(𝑘)−𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)′+𝑅𝑟,𝑓. (2.12) This, together with Lemma 2.1, yields (𝑛−1)𝑆_𝛼,𝛽𝑟,𝑓^(𝑘)≤1𝐶𝑘+1^𝛼,𝛽𝑟,𝑓^(𝑘)+(𝑛−1)𝐶_𝛼,𝛽1𝑟,𝑓^(𝑘)+𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)+1−𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)′+𝑅𝑟,𝑓. (2.13)
Next we divide into two cases.
Case  1 (𝑛≥𝑘+2). Suppose that 𝑧₀ is a zero of 𝑓^(𝑘) of multiplicity 𝑚. Then we can know that 𝑧₀ is a zero of (𝑄(𝑓^(𝑘)))^′ of multiplicity at least (𝑛−𝑘−1)𝑚+𝑘(𝑚−1)−1=(𝑛−1)𝑚−𝑘−1. Thus, we have (𝑛−1)𝐶_𝛼,𝛽1𝑟,𝑓^(𝑘)+𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)+1−𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)′=𝐶_𝛼,𝛽1𝑟,𝑓^{(𝑘)𝑛−1}+𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)+1−𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘)′≤(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓^(𝑘)+𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘).+1 (2.14) Substituting this into (2.13) gives (𝑛−1)𝑆_𝛼,𝛽𝑟,𝑓^(𝑘)≤1𝐶𝑘+1^𝛼,𝛽𝑟,𝑓^(𝑘)+(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓^(𝑘)+𝐶_𝛼,𝛽1𝑟,𝑄𝑓^(𝑘).+1+𝑅𝑟,𝑓 (2.15) From this and (2.11) it follows that (𝑛−1)𝑆_𝛼,𝛽𝑟,𝑓^(𝑘)≤𝑘²+𝑘+1𝐶𝑘+1^𝛼,𝛽𝑟,𝑓^(𝑘)+(𝑘+1)²𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛,−𝑏+𝑅𝑟,𝑓 (2.16) implying that 𝑆_𝛼,𝛽𝑟,𝑓^(𝑘)≤𝑘²+𝑘+1()𝐶𝑛−1)(𝑘+1^𝛼,𝛽𝑟,𝑓^(𝑘)+(𝑘+1)²𝐶𝑛−1^𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛.−𝑏+𝑅𝑟,𝑓 (2.17) Thereby, noting 𝑛≥𝑘+2, we get 𝑆_𝛼,𝛽𝑟,𝑓^(𝑘)≤𝑘²+𝑘+1(𝑘+1)²𝐶_𝛼,𝛽𝑟,𝑓^(𝑘)+(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛−𝑏+𝑅𝑟,𝑓≤𝐶_𝛼,𝛽𝑟,𝑓^(𝑘)+(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛,−𝑏+𝑅𝑟,𝑓 (2.18) so that 𝐵_𝛼,𝛽𝑟,𝑓^(𝑘)≤𝐴_𝛼,𝛽𝑟,𝑓^(𝑘)+𝐵_𝛼,𝛽𝑟,𝑓^(𝑘)≤(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛.−𝑏+𝑅𝑟,𝑓 (2.19) This is the desired result.
Case  2 (𝑛=𝑘+1). Then, by (2.11) and (2.13), we have 𝑘𝑆_𝛼,𝛽𝑟,𝑓^(𝑘)≤𝑘²+1𝐶𝑘+1^𝛼,𝛽𝑟,𝑓^(𝑘)+𝑘(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛,−𝑏+𝑅𝑟,𝑓 (2.20) which leads to 𝑆_𝛼,𝛽𝑟,𝑓^(𝑘)≤𝑘²+1𝑘()𝐶𝑘+1^𝛼,𝛽𝑟,𝑓^(𝑘)+(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛−𝑏+𝑅𝑟,𝑓≤𝐶_𝛼,𝛽𝑟,𝑓^(𝑘)+(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛.−𝑏+𝑅𝑟,𝑓 (2.21) This yields 𝐵_𝛼,𝛽𝑟,𝑓^(𝑘)≤𝐴_𝛼,𝛽𝑟,𝑓^(𝑘)+𝐵_𝛼,𝛽𝑟,𝑓^(𝑘)≤(𝑘+1)𝐶_𝛼,𝛽1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛,−𝑏+𝑅𝑟,𝑓 (2.22) obtaining the desired result.
This completes the proof of Lemma 2.4.

The following auxiliary results regarding Pólya peaks and the spread relation are necessary in the proofs of our theorems.

Lemma 2.5 (see [14–16]). Let 𝑓 be a transcendental meromorphic function of finite lower order 𝜇 and order 𝜆(0<𝜆≤∞) in ℂ. Then, for an arbitrary positive number 𝜎 satisfying 𝜇≤𝜎≤𝜆 and any set 𝐸 of finite linear measure, there exist Pólya peaks {𝑟_𝑛} satisfying the following:(i)𝑟_𝑛∉𝐸, lim_𝑛→∞(𝑟_𝑛/𝑛)=∞;(ii)liminf_𝑛→∞(log𝑇(𝑟_𝑛,𝑓))/log𝑟_𝑛≥𝜎;(iii)𝑇(𝑡,𝑓)<(1+𝑜(1))(𝑡/𝑟_𝑛)^𝜎𝑇(𝑟_𝑛,𝑓),𝑡∈[𝑟_𝑛/𝑛,𝑛𝑟_𝑛].
A sequence of {𝑟_𝑛} satisfying (i), (ii), and (iii) in Lemma 2.5 is called a Pólya peak of order 𝜎 of 𝑓 outside 𝐸. Given a positive function Λ=Λ(𝑟) on (0,∞) with Λ→0 as 𝑟→∞, we define 𝐷_Λ(𝑟,𝑎)=𝜃∈[−𝜋,𝜋)∣log⁺1𝑓𝑟𝑒^𝑖𝜃(𝑟)𝑇,𝐷−𝑎>Λ𝑟,𝑓_Λ(𝑟,∞)=𝜃∈[−𝜋,𝜋)∣log⁺𝑓𝑟𝑒^𝑖𝜃>Λ(𝑟)𝑇𝑟,𝑓. (2.23)

Lemma 2.6 (see [17]). Let 𝑓 be a transcendental meromorphic function of finite lower order 𝜇 and order 𝜆(0<𝜆≤∞) in ℂ. Suppose that 𝛿=𝛿(𝑎,𝑓)>0 for some 𝑎∈ℂ. Then for an arbitrary Pólya peak {𝑟_𝑛} of order 𝜎(𝜇≤𝜎≤𝜆) and an arbitrary positive function Λ=Λ(𝑟) with Λ→0 as 𝑟→∞, we have liminf_𝑛→∞meas𝐷_Λ𝑟_𝑛4,𝑎≥min2𝜋,𝜎arcsin𝛿2. (2.24)
Now a more precise estimation of 𝑇(𝑟,𝑓) in terms of 𝑇(𝑟,𝑓^(𝑘)) is introduced as follows.

Lemma 2.7 (see [18]). Let 𝑓 be transcendental meromorphic in ℂ. Then, for a positive integer 𝑘 and a real number 𝜏>1, we have 𝑇𝑟,𝑓<𝐾_𝜏,𝑘𝑇𝜏𝑟,𝑓^(𝑘)+log(𝜏𝑟)+𝑂(1), (2.25) where 𝐾_𝜏,𝑘 is a positive number depending on only 𝜏 and 𝑘.

At last, we state the following results due to Edrei, Hayman, and Miles, respectively.

Lemma 2.8 (see [19]). Let 𝑓 be a transcendental meromorphic function with 𝛿=𝛿(∞,𝑓)>0 in ℂ. Then, given 𝜀>0, we have >1meas𝐸𝑟,𝑓𝑇^𝜀𝑟,𝑓log𝑟^1+𝜀,∀𝑟∉𝐹, (2.26) where 𝐸𝑟,𝑓=𝜃∈[−𝜋,𝜋)∶log⁺𝑓𝑟𝑒^𝑖𝜃>𝛿4𝑇𝑟,𝑓 (2.27) and 𝐹 is a set of positive real numbers with finite logarithmic measure (i.e., ∫_𝐹(𝑑𝑡/𝑡)<∞) depending on 𝜀 only.

Lemma 2.9 (see [20]). Let 𝑓 be a transcendental meromorphic function in ℂ. Then for each 𝐾>1 there exists a set 𝑀(𝐾) of the lower logarithmic density at least 𝑑(𝐾)=1−(2𝑒^𝐾−1−1)⁻¹>0, that is, logdens𝑀(𝐾)∶=liminf_𝑟→∞1log𝑟_{𝑀(𝐾)∩[1,𝑟]}𝑑𝑡𝑡≥𝑑(𝐾), (2.28) such that, for every positive integer 𝑝, we have limsup_{𝑟→∞,𝑟∈𝑀(𝐾)}𝑇𝑟,𝑓𝑇𝑟,𝑓^(𝑝)≤3𝑒𝐾. (2.29)

3. Proofs of Theorems 1.1 and 1.4

In this section, we state the detailed proofs of Theorems 1.1 and 1.4 by using the method in [14]. To begin with, we give the proof of Theorem 1.4. Finally the proof of Theorem 1.1 can be derived from Theorem 1.4.

3.1. Proof of Theorem 1.4

Assume on the contrary that Theorem 1.4 does not hold. Then 𝜆(𝑓)>max{𝜔,𝜌}. Now by (1.12), we have 𝑛𝑟,Ω𝛼_𝑗,𝛽_𝑗,𝑓+𝑎𝑓^(𝑘)𝑛=𝑏≤𝑟^𝜌+𝜀,𝑗=1,2,…,𝑞 (3.1) for arbitrarily small 𝜀>0 and sufficiently large 𝑟≥𝑟₀. Let 𝜉_𝑚 be the zeros of 𝑓+𝑎(𝑓^(𝑘))^𝑛−𝑏 on Ω(𝛼_𝑗,𝛽_𝑗) appearing according to their multiplicities, and set 𝜔_𝑗=𝜋/(𝛽_𝑗−𝛼_𝑗). By the definition of 𝐶_𝛼,𝛽(𝑟,∗), we deduce that 𝐶_{𝛼𝑗,𝛽𝑗}1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛−𝑏≤2_{𝜉1<𝑚<𝑟}1𝜉_𝑚^𝜔𝑗=2^𝑟₁𝑑𝑛𝑡,Ω𝛼_𝑗,𝛽_𝑗,𝑓+𝑎𝑓^(𝑘)𝑛=𝑏𝑡^𝜔𝑗𝑛≤2𝑟,Ω𝛼_𝑗,𝛽_𝑗,𝑓+𝑎𝑓^(𝑘)𝑛=𝑏𝑟^𝜔𝑗+2𝜔_𝑗^𝑟₁𝑛𝑡,Ω𝛼_𝑗,𝛽_𝑗,𝑓+𝑎𝑓^(𝑘)𝑛=𝑏𝑡^𝜔𝑗+1𝑑𝑡≤2𝑟^{𝜌+𝜀−𝜔𝑗}+𝑂(1)+2𝜔_𝑗^𝑟_𝑟0𝑡^𝜌+𝜀𝑡^𝜔𝑗+1𝑑𝑡≤𝐾_𝑗,𝜀𝑟^{𝜌+𝜀−𝜔𝑗}+𝑂log𝑟, (3.2) where 𝐾_𝑗,𝜀 is a positive number depending on only 𝑗 and 𝜀, which is not necessarily the same for every occurrence in the context. From Lemma 2.4, we have 𝐵_{𝛼𝑗,𝛽𝑗}𝑟,𝑓^(𝑘)≤(𝑘+1)𝐶_{𝛼𝑗,𝛽𝑗}1𝑟,𝑓+𝑎𝑓^(𝑘)𝑛.−𝑏+𝑅𝑟,𝑓 (3.3) Thus, it follows by (3.2) and (3.3) that 𝐵_{𝛼𝑗,𝛽𝑗}𝑟,𝑓^(𝑘)≤𝐾_𝑗,𝜀𝑟^{𝜌+𝜀−𝜔𝑗}+𝑂log𝑟𝑇𝑟,𝑓,∀𝑟∉𝐸, (3.4) where the exceptional set 𝐸 associated with 𝑅(𝑟,𝑓) is of at most finite linear measure.

Now we discuss two cases separately.

Case 1 (𝜆(𝑓)>𝜇(𝑓)). Then by the assumption 𝜎=max{𝜔,𝜌,𝜇} and 𝜆(𝑓)>max{𝜔,𝜌}, we have 𝜆(𝑓^(𝑘))=𝜆(𝑓)>𝜎≥𝜇(𝑓)=𝜇(𝑓^(𝑘)). Now from (1.13), we can find a real number 𝜀>0 such that ^𝑞_𝑗=1𝛼_𝑗+1−𝛽_𝑗4+2𝜀+2𝜀<𝜎+2𝜀arcsin𝛿∞,𝑓^(𝑘)2,𝛼_𝑞+1=𝛼₁𝜆𝑓+2𝜋,_(𝑘)>𝜎+2𝜀>𝜇𝑓^(𝑘). (3.5)(3.6) Applying Lemma 2.5 to 𝑓^(𝑘) gives the existence of the Pólya peak {𝑟_𝑛} of order 𝜎+2𝜀 of 𝑓^(𝑘) outside the set 𝐸. Then, noting that 𝜎+2𝜀>𝜔_𝑗≥1/2 and 0<𝛿(∞,𝑓^(𝑘))≤1, by applying Lemma 2.6 to the Pólya peak {𝑟_𝑛}, for sufficiently large 𝑛 we have meas𝐷_Λ𝑟_𝑛>4,∞𝜎+2𝜀arcsin𝛿∞,𝑓^(𝑘)2−𝜀. (3.7) Without loss of generality, we can assume that (3.7) holds for all the 𝑛. Set 𝐾_𝑛∶=meas𝐷_Λ𝑟_𝑛,∞∩^𝑞_𝑗=1𝛼_𝑗+𝜀,𝛽_𝑗−𝜀. (3.8) It then follows from (3.5) and (3.7) that 𝐾_𝑛≥meas𝐷_Λ𝑟_𝑛,∞−meas[−𝜋,𝜋)⧵^𝑞_𝑗=1𝛼_𝑗+𝜀,𝛽_𝑗𝐷−𝜀=meas_Λ𝑟_𝑛,∞−meas^𝑞_𝑗=1𝛽_𝑗−𝜀,𝛼_𝑗+1𝐷+𝜀=meas_Λ𝑟_𝑛,∞−^𝑞_𝑗=1𝛼_𝑗+1−𝛽_𝑗+2𝜀>𝜀>0. (3.9) By (3.9), it is easy to see that there exists a 𝑗₀ such that, for infinitely many 𝑛, we have meas𝐷_Λ𝑟_𝑛,∞∩𝛼_𝑗0+𝜀,𝛽_𝑗0≥𝐾−𝜀^𝑛𝑞>𝜀𝑞. (3.10) Without loss of generality, we can assume that (3.10) holds for all the 𝑛. Set 𝐸_𝑛=𝐷_Λ(𝑟_𝑛,∞)∩(𝛼_𝑗0+𝜀,𝛽_𝑗0−𝜀) and Λ(𝑟)=[log𝑟]⁻¹. From the definition of 𝐷_Λ(𝑟_𝑛,∞), we deduce that ^𝛽𝑗0_𝛼^−𝜀_𝑗0+𝜀log⁺𝑓^(𝑘)𝑟_𝑛𝑒^𝑖𝜃𝑑𝜃≥_𝐸𝑛log⁺𝑓^(𝑘)𝑟_𝑛𝑒^𝑖𝜃≥𝜀𝑑𝜃𝑞𝑇𝑟_𝑛,𝑓^(𝑘)log𝑟_𝑛. (3.11)
On the other hand, by the definition of 𝐵_𝛼,𝛽(𝑟,∗) and (3.4), it follows that ^𝛽𝑗0_𝛼^−𝜀_𝑗0+𝜀log⁺𝑓^(𝑘)𝑟_𝑛𝑒^𝑖𝜃𝑑𝜃≤𝜋𝑟^𝜔𝑗0_𝑛2𝜔_𝑗0sin𝜀𝜔_𝑗0𝐵_{𝛼𝑗0,𝛽𝑗0}𝑟_𝑛,𝑓^(𝑘)≤𝐾_𝑗0,𝜀𝑟_𝑛^𝜌+𝜀+𝑟^𝜔𝑗0_𝑛log𝑟_𝑛𝑇𝑟_𝑛,𝑓, (3.12) where 𝑟_𝑛∉𝐸, 𝜔_𝑗0=𝜋/(𝛽_𝑗0−𝛼_𝑗0), and 𝐾_𝑗0,𝜀 is a positive number depending on only 𝑗₀ and 𝜀. Combining (3.11) with (3.12) gives 𝑇𝑟_𝑛,𝑓^(𝑘)≤𝑞𝐾_𝑗0,𝜀log𝑟_𝑛𝜀𝑟_𝑛^𝜌+𝜀+𝑟^𝜔𝑗0_𝑛log𝑟_𝑛𝑇𝑟_𝑛,𝑓, (3.13) implying together with (iii) in Lemma 2.5 and Lemma 2.7 that log𝑇𝑟_𝑛,𝑓^(𝑘)≤3loglog𝑟_𝑛+max𝜌+𝜀,𝜔_𝑗0log𝑟_𝑛+loglog𝑇𝑟_𝑛,𝑓^(𝑘)+𝑂(1). (3.14) Thus, from (ii) in Lemma 2.5 for 𝜎+2𝜀, we have 𝜎+2𝜀≤limsup_𝑛→∞log𝑇𝑟_𝑛,𝑓^(𝑘)log𝑟_𝑛≤max𝜌+𝜀,𝜔_𝑗0≤𝜎+𝜀, (3.15) which is impossible.

Case 2 (𝜆(𝑓)=𝜇(𝑓)). Then by the assumption 𝜎=max{𝜔,𝜌,𝜇} and 𝜆(𝑓)>max{𝜔,𝜌}, we have 𝜎=𝜇(𝑓)=𝜆(𝑓)=𝜆(𝑓^(𝑘))=𝜇(𝑓^(𝑘)). By the same argument as in Case 1 with all the 𝜎+2𝜀 replaced by 𝜎=𝜇(𝑓), we can derive 𝜇𝑓=𝜎≤max𝜌+𝜀,𝜔_𝑗0<𝜆𝑓, (3.16) which is also impossible.

This completes the proof of Theorem 1.4.

3.2. Proof of Theorem 1.1

By Theorem 1.4, it suffices to prove that the lower order 𝜇(𝑓) of 𝑓 is finite. As in the proof of Theorem 1.4, we have, for each 𝑗∈{1,2,…,𝑞}, 𝐵_{𝜃𝑗,𝜃𝑗+1}𝑟,𝑓^(𝑘)≤𝐾_𝑗,𝜀𝑟^{𝜌+𝜀−𝜔𝑗}+𝑂log𝑟𝑇𝑟,𝑓,𝜔_𝑗=𝜋𝜃_𝑗+1−𝜃_𝑗,∀𝑟∉𝐸, (3.17) where the exceptional set 𝐸 associated with 𝑅(𝑟,𝑓) is of at most finite linear measure.

For 𝐹 in Lemma 2.8 and 𝐸 in (3.17), logdens(𝐹∪𝐸)=0 and hence for 𝑀(2) in Lemma 2.9 when 𝐾=2, logdens(𝑀(2)⧵(𝐹∪𝐸))≥𝑑(2)>0. Applying Lemma 2.8 to 𝑓^(𝑘) gives the existence of a sequence {𝑟_𝑛} of positive numbers such that 𝑟_𝑛→∞(𝑛→∞), 𝑟_𝑛∈𝑀(2)⧵(𝐹∪𝐸), and meas𝐸𝑟_𝑛,𝑓^(𝑘)>1𝑇^𝜀𝑟_𝑛,𝑓^(𝑘)log𝑟_𝑛^1+𝜀. (3.18) Set 𝜀_𝑛=112𝑞+1𝑇^𝜀𝑟_𝑛,𝑓^(𝑘)log𝑟_𝑛^1+𝜀. (3.19) Then, from (3.18) and (3.19), it follows that meas𝐸𝑟_𝑛,𝑓^(𝑘)∩^𝑞_𝑗=1𝜃_𝑗+𝜀_𝑛,𝜃_𝑗+1−𝜀_𝑛≥meas𝐸𝑟_𝑛,𝑓^(𝑘)−meas^𝑞_𝑗=1𝜃_𝑗−𝜀_𝑛,𝜃_𝑗+𝜀_𝑛≥2𝑞+1𝜀_𝑛−2𝑞𝜀_𝑛=𝜀_𝑛>0. (3.20) Hence, there exists a 𝑗∈{1,2,…,𝑞} such that, for infinitely many 𝑛, we have meas𝐸𝑟_𝑛,𝑓^(𝑘)∩𝜃_𝑗+𝜀_𝑛,𝜃_𝑗+1−𝜀_𝑛≥𝜀_𝑛𝑞. (3.21) Without loss of generality, we can assume that this holds for all the 𝑛. Let 𝐸_𝑛=𝐸(𝑟_𝑛,𝑓^(𝑘))∩(𝜃_𝑗+𝜀_𝑛,𝜃_𝑗+1−𝜀_𝑛). Thus, by the definition of 𝐸(𝑟,𝑓) and (3.21), it follows that ^{𝜃𝑗+1−𝜀𝑛}_{𝜃𝑗+𝜀𝑛}log⁺𝑓^(𝑘)𝑟_𝑛𝑒^𝑖𝜃𝑑𝜃≥_𝐸𝑛log⁺𝑓^(𝑘)𝑟_𝑛𝑒^𝑖𝜃𝐸𝑑𝜃≥meas_𝑛𝛿∞,𝑓^(𝑘)4𝑇𝑟_𝑛,𝑓^(𝑘)≥𝜀_𝑛𝛿∞,𝑓^(𝑘)𝑇𝑟4𝑞^𝑛,𝑓^(𝑘). (3.22)

On the other hand, by the definition of 𝐵_𝛼,𝛽(𝑟,∗) and (3.17), we have ^{𝜃𝑗+1−𝜀𝑛}_{𝜃𝑗+𝜀𝑛}log⁺𝑓^(𝑘)𝑟_𝑛𝑒^𝑖𝜃𝑑𝜃≤𝜋𝑟^𝜔𝑗_𝑛2𝜔_𝑗sin𝜀_𝑛𝜔_𝑗𝐵_{𝜃𝑗,𝜃𝑗+1}𝑟_𝑛,𝑓^(𝑘)≤𝜋²𝐾_𝑗,𝜀4𝜔²_𝑗𝜀_𝑛𝑟_𝑛^𝜌+𝜀+𝑟^𝜔𝑗_𝑛log𝑟_𝑛𝑇𝑟_𝑛,𝑓, (3.23) where 𝑟_𝑛∉𝐸∪𝐹, 𝜔_𝑗=𝜋/(𝜃_𝑗+1−𝜃_𝑗), and 𝐾_𝑗,𝜀 is a positive number depending on only 𝑗 and 𝜀. Combining (3.22) with (3.23) now yields 𝜀²_𝑛𝑇𝑟_𝑛,𝑓^(𝑘)≤𝑞𝜋²𝐾_𝑗,𝜀𝜔²_𝑗𝛿∞,𝑓^(𝑘)𝑟_𝑛^𝜌+𝜀+𝑟^𝜔𝑗_𝑛log𝑟_𝑛𝑇𝑟_𝑛,𝑓, (3.24) so that, together with (3.19) and Lemma 2.9, we have 𝑇^1−2𝜀𝑟_𝑛,𝑓^(𝑘)≤𝑞𝜋²2𝑞+1²log𝑟_𝑛^2+2𝜀𝐾_𝑗,𝜀𝜔²_𝑗𝛿∞,𝑓^(𝑘)𝑟_𝑛^𝜌+𝜀+𝑟^𝜔𝑗_𝑛log𝑟_𝑛+log𝑇𝑟_𝑛,𝑓^(𝑘)+log(6𝑒). (3.25) Thus 𝜇(𝑓)=𝜇(𝑓^(𝑘))≤max{𝜌+𝜀,𝜔_𝑗}/(1−2𝜀)<∞ and so Theorem 1.1 follows from Theorem 1.4.

This completes the proof of Theorem 1.1.

Acknowledgments

The author is extremely grateful to the referees for their many valuable suggestions to improve the presentation. The project was supported by the National Natural Science Foundation of China (Grant no. 11126351) and the Natural Science Foundation of Fujian Province, China (Grant no. 2010J05003).