Value Distribution for a Class of Small Functions in the Unit Disk

Abstract

If 𝑓 is a meromorphic function in the complex plane, R. Nevanlinna noted that its characteristic function 𝑇(𝑟,𝑓) could be used to categorize 𝑓 according to its rate of growth as |𝑧|=𝑟→∞. Later H. Milloux showed for a transcendental meromorphic function in the plane that for each positive integer 𝑘, 𝑚(𝑟,𝑓^(𝑘)/𝑓)=𝑜(𝑇(𝑟,𝑓)) as 𝑟→∞, possibly outside a set of finite measure where 𝑚 denotes the proximity function of Nevanlinna theory. If 𝑓 is a meromorphic function in the unit disk 𝐷={𝑧∶|𝑧|<1}, analogous results to the previous equation exist when limsup_𝑟→1−(𝑇(𝑟,𝑓)/log(1/(1−𝑟)))=+∞. In this paper, we consider the class of meromorphic functions 𝒫 in 𝐷 for which limsup_𝑟→1−(𝑇(𝑟,𝑓)/log(1/(1−𝑟)))<∞, lim_𝑟→1⁻𝑇(𝑟,𝑓)=+∞, and 𝑚(𝑟,𝑓′/𝑓)=𝑜(𝑇(𝑟,𝑓)) as 𝑟→1. We explore characteristics of the class and some places where functions in the class behave in a significantly different manner than those for which limsup_𝑟→1−(𝑇(𝑟,𝑓)/log(1/(1−𝑟)))=+∞ holds. We also explore connections between the class 𝒫 and linear differential equations and values of differential polynomials and give an analogue to Nevanlinna's five-value theorem.

1. Introduction

This paper uses notation from Nevanlinna theory which is summarized here for the reader's convenience. We denote by 𝑛(𝑟,𝑓) the number of poles of 𝑓 in |𝑧|≤𝑟<1, where each pole is counted according to its multiplicity. Also, 𝑛(𝑟,𝑓) counts the number of distinct poles of 𝑓 in |𝑧|≤𝑟<1 disregarding multiplicity. If 𝑥≥0, then log⁺𝑥=max(0,log𝑥). We define the proximity function 𝑚(𝑟,𝑓), the counting function 𝑁(𝑟,𝑓), and the Nevanlinna characteristic function 𝑇(𝑟,𝑓) as follows: 𝑚=1𝑟,𝑓2𝜋₀^2𝜋log⁺𝑓𝑟𝑒^𝑖𝜃𝑁=𝑑𝜃,𝑟,𝑓^𝑟₀𝑛𝑡,𝑓−𝑛0,𝑓𝑡𝑇.𝑑𝑡+𝑛0,𝑓log𝑟,𝑟,𝑓=𝑚𝑟,𝑓+𝑁𝑟,𝑓 (1.1)(1) Also, we have that𝑁𝑟,𝑓=^𝑟₀𝑛𝑡,𝑓−𝑛0,𝑓𝑡𝑑𝑡+𝑛0,𝑓log𝑟. (1.2)

A meromorphic function 𝑓 in the unit disk 𝐷={𝑧∶|𝑧|<1} can be categorized according to the rate of growth of its Nevanlinna characteristic 𝑇(𝑟,𝑓) as |𝑧|=𝑟 approaches one. Iflimsup_𝑟→1𝑇𝑟,𝑓−log(1−𝑟)=+∞, (1.3) many value distribution theorems analogous to those for transcendental meromorphic functions in the complex plane can be derived. In particular, results useful in studying solutions of linear differential equations which are analogous to theorems of H. Milloux can be shown—namely, for each positive integer 𝑘,𝑚𝑓𝑟,^(𝑘)𝑓=𝑜𝑇𝑟,𝑓 (1.4) as 𝑟 approaches one, possibly outside a set of finite measure where 𝑚 denotes the proximity function of Nevanlinna theory. For meromorphic functions of lesser growth than (1.3), analogous theorems need not hold. If ℱ consists of those meromorphic functions 𝑓 in 𝐷 for whichlimsup_𝑟→1𝑇𝑟,𝑓()𝑓−log1−𝑟=𝛼<∞, (1.5) Shea and Sons [1] showed that 𝑓^′ is in ℱ for each 𝑓 in ℱ, but also that there exist functions 𝑓 in ℱ with unbounded characteristic for which𝑚𝑓𝑟,^′𝑓1>log11−𝑟+loglog1−𝑟 (1.6) on a sequence of 𝑟 approaching one. Thus all functions in ℱ with unbounded characteristic need not satisfy (1.4) for 𝑘=1.

In this paper, we denote by 𝒫 those functions 𝑓 in ℱ for which 𝑇(𝑟,𝑓) is unbounded as 𝑟 approaches one and for which (1.4) does hold for 𝑘=1. We derive striking properties of class 𝒫 and make some connections between functions in class 𝒫 and solutions of linear differential equations defined in 𝐷. For functions in ℱ with 𝛼(𝑓)>0, we develop an interesting theorem analogous to Nevanlinna's five-value theorem for functions in the plane. Further, we prove a value distribution theorem for differential polynomials Φ=^𝑛_𝑘=0𝑎_𝑘𝑓^(𝑘), (1.7) where 𝑓 is in 𝒫, the 𝑎_𝑘 are meromorphic functions in 𝐷, and 𝑇(𝑟,𝑎_𝑘)=𝑜(𝑇(𝑟,𝑓)), as 𝑟→1.

Our paper proceeds as follows. In Section 2, we note examples of functions in 𝒫 and properties of the class. In Section 3, we prove a uniqueness theorem for functions in class ℱ (and hence in class 𝒫). In Section 4, we look at differential equations for which functions in 𝒫 are either coefficients or solutions, and in Section 5 we consider differential polynomials.

Much of the research reported here was part of the author's Ph.D. dissertation written at Northern Illinois University [2].

2. Properties and Examples of Functions in Class 𝒫

First we note that 𝒫 is not empty. For 𝛽>0, the function 𝑓 defined in 𝐷 by𝑓(𝑧)=exp𝛽𝑖1−𝑧 (2.1) is in class 𝒫, since𝑇=𝛽𝑟,𝑓2𝜋log1+𝑟1−𝑟 (2.2) by a calculation in Benbourenane [3] and by properties of 𝑚 and a lemma of Tsuji ([4, page 226]),𝑚𝑓𝑟,^′𝑓1=𝑂loglog,().1−𝑟𝑟⟶1 (2.3) Clearly 𝛼(𝑓) as defined in (1.5) is 𝛽/2𝜋 for this function.

The following proposition gives some simple closure properties of 𝒫.

Proposition 2.1. If 𝑓 and 𝑔 are in 𝒫 and 𝑐 is a nonzero complex number, we have (i)𝑐𝑓 is in 𝒫;(ii)1/𝑓 is in 𝒫;(iii)𝑓^𝑛 is in 𝒫 for each positive integer 𝑛;(iv)𝑓𝑔 may not be in 𝒫;(v)𝑓+𝑔 may not be in 𝒫.

The proof of (i), (ii), and (iii) in Proposition 2.1 follows by easy calculation. To see (iv), let 𝑔=1/𝑓, and to see (v), let 𝑔=−𝑓.

The complicated nature of class 𝒫 is demonstrated by the following theorem whereby some sums and products are in 𝒫.

Theorem 2.2. Let 𝑓 be a meromorphic function in class 𝒫.(i)If 𝑐 is a nonzero complex number for which liminf_𝑟→1𝑁(𝑟,−𝑐)𝑇𝑟,𝑓=0, (2.4) then 𝑓+𝑐 is in 𝒫.(ii)If 𝑔 is a meromorphic function in 𝐷 which is not identically zero and such that 𝑇(𝑟,𝑔)=𝑜(𝑇(𝑟,𝑓)), (𝑟→1), and 𝑚(𝑟,𝑔^′/𝑔)=𝑜(𝑇(𝑟,𝑓)), (𝑟→1), then 𝑓𝑔 is in 𝒫.(iii)There exists a Blaschke product 𝐵 such that 𝐵𝑓 is not in 𝒫.(iv)There exists a Blaschke product 𝐵 such that 𝐵𝑓 is in 𝒫.

Remark 2.3. In Nevanlinna theory, the Valiron deficiency of a complex value 𝑐 for a meromorphic function 𝑓 in 𝐷 is defined by Δ𝑐,𝑓=liminf_𝑟→1𝑁(𝑟,𝑐)𝑇.𝑟,𝑓 (2.5) It is known (cf. Theorem 2.20 on page 210 in [5]) that if lim_𝑟→1𝑇𝑟,𝑓=+∞, (2.6) then lim_𝑟→1𝑁(𝑟,𝑎)𝑇𝑟,𝑓=1 (2.7) except for at most a set of 𝑎-values of vanishing inner capacity. This fact enables us to show that the function 𝐹 in 𝒫 defined by 𝐹(𝑧)𝑖=exp1−𝑧 (2.8) has 𝐹+𝑐 in 𝒫 for all complex numbers 𝑐, because Δ(−𝑐,𝐹)=0 (2.9) for all 𝑐≠0.

Remark 2.4. The following example illustrates part (ii) of Theorem 2.2.

Example 2.5. Let 𝑓(𝑧)=𝑒^{𝑖/(1−𝑧)} and 𝑔(𝑧)=𝑒^{(1+𝑧)/(1−𝑧)}. Then 𝑔 is not identically zero, and it is well known that 𝑇(𝑟,𝑔)=𝑂(1). Therefore, 𝑇(𝑟,𝑔)=𝑜(𝑇(𝑟,𝑓)) as 𝑟→1. Also we have that 𝑚𝑔𝑟,^′𝑔2≤𝑇𝑟,(1−𝑧)²=𝑂(1)as𝑟⟶1, (2.10) since 2/(1−𝑧)² is the quotient of two bounded, analytic functions in the unit disk. And so we have that 𝑓𝑔=𝑒^{(𝑖+1+𝑧)/(1−𝑧)}∈𝒫.

We turn to the proof of Theorem 2.2.

Proof of Part (i). Let 𝑔=𝑓+𝑐. Then 𝑔^′=𝑓^′. We will show that 𝑔∈𝒫.
First, by calculation and properties of the Nevanlinna characteristic, note that 𝑇𝑟,𝑔=𝑇𝑟,𝑓+𝑂(1)as𝑟⟶1. (2.11)
Also, by calculation and properties of the proximity function, we get 𝑚𝑔𝑟,^′𝑔𝑓=𝑚𝑟,^′𝑓𝑓+𝑐≤𝑚𝑟,^′𝑓𝑓+𝑚𝑟,𝑓𝑓+𝑐≤𝑚𝑟,^′𝑓𝑐+𝑚𝑟,1+𝑓𝑓+𝑐≤𝑚𝑟,^′𝑓1+𝑚𝑟,(1)𝑓+𝑐+𝑂as𝑟⟶1. (2.12) Now, since 𝑓∈𝒫, 𝑚𝑓𝑟,^′𝑓𝑇𝑇=𝑜𝑟,𝑓as𝑟⟶1,=𝑜𝑟,𝑔as𝑟⟶1. (2.13) Also, since Δ(−𝑐,𝑓)=0, limsup_𝑟→1𝑚𝑟,1/𝑓+𝑐𝑇𝑟,𝑓=0, (2.14) and so 𝑚1𝑟,𝑇𝑇𝑓+𝑐=𝑜𝑟,𝑓as𝑟⟶1,=𝑜𝑟,𝑔as𝑟⟶1. (2.15)
Therefore, 𝑔∈𝒫 since 𝑚(𝑟,𝑔^′/𝑔)=𝑜(𝑇(𝑟,𝑔)) as 𝑟→1, and 𝑇(𝑟,𝑔) is unbounded as 𝑟→1.

Proof of Part (ii). First, 𝑇(𝑟,𝑓𝑔) is unbounded, since it can be shown that 𝑇𝑟,𝑓𝑔=𝑇𝑟,𝑓+𝑂(1)as𝑟⟶1, (2.16) since 𝑇(𝑟,𝑓) is unbounded.
Now note that (𝑓𝑔)^′/(𝑓𝑔)=𝑔^′/𝑔+𝑓^′/𝑓. Therefore, 𝑚𝑟,𝑓𝑔^′𝑔𝑓𝑔≤𝑚𝑟,^′𝑔𝑓+𝑚𝑟,^′𝑓,𝑇𝑇=𝑜𝑟,𝑓as𝑟⟶1,=𝑜𝑟,𝑓𝑔as𝑟⟶1. (2.17) Thus 𝑓𝑔∈𝒫.

Proof of Part (iii). Let 𝐵 be the Blaschke product defined in [6, Proposition 6.1, page 273]. This Blaschke product has the feature that for any 𝜖>0 there exists an exceptional set 𝐸₁⊂[0,1) satisfying _𝐸1𝑑𝑟1−𝑟<∞, (2.18) such that 𝐵^′(𝑧)𝐵(𝑧)1=𝑂1−|𝑧|^1.5+𝜖,|𝑧|∉𝐸₁, (2.19) and there exists a set 𝐹₁∈[0,1), satisfying _𝐹1𝑑𝑟1−𝑟=∞, (2.20) and a constant 𝐶>0, such that 𝐵^′(𝑥)𝐵(𝑥)≥𝐶(1−𝑥)^1.51log1−𝑥,𝑥∈𝐹₁⧵𝐸₁. (2.21)
Let 𝑔(𝑧)=𝑖/(1−𝑧). Then 𝑒^𝑔(𝑧)∈𝒫. Now define 𝑞(𝑧)=𝐵(𝑧)𝑒^𝑔(𝑧). We will now show that 𝑞(𝑧)∉𝒫.
First, it is easily shown that 𝑇(𝑟,𝑒^𝑔(𝑧))=𝑇(𝑟,𝑞)+𝑂(1) as 𝑟→1.
Note that since (𝑒^𝑔(𝑧))^′=𝑔^′𝑒^𝑔(𝑧), we have 𝑔^′=(𝑒^𝑔(𝑧))^′/𝑒^𝑔(𝑧). Therefore, since 𝑒^𝑔(𝑧)∈𝒫, 𝑚𝑟,𝑔^′=𝑜𝑇𝑟,𝑒^𝑔(𝑧)as𝑟⟶1. (2.22) And so 𝑚𝐵𝑟,^′𝐵𝑞=𝑚𝑟,^′𝑞−𝑔^′𝑞≤𝑚𝑟,^′𝑞+𝑚𝑟,−𝑔+log2. (2.23) Therefore, 𝑚𝑞𝑟,^′𝑞𝐵≥𝑚𝑟,^′𝐵−𝑚𝑟,−𝑔−log2. (2.24) Using (2.21), we have on a small exceptional set with |𝑧|=𝑟 for 𝑟∈𝐹₁⧵𝐸₁, 𝐵^′𝐵≥𝐶(1−𝑟)^1.51log.1−𝑟 (2.25) Taking the log⁺ of both sides, we get log⁺𝐵^′𝐵≥log⁺𝐶(1−𝑟)^1.51log𝐶1−𝑟≥log(1−𝑟)^1.51≥log𝐶+1.5log.1−𝑟 (2.26) So, calculating the following ratio yields 𝑚𝑟,𝑞^′/𝑞𝑇≥𝑚𝑟,𝑞𝑟,𝐵^′/𝐵𝑇𝑟,𝑒^𝑔(𝑧)−𝑚𝑟,𝑔^′𝑇𝑟,𝑒^𝑔(𝑧)−log2𝑇𝑟,𝑒^𝑔(𝑧)⟶3𝜋>0as𝑟⟶1. (2.27)
Therefore, 𝑞∉𝒫.

Proof of Part (iv). Let 𝐵 be a Blaschke product with zeros {𝑧_𝑛} such that |𝑧_𝑛|=1−1/𝑛⁵ for all integers 𝑛≥2. Theorem B in Heittokangas [6] shows that 𝐵^′ is in 𝐻^𝑝 for 𝑝 in (0,3/4), so 𝐵^′/𝐵 is of bounded characteristic. Hence 𝐵𝑓 is in 𝒫 for 𝑓 in 𝒫.

Remark 2.6. Since a Blaschke product is a bounded, analytic function, we see from parts (iii) and (iv) above that multiplication by such functions may or may not yield a function in 𝒫.

Further study of examples in class 𝒫 shows that the function 𝑓 defined by 𝑓(𝑧)=exp(𝑖/1−𝑧) has (1.4) holding for all 𝑘. However, there are also 𝑓 in 𝒫 for which (1.4) does not hold for 𝑘=2. We have the following theorem.

Theorem 2.7. There exists an analytic function ℎ in 𝒫 such that 𝑚(𝑟,ℎ^′′/ℎ)≠𝑜(𝑇(𝑟,ℎ)), as 𝑟→1.

Proof. First, we begin by constructing a function which has unbounded characteristic as 𝑟→1, but its derivative is of bounded characteristic. This construction is from [7, page 557].
Let 𝑎 be an integer greater than or equal to 2. Define for 𝑚≥1, 𝑄_𝑚=^𝑚_𝑘=1𝑎^𝑘=𝑎𝑎𝑎−1^𝑚,𝑡−1_𝑚𝛾=1−𝑄_𝑚, (2.28) where 𝛾 is a constant such that 0<𝛾<1. Now define 𝐹(𝑧)=^𝑧₀𝑓(𝑤)𝑑𝑤, (2.29) where 𝑓(𝑧)=^∞_𝑚=1𝑧1+𝑡_𝑚^𝑎𝑚. (2.30) Shea showed in [7] that 𝑓 is analytic in the unit disk and satisfies 1𝛼log11−𝑟<𝑇𝑟,𝑓≤log𝑀𝑟,𝑓<𝛽log1−𝑟as𝑟⟶1, (2.31) where 𝛼 and 𝛽 are constants such that 𝛾0<𝛼<𝑎,𝛽>𝛾+log2,log𝑎 (2.32) and 𝑀(𝑟,𝑓) is the maximum modulus function for 𝑓. Choose 𝛾 and 𝑎 such that 𝛾+log2/log𝑎<1, so we can take 𝛽<1. This implies that 𝐹 is bounded in the unit disk by the following argument: for 𝑧=𝑟𝑒^𝑖𝜃, |𝐹(𝑧)|=^𝑧₀𝑓(𝑤)𝑑𝑤≤^𝑟₀𝑓𝜌𝑒^𝑖𝜃𝑑𝜌≤^𝑟₀𝑀𝜌,𝑓𝑑𝜌. (2.33) By (2.31), we have that 𝑀<1𝜌,𝑓1−𝜌^𝛽. (2.34) Therefore, since 𝛽<1, we have by a simple integration that |𝐹(𝑧)|<^𝑟₀11−𝜌^𝛽𝑑𝜌=𝑂(1)as𝑟⟶1. (2.35)
Let 𝐾(𝑧)=∫^𝑧₀𝐹(𝑤)𝑑𝑤. Define ℎ(𝑧)=𝐾(𝑧)𝑒^𝑔(𝑧) where 𝑔(𝑧)=𝑖/(1−𝑧). Recall that 𝑒^𝑔(𝑧)∈𝒫. We now show that ℎ(𝑧)∈𝒫. First, we see that 𝑇(𝑟,𝐾)=𝑂(1) as 𝑟→1, by the following: 𝑇()()≤1𝑟,𝐾=𝑚𝑟,𝐾2𝜋₀^2𝜋log^+𝑟₀𝐹𝜌𝑒^𝑖𝜃𝑑𝜌𝑑𝜃, (2.36) and since 𝐹 is bounded, there exists a constant 𝑀 such that 𝑇()≤1𝑟,𝐾2𝜋₀^2𝜋log^+𝑟₀𝑀𝑑𝜌𝑑𝜃≤log𝑀. (2.37) Thus, 𝑇(𝑟,ℎ)≤𝑇(𝑟,𝐾)+𝑇(𝑟,𝑒^𝑔(𝑧))≤𝑇(𝑟,𝑒^𝑔(𝑧))+𝑂(1), as 𝑟→1. On the other hand, 𝑇𝑟,𝑒^𝑔(𝑧)ℎ=𝑇𝑟,𝐾≤𝑇(𝑟,ℎ)+𝑂(1)as𝑟⟶1. (2.38) Therefore, 𝑇(𝑟,ℎ)∼𝑇(𝑟,𝑒^𝑔(𝑧)) as 𝑟→1.
Now, we bound 𝑚(𝑟,ℎ^′/ℎ) from above by using properties of the Nevanlinna characteristic: 𝑚ℎ𝑟,^′ℎ=𝑚𝑟,𝐾𝑔^′𝑒^𝑔+𝐾^′𝑒^𝑔𝐾𝑒^𝑔𝐾≤𝑚𝑟,^′𝐾+𝑚𝑟,𝑔^′+log2≤𝑚𝑟,𝑔^′(1)𝑇+𝑂as𝑟⟶1=𝑜𝑟,𝑒^𝑔(𝑧)(𝑇(as𝑟⟶1=𝑜𝑟,ℎ))as𝑟⟶1. (2.39) And so we have 𝑚(𝑟,ℎ^′/ℎ)=𝑜(𝑇(𝑟,ℎ)) as 𝑟→1 and thus ℎ∈𝒫.
Now we show that 𝑚(𝑟,ℎ^′′/ℎ)≠𝑜(𝑇(𝑟,ℎ)) as 𝑟→1. By a quick calculation, we have 𝑚𝐾𝑟,^′′𝐾ℎ≤𝑚𝑟,^′′ℎ−2𝑔^′𝐾^′𝐾−𝑔^′′−𝑔^′2ℎ≤𝑚𝑟,^′′ℎ+𝑚(𝑟,2)+𝑚𝑟,𝑔^′𝐾+𝑚𝑟,^′𝐾+𝑚𝑟,𝑔^′′+2𝑚𝑟,𝑔^′ℎ+log4≤𝑚𝑟,^′′ℎ+𝑜(𝑇(𝑟,ℎ))as𝑟⟶1. (2.40) Also we have from the above construction 𝐾^′′=𝑓 so there exists an 𝛼>0 such that 𝑚(𝑟,𝐾^′′)>𝛼log(1/(1−𝑟)). And so, 𝑚𝐾𝑟,^′′𝐾≥𝑚𝑟,𝐾^′′()1−𝑚𝑟,𝐾≥𝛼log(1)1−𝑟+𝑂as𝑟⟶1. (2.41) Combining (2.40) and (2.41), we have 𝑚𝑟,ℎ^′′/ℎ𝑇(𝑟,ℎ)≥2𝜋𝛼≠0as𝑟⟶1. (2.42)

Remark 2.8. If we define 𝐴 to be the set of functions in 𝐹 such that (1.4) holds for all positive integers 𝑘, Theorem 2.7 shows 𝐴 is properly contained in 𝐹. Further, we note that in the proof of Theorem 2.7 above ℎ=𝐾𝑒^𝑔 can be replaced with ℎ=𝐾𝑝 where 𝑝∈𝐴. Also the idea of the proof of Theorem 2.7 can be used to show that for 𝑘>1 there exist functions ℎ in 𝐹 for which 𝑚ℎ𝑟,^(𝑗)ℎ=𝑜(𝑇(𝑟,ℎ))as𝑟⟶1 (2.43) for all integers 1<𝑗≤𝑘, but 𝑚ℎ𝑟,^(𝑘+1)ℎ≠𝑜(𝑇(𝑟,ℎ))as𝑟⟶1. (2.44)
The function ℎ in the proof of Theorem 2.7 provides us with further information about 𝒫.

Theorem 2.9. There exists a function ℎ in 𝒫 such that ℎ^′ is not in 𝒫.

Proof. The function ℎ=𝐾𝑒^𝑔 of Theorem 2.7 is in 𝒫. Using the Nevanlinna calculus and properties of 𝐾, one can show 𝑚(𝑟,ℎ^′′/ℎ^′)≠𝑜(𝑇(𝑟,ℎ^′)) as 𝑟→1. We omit the details here (cf. [2]).

For functions 𝑓 in class ℱ, we may call 𝛼(𝑓) defined in (1.5) the index of 𝑓. In [1], Shea and Sons showed for 𝑓 in ℱ that𝛼𝑓^′≤𝛼𝑓1+𝑘𝑓+1, (2.45) where 𝑘𝑓=limsup_𝑟→1𝑁𝑟,𝑓𝑇,𝑟,𝑓+1 (2.46) and this inequality is best possible. For analytic functions in class 𝒫, we get

Theorem 2.10. If 𝑓 is an analytic function in class 𝒫, then (i)𝑇(𝑟,𝑓^′)≤𝑇(𝑟,𝑓)+𝑜(𝑇(𝑟,𝑓)) as 𝑟→1;(ii)𝛼(𝑓^′)≤𝛼(𝑓);(iii)if 𝑁(𝑟,1/𝑓)=𝑜(log1/(1−𝑟)) as 𝑟→1, then 𝛼(𝑓)=𝛼(𝑓^′).

Proof. For part (i) since 𝑓 is analytic in 𝒫, we have 𝑇𝑟,𝑓^′=𝑚𝑟,𝑓^′𝑓≤𝑚𝑟,^′𝑓𝑓+𝑚𝑟,𝑓=𝑚𝑟,^′𝑓+𝑇𝑟,𝑓 (2.47) from which the result follows. Using the definition of the index of 𝑓^′ and of 𝑓, part (ii) comes from (i).
To see (iii), we observe 𝑇𝑟,𝑓≤𝑇𝑟,𝑓^′𝑓+𝑇𝑟,𝑓^′=𝑇𝑟,𝑓^′𝑓+𝑇𝑟,^′𝑓+𝑂(1)as𝑟⟶1. (2.48) Thus, 𝑇𝑟,𝑓≤𝑇𝑟,𝑓^′𝑓+𝑚𝑟,^′𝑓+𝑁𝑟,𝑓^′1+𝑁𝑟,𝑓+𝑂(1)as𝑟⟶1. (2.49) Dividing both sides of (2.49) by log1/(1−𝑟) and taking the limit superior as 𝑟 approaches one, we get 𝛼(𝑓)≤𝛼(𝑓^′).

3. Connections between Class 𝒫 and Differential Equations

We discuss some relationships between the coefficients of the linear differential equation and its solutions and how the coefficients and solutions relate to class 𝒫. We consider the complex linear differential equation𝑓^(𝑛)+𝑎_𝑛−1(𝑧)𝑓^(𝑛−1)+⋯+𝑎₀(𝑧)𝑓=0 (3.1) in the unit disk, with analytic coefficients.

There has been a tremendous amount of recent research on the relationship between the growth of the solutions of (3.1) and the growth of the analytic coefficients in the unit disk. Some recent papers include [8–10]. We now quote some of the important results that have a connection with class ℱ and, therefore, class 𝒫. The theorems use the definitions of the weighted Hardy space and weighted Bergman space which are stated below for convenience.

Definition 3.1. We say that an analytic function 𝑓 in the unit disk is in the weighted Hardy space 𝐻^𝑝_𝑞 for 0<𝑝<∞ and 0≤𝑞<∞ if sup_0≤𝑟<11−𝑟^2𝑞12𝜋₀^2𝜋𝑓𝑟𝑒^𝑖𝜃𝑝𝑑𝜃^1/𝑝<∞. (3.2) We say that 𝑓 is in 𝐻^∞_𝑞 if sup_𝑧∈𝐷1−|𝑧|^2𝑞𝑓(𝑧)<∞. (3.3)

Definition 3.2. We say that an analytic function 𝑓 in the unit disk, 𝐷, is in the weighted Bergman space 𝐴^𝑝_𝑞 if the area integral over 𝐷 satisfies _𝐷𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝑑𝜎^1/𝑝<∞ (3.4) for 0<𝑝<∞ and −1<𝑞<∞.

The theorems below also mention the Nevanlinna class 𝑁, the meromorphic functions of bounded characteristic in 𝐷. If a function is in 𝑁, then it is not in 𝒫, since 𝒫 only has functions of unbounded characteristic.

Theorem 3.3 (see [10, page 320]). Let f be a nontrivial solution of (3.1) with analytic coefficients 𝑎_𝑗, 𝑗=0,…,𝑛−1, in the unit disk. Then we have that(i)if −1<𝛼<0 and 𝑎_𝑗∈𝐻^{1/(𝑛−𝑗)}_{(𝑎+1)(𝑛−𝑗)} for all 𝑗=0,…,𝑛−1, then 𝑓∈𝑁;(ii)if 𝑎_𝑗∈𝐴^{1/(𝑛−𝑗)} for all 𝑗=0,…,𝑛−1, or 𝑎_𝑗∈𝐴¹_{(𝑛−𝑗−1)} for all 𝑗=0,…,𝑛−1, then 𝑓∈𝑁;(iii)if 𝑎_𝑗∈𝐻^{1/(𝑛−𝑗)}_𝑛−𝑗 for all 𝑗=0,…,𝑛−1, then 𝑓∈ℱ.

Theorem 3.4 (see [10, page 320]). We have that(i)if all nontrivial solutions 𝑓∈𝑁, then the coefficients 𝑎_𝑗∈⋂_{0<𝑝<1/(𝑛−𝑗)}𝐴^𝑝 for all 𝑗=0,…,𝑛−1;(ii)if all nontrivial solutions 𝑓∈ℱ, then the coefficients 𝑎_𝑗∈⋂_{0<𝑝<1/(𝑛−𝑗)}𝐻^𝑝_1/𝑝 for all 𝑗=0,…,𝑛−1.

We also have the following characterization, which uses the order of growth of 𝑓 in the unit disk defined as𝜌𝑓=limsup_𝑟→1−log⁺𝑇𝑟,𝑓().−log1−𝑟 (3.5)

Theorem 3.5 (see [9, page 44]). All solutions 𝑓 of (3.1), where 𝑎_𝑗 is analytic in 𝐷 for all 𝑗=0,…,𝑘−1, satisfy 𝜌(𝑓)=0 if and only if 𝑎_𝑗∈⋂_{0<𝑝<1/(𝑛−𝑗)}𝐴^𝑝 for all 𝑗=0,…,𝑛−1.

When 𝑛=1 in (3.1), we observe using Theorem 3.3, if 𝑓^′/𝑓=−𝑎₀∈𝐻¹_(𝛼+1) with −1<𝛼<0, then 𝑓∈𝑁 and, therefore, 𝑓∉𝒫. We can also conclude that if 𝑓^′/𝑓=−𝑎₀∈𝐴¹, then 𝑓∈𝑁 and so 𝑓∉𝒫. Also, if 𝑓^′/𝑓=−𝑎₀∈𝐻¹₁, then 𝑓∈ℱ, which means 𝑓 may be in 𝒫. On the other hand, using Theorem 3.4, we have if 𝑓∈𝒫, then 𝑓^′/𝑓=−𝑎₀∈⋂_0<𝑝<1𝐻^𝑝_1/𝑝.

If 𝑎₀(𝑧)=−𝛽𝑖/(1−𝑧)², then 𝑓(𝑧)=𝑒^{𝛽𝑖/(1−𝑧)} is a solution of the differential equation. However, if 𝑎₀=𝛽𝑖/(1−𝑧)^𝑘 for an integer 𝑘≥3, then 𝑎₀ is of bounded characteristic, but the solution𝑓=𝐶𝑒^{𝛽𝑖/(1−𝑧)𝑘−1} (3.6) has order 𝜌=𝑘−2>0 and, therefore, 𝑓∉ℱ. This shows the delicate nature between the growth of the coefficient and the solution; that is, a subtle change in growth of the coefficient can result in a solution that is no longer considered slow growth.

When 𝑛=2 in (3.1), Theorems 3.3 and 3.4 have the following corollary.

Corollary 3.6. Let 𝑓 be a non-trivial solution of (3.1) with analytic coefficients 𝑎₀ and 𝑎₁ in the unit disk. Then(i)if 𝑎₁∈𝐻¹_(𝛼+1) and 𝑎₀∈𝐻^1/2_2(𝛼+1) for −1<𝛼<0, then 𝑓∈𝑁 and 𝑓∉𝒫;(ii)if 𝑎₁∈𝐴¹ and 𝑎₀∈𝐴^1/2 or 𝑎₀ and 𝑎₁∈𝐴¹₁, then 𝑓∈𝑁 and 𝑓∉𝒫;(iii)if 𝑎₁∈𝐻¹₁ and 𝑎₀∈𝐻₂^1/2, then 𝑓∈ℱ and, therefore, could be in 𝒫;(iv)if all non-trivial solutions 𝑓∈ℱ, then 𝑎₁∈⋂_0<𝑝<1𝐻^𝑝_1/𝑝 and 𝑎₀∈⋂_0<𝑝<1/2𝐻^𝑝_1/𝑝.

The function 𝑤=𝑒^{𝛽𝑖/(1−𝑧)} is a solution to the equation𝑤^′′−𝛽𝑖(1−𝑧)²𝑤^′−2𝛽𝑖(1−𝑧)³𝑤=0. (3.7)

Since 𝑤 has no zeros, another solution of the above second-order differential equation that is linearly independent of 𝑤=𝑓₁(𝑧) is𝑓₂(𝑧)=𝑓₁(𝑧)𝑑𝑧𝑓₁(𝑧)²=𝑒^{𝛽𝑖/(1−𝑧)}𝑒^{−2𝛽𝑖/(1−𝑧)}𝑑𝑧. (3.8)

Computation shows𝑓₂(𝑧)=𝑒^{𝛽𝑖/(1−𝑧)}−(1−𝑧)−2𝛽𝑖log−2𝛽𝑖+1−𝑧^∞_𝑛=1−2𝛽𝑖𝑛(𝑛+1)!−2𝛽𝑖1−𝑧^𝑛. (3.9)

We know that 𝑓₁∈𝒫, but what can be said about 𝑓₂? The above form of 𝑓₂ makes it difficult to calculate the growth, but we do know that, by (iii) in Corollary 3.6, if 𝑎₀∈𝐻₂^1/2 and 𝑎₁∈𝐻¹₁, then 𝑓₂∈ℱ.

Example 3.7. We show for 𝑎₀=−2𝛽𝑖/(1−𝑧)³ and 𝑎₁=−𝛽𝑖/(1−𝑧)², 𝑎₀∈𝐻₂^1/2 and 𝑎₁∈𝐻¹₁. To see 𝑎₀∈𝐻₂^1/2, we first note 𝑎₀^1/2=−2𝛽𝑖(1−𝑧)^31/2=2𝛽|1−𝑧|^3/2. (3.10) We integrate and apply a lemma from Tsuji [4, page 226] which states that, in particular, ₀^2𝜋𝑑𝜃1−𝑟𝑒^𝑖𝜃3/21=𝑂(1−𝑟)^1/2, (3.11) and get that there exists a constant 𝑀 such that 12𝜋₀^2𝜋2𝛽|1−𝑧|^3/2𝑑𝜃=2𝛽2𝜋₀^2𝜋1|1−𝑧|^3/2𝑑𝜃≤2𝛽𝑀2𝜋(1−𝑟)^1/2. (3.12) And so 1−𝑟²²12𝜋₀^2𝜋𝑎₀^1/2𝑑𝜃²≤(1+𝑟)²(1−𝑟)²2𝛽𝑀2𝜋(1−𝑟)^1/22=(1+𝑟)²(1−𝑟)𝐶, (3.13) which goes to zero as 𝑟→1. Therefore, 𝑎₀∈𝐻₂^1/2.
A similar calculation shows that 𝑎₁∈𝐻¹₁.

The question as to whether 𝑓₂∈𝒫 is not a trivial question as there exist examples, such as Example 3.9 below, where at least one solution is in class 𝒫 and at least one solution is not in class 𝒫.

Example 3.8. The function ℎ in Theorem 2.7 is a solution of (3.1) with 𝑛=2 when 𝑎₀=−𝑔^′′−(𝑔^′)² and 𝑎₁=−(𝐾^′(2𝑔^′)+𝐾^′′)/(𝐾𝑔^′+𝐾^′). Then since ℎ^′′ℎ=−𝑎₁ℎ^′ℎ−𝑎₀, (3.14) we have that 𝑚ℎ𝑟,^′′ℎ≤𝑚𝑟,−𝑎₁ℎ+𝑚𝑟,^′ℎ+𝑚𝑟,−𝑎₀+log2. (3.15)
Now, recall from the proof of Theorem 2.7 that 1𝛼logℎ1−𝑟≤𝑚𝑟,^′′ℎ+𝑜(𝑇(𝑟,ℎ))as𝑟⟶1, (3.16) and since ℎ∈𝒫, by (3.15), we conclude that at least one of 𝑎₀ or 𝑎₁ has index greater than or equal to 𝛼. Therefore, as a consequence of Theorem 2.7, we have a growth estimate for the coefficients of this differential equation.
It can also be shown that 𝑎₀∈𝐻₂^1/2. However, 𝑎₁ is not in 𝐻¹₁, and thus the converse of Corollary 3.6(iii) is not true.

For differential equations of the form (3.1) where 𝑛≥3, we first quote two examples.

The first example has some solutions of (3.1) in 𝒫 and some not.

Example 3.9 (see [9, Example 10, page 52]). The functions 𝑓₁(𝑧)=𝑒^{𝑖((1+𝑧)/(1−𝑧))}−𝑒^{−𝑖((1+𝑧)/(1−𝑧))},𝑓₂(𝑧)=𝑒^{𝑖((1+𝑧)/(1−𝑧))},𝑓₃(𝑧)=1+𝑧1−𝑧 (3.17) are linearly independent solutions of 𝑓^′′′+𝑎₁(𝑧)𝑓^′′+𝑎₁(𝑧)𝑓^′+𝑎₀(𝑧)𝑓=0, (3.18) where 𝑎₀(𝑧)=−8(1+𝑧)(1−𝑧)⁵,𝑎₁(𝑧)=4(1−𝑧)⁴+23𝑧²+8𝑧+5(1+𝑧)²(1−𝑧)²,𝑎₂(𝑧)=−23𝑧+4().1+𝑧)(1−𝑧 (3.19) It can be shown that 𝑓₂∈𝒫. However, 𝑓₃ is of bounded characteristic and, therefore, 𝑓₃∉𝒫. (It is known that 𝑓₁ has order zero but unknown if 𝑓₁∈𝒫.) Also, according to [9], we have that 𝑎₀∈⋂_0<𝑝<1/3𝐴^𝑝_−1/3, 𝑎₁∈⋂_0<𝑝<1/2𝐴^𝑝₀, and 𝑎₂∈⋂_0<𝑝<1𝐴^𝑝₀.

This next example is also from [9].

Example 3.10 (see [9, Example 11, page 53]). The functions 𝑓₁(𝑧)=𝑒^{𝑖((1+𝑧)/(1−𝑧))}−𝑒^{−𝑖((1+𝑧)/(1−𝑧))},𝑓₂(𝑧)=𝑒^{𝑖((1+𝑧)/(1−𝑧))},𝑓_3,4(𝑧)=1+𝑧1−𝑧×𝑒^{±𝑖((1+𝑧)/(1−𝑧))} (3.20) are linearly independent solutions of 𝑓⁽⁴⁾+𝑎₃(𝑧)𝑓⁽³⁾+𝑎₂(𝑧)𝑓^′′+𝑎₁(𝑧)𝑓^′+𝑎₀(𝑧)𝑓=0, (3.21) where 𝑎₀(𝑧)=16(1−𝑧)⁸,𝑎₁(𝑧)=161+𝑧²(1+𝑧)(1−𝑧)⁵−3+9𝑧+9𝑧²+3𝑧³(1+𝑧)³(1−𝑧)³,𝑎₂(𝑧)=8(1−𝑧)⁴+49+18𝑧+9𝑧²(1+𝑧)²(1−𝑧)²,𝑎₃(𝑧)=−121+𝑧().1+𝑧)(1−𝑧 (3.22) It is known that 𝑓₂, 𝑓₃, and 𝑓₄ are all in 𝒫, and it is known that 𝑓₁ has order zero. Also, it can be shown that 𝑎₀∈⋂_0<𝑝<1/4𝐴^𝑝₀, 𝑎₁∈⋂_0<𝑝<1/3𝐴^𝑝_−1/3, 𝑎₂∈⋂_0<𝑝<1/2𝐴^𝑝₀, and 𝑎₃∈⋂_0<𝑝<1𝐴¹₀.

Proceeding as in the discussion of Example 3.8, we have the following theorem.

Theorem 3.11. If a function 𝑓 in 𝒫 satisfies a differential equation of the form (3.1) such that 𝑚𝑓𝑟,^(𝑛)𝑓𝑇𝑚𝑓≠𝑜𝑟,𝑓as𝑟⟶1,𝑟,^(𝑗)𝑓=𝑜𝑇𝑟,𝑓as𝑟⟶1 (3.23) for all integers 𝑗 such that 1≤𝑗<𝑛, then at least one of the analytic coefficients 𝑎_𝑗 has index greater than or equal to 𝛼 as 𝑟→1 for some 𝛼>0.

For nonhomogeneous differential equations of the form𝑓^(𝑛)+𝑎_𝑛−1(𝑧)𝑓^(𝑛−1)+⋯+𝑎₁(𝑧)𝑓^′+𝑎₀(𝑧)𝑓=𝑎_𝑛(𝑧), (3.24) where 𝑎_𝑗 is analytic in the disk for all 𝑗=0,…,𝑛, we state a result from [9] that applies to class ℱ.

Theorem 3.12 (see [9, Theorem 7, page 46]). All solutions 𝑓 of (3.24) satisfy 𝜌(𝑓)=0 if and only if 𝜌(𝑎_𝑛)=0 and 𝑎_𝑗∈⋂_{0<𝑝<1/(𝑘−𝑗)}𝐴^𝑝 for all 𝑗=0,…,𝑛−1. Therefore, if all solutions 𝑓 of (3.24) are in 𝒫, then 𝜌(𝑎_𝑛)=0 and 𝑎_𝑗∈⋂_{0<𝑝<1/(𝑘−𝑗)}𝐴^𝑝 for all 𝑗=0,…,𝑛−1.

Theorems 3.3 and 3.4 do not tell the whole story regarding class ℱ. Instead of the coefficients being in a certain function class, what can we say about the solutions of (3.1) if we know the coefficients have a certain index in class ℱ? We show the following proposition.

Proposition 3.13. Let 𝑘 be a positive integer. If 𝑎₀ is an analytic function in 𝐷 for which the index of 𝑎₀ is 𝛼(𝑎₀)>𝑘, then 𝑓∉𝒫 for 𝑓∈ℱ when 𝑓^(𝑘)−𝑎₀𝑓=0.

Proof. Note that since 𝑓^(𝑘)−𝑎₀𝑓=0, we have 𝑎₀=𝑓^(𝑘)/𝑓, and we want to show that limsup_𝑟→1−𝑚𝑟,𝑓^′/𝑓𝑇𝑟,𝑓≠0. (3.25) Since 𝛼(𝑎₀)>𝑘, there exists a real number 𝑠>𝑘 such that 𝑇(𝑟,𝑎₀)≥𝑠log(1/(1−𝑟)) on some sequence of 𝑟's as 𝑟→1. Also, since 𝑓∈ℱ, we have 𝑇(𝑟,𝑓)≤𝑡log(1/(1−𝑟)). Now, since 𝑓∈ℱ, 𝑓^(𝑘)∈ℱ and so 𝑓^(𝑖)/𝑓^(𝑖−1)∈ℱ for 1≤𝑖≤𝑘. By Shea and Sons [1], 𝑚(𝑟,𝑓^(𝑖)/𝑓^(𝑖−1))≤log(1/(1−𝑟))+(2+𝑜(1))loglog(1/(1−𝑟)) for 1≤𝑖≤𝑘 as 𝑟→1. Now, we have the following: 𝑚𝑓𝑟,^(𝑘)𝑓𝑓=𝑚𝑟,^(𝑘)𝑓^(𝑘−1)𝑓^(𝑘−1)𝑓^(𝑘−2)⋯𝑓^′𝑓𝑓≤𝑚𝑟,^(𝑘)𝑓^(𝑘−1)𝑓+⋯+𝑚𝑟,^′′𝑓^′𝑓+𝑚𝑟,^′𝑓,1≤𝑘log((111−𝑟+𝑘2+𝑜))loglog1−𝑟as𝑟⟶1. (3.26) So, since 𝑎₀ is analytic, we have 𝑚𝑓𝑟,^′𝑓𝑓≥𝑚𝑟,^(𝑘)𝑓𝑓−𝑚𝑟,^(𝑘)𝑓^(𝑘−1)𝑓−⋯−𝑚𝑟,^′′𝑓^′1≥𝑠log11−𝑟−𝑘log((111−𝑟−𝑘2+𝑜))loglog=()11−𝑟as𝑟⟶1𝑠−𝑘log((111−𝑟−𝑘2+𝑜))loglog1−𝑟as𝑟⟶1. (3.27) Therefore, 𝑚(𝑟,𝑓^′/𝑓)/𝑇(𝑟,𝑓)≥(𝑠−𝑘)/𝑡>0 as 𝑟→1.

With a similar argument as above, Proposition 3.13 is also true if 𝑎₀ is meromorphic and 𝑁(𝑟,𝑎₀)=𝑜(log(1/(1−𝑟))) as 𝑟→1.

4. The Identical Function Theorem

For functions in class ℱ, we have an analogue to the Nevanlinna five-value theorem which we quote as stated in [11].

Theorem 4.1 (see [11, page 48]). Suppose that 𝑓₁ and 𝑓₂ are meromorphic in the plane and let 𝐸_𝑗(𝑎) be the set of points 𝑧 such that 𝑓_𝑗(𝑧)=𝑎(𝑗=1,2). Then if 𝐸₁(𝑎)=𝐸₂(𝑎) for five distinct values of 𝑎, 𝑓₁(𝑧)≡𝑓₂(𝑧) or 𝑓₁ and 𝑓₂ are both constant.

Our analogue and proof follow. The proof has a subtle difference from the direct analogue of the proof of Theorem 4.1 in [11].

Theorem 4.2. Let 𝑓₁(𝑧) and 𝑓₂(𝑧) be meromorphic functions in class ℱ such that 𝛼(𝑓₁)≥𝛼(𝑓₂)>0 and let 𝐸_𝑗(𝑎) be the set of points 𝑧 such that 𝑓_𝑗(𝑧)=𝑎 for 𝑗=1,2. Then if 𝐸₁(𝑎)=𝐸₂(𝑎) for 𝑞 distinct values of 𝑎 such that 𝑞 is an integer and 𝑞>4+2/𝛼(𝑓₁), then 𝑓₁(𝑧)≡𝑓₂(𝑧).

Proof. Suppose 𝑓₁ and 𝑓₂ are not identical and that {𝑎₁,𝑎₂,…,𝑎_𝑞} are 𝑞 distinct complex numbers such that 𝐸₁(𝑎_𝜈) and 𝐸₂(𝑎_𝜈) are identical for 𝜈=1,2,…,𝑞 and 𝑞 is an integer greater than or equal to 4+2/𝛼(𝑓₁). We write the following notations: 𝑁_𝜈(𝑟)=𝑁1𝑟,𝑓₁(𝑧)−𝑎_𝜈=𝑁1𝑟,𝑓₂(𝑧)−𝑎_𝜈𝜈=1,2,…,𝑞. (4.1) Now using a reformulation of Nevanlinna's inequality for functions in class ℱ [1], we have the following for all 𝑟<1: 𝑞−2𝑇𝑟,𝑓₁≤^𝑞_𝜈=1𝑁_𝜈(𝑟)1+log11−𝑟+𝑂loglog𝑇1−𝑟as𝑟⟶1,𝑞−2𝑟,𝑓₂≤^𝑞_𝜈=1𝑁_𝜈(𝑟)1+log11−𝑟+𝑂loglog1−𝑟as𝑟⟶1. (4.2)(4.3) Now assume that 𝛼(𝑓₁)≥𝛼(𝑓₂)>0. Then from the definition of index we have that for 0<𝜖<𝛼(𝑓₁), there exists a sequence {𝑟_𝑚}→1 such that 𝑇𝑟_𝑚,𝑓₁>𝛼𝑓₁1−𝜖log1−𝑟_𝑚1∀𝑚⟶∞,orlog1−𝑟_𝑚<1𝛼𝑓₁𝑇𝑟−𝜖^𝑚,𝑓₁. (4.4) So, combining (4.2) and (4.3) and using (4.4), we get 𝑞−2𝑇𝑟_𝑚,𝑓₁+𝑇𝑟_𝑚,𝑓₂≤2^𝑞_𝜈=1𝑁_𝜈𝑟_𝑚+((112+𝑜))log1−𝑟_𝑚as𝑚⟶∞<2^𝑞_𝜈=1𝑁_𝜈𝑟_𝑚+2+𝑜(1)𝛼𝑓₁𝑇𝑟−𝜖^𝑚,𝑓₁as𝑚⟶∞, (4.5) which leads to (1)𝑞−2−2+𝑜𝛼𝑓₁𝑇𝑟−𝜖^𝑚,𝑓₁+𝑇𝑟_𝑚,𝑓₂≤2^𝑞_𝜈=1𝑁_𝜈𝑟_𝑚. (4.6)
Since 𝑓₁ and 𝑓₂ are not identical, we have 𝑇𝑟_𝑚,1𝑓₁−𝑓₂=𝑇𝑟_𝑚,𝑓₁−𝑓₂(1)𝑟+𝑂as𝑚⟶∞≤𝑇_𝑚,𝑓₁+𝑇𝑟_𝑚,𝑓₂(1)≤2+𝑂as𝑚⟶∞𝑞−2−(2+𝑜(1))/𝛼𝑓₁−𝜖^𝑞_𝜈=1𝑁_𝜈𝑟_𝑚+𝑂(1)as𝑚⟶∞. (4.7) On the other hand, every common root of the equations 𝑓_𝜈(𝑧)=𝑎 is a pole of 1/(𝑓₁−𝑓₂), and so we have ^𝑞_𝜈=1𝑁_𝜈𝑟_𝑚≤𝑁𝑟_𝑚,1𝑓₁−𝑓₂(1)≤2+𝑂𝑞−2−(2+𝑜(1))/𝛼𝑓₁−𝜖^𝑞_𝜈=1𝑁_𝜈𝑟_𝑚+𝑂(1)as𝑟⟶1, (4.8) which gives a contradiction since 𝑞>4+2/𝛼(𝑓₁) implies 2/(𝑞−2−(2+𝑜(1))/(𝛼(𝑓₁)−𝜖))<1 as 𝑟→1, unless ∑^𝑞_𝜈=1𝑁_𝜈(𝑟_𝑚)=𝑂(1). This, however, cannot occur since 𝑇(𝑟,𝑓₁) and 𝑇(𝑟,𝑓₂) are unbounded. Therefore, the result follows.

Remark 4.3. Since 𝒫⊂ℱ, Theorem 4.2 gives conditions for when two functions in 𝒫 are identical.

5. Values of Differential Polynomials

We now turn our focus on determining values for differential polynomials in the disk as it relates to class 𝒫. In a preliminary report by Sons [12], the author explores various results for functions satisfying (1.3) in the disk and their analogues for functions in class ℱ. Some of these results for class ℱ can be refined further if we restrict the functions to class 𝒫. We state a theorem from Sons (without proof) and follow it with a refinement for class 𝒫.

Theorem 5.1 (see [12, Theorem 4]). Let 𝑓 be a meromorphic function in 𝐷 which is in class ℱ and for which 𝑁1𝑟,𝑓+𝑁𝑟,𝑓=𝑜𝑇𝑟,𝑓,as𝑟⟶1. (5.1) Let 𝑛 be a positive integer, and for 𝑘=0,1,2,…,𝑛 let 𝑎_𝑘 be a meromorphic function in 𝐷 for which 𝑇𝑟,𝑎_𝑘=𝑜𝑇𝑟,𝑓,as𝑟⟶1. (5.2) If 𝜓 is defined in 𝐷 by 𝜓=^𝑛_𝑘=0𝑎_𝑘𝑓^(𝑘) (5.3) and 𝜓 is nonconstant, then 𝜓 assumes every complex number except possibly zero infinitely often provided the index of 𝑓 is 𝛼>1+𝑛(𝑛+1)/2.

Theorem 5.2. Let 𝑓 be a meromorphic function in 𝐷 which is in class 𝒫 and for which 𝑁1𝑟,𝑓+𝑁𝑟,𝑓=𝑜𝑇𝑟,𝑓,as𝑟⟶1. (5.4) Let 𝑛 be a positive integer, and for 𝑘=0,1,2,…,𝑛, let 𝑎_𝑘 be a meromorphic function in 𝐷 for which 𝑇𝑟,𝑎_𝑘=𝑜𝑇𝑟,𝑓,as𝑟⟶1. (5.5) Also, define 𝐸 to be the set {𝑘∶𝑚(𝑟,𝑓^(𝑘)/𝑓)=𝑜(𝑇(𝑟,𝑓))as𝑟→1}. If 𝜓 is defined in 𝐷 by 𝜓=^𝑛_𝑘=0𝑎_𝑘𝑓^(𝑘) (5.6) and 𝜓 is nonconstant, then 𝜓 assumes every complex number except possibly zero infinitely often, provided the index of 𝑓 is 𝛼>1+𝑛(𝑛+1)/2−∑𝐸, where ∑𝐸 is the sum of the values of 𝐸.

Proof. Since class ℱ is closed under differentiation, addition, and multiplication, we know that 𝜓 is in class ℱ. Therefore, we can apply the reformulation of the Second Fundamental theorem for class ℱ [1] to 𝜓. Thus, using 0, ∞, and 𝑐, a nonzero complex number, we get 𝑇𝑟,𝜓≤𝑁1𝑟,𝜓+𝑁1𝑟,+𝜓−𝑐𝑁1𝑟,𝜓+log11−𝑟+𝑂loglog1−𝑟,as𝑟⟶1. (5.7) Since poles of 𝜓 come from poles of 𝑎_𝑘 or 𝑓, we have an upper bound for 𝑁(𝑟,𝜓): 𝑁𝑟,𝜓≤𝑁𝑟,𝑓+^𝑛_𝑘=0𝑁𝑟,𝑎_𝑘. (5.8) From the hypothesis, we then have 𝑁𝑟,𝜓≤𝑁𝑟,𝑓+𝑜𝑇𝑟,𝑓,as𝑟⟶1. (5.9) Therefore, using (5.9) and the First Fundamental theorem, we get 𝑇1𝑟,𝜓=𝑚𝑟,𝜓1+𝑁𝑟,𝜓(1),()≤+𝑂𝑟⟶1𝑁1𝑟,𝜓+𝑁1𝑟,+𝜓−𝑐𝑁1𝑟,𝜓+log11−𝑟+𝑂loglog≤1−𝑟,as𝑟⟶1,𝑁1𝑟,𝜓+𝑁1𝑟,+𝜓−𝑐𝑁1𝑟,𝑓+log11−𝑟+𝑂loglog1−𝑟,as𝑟⟶1. (5.10) Now, solving for 𝑚(𝑟,1/𝜓) in the above calculation, we have the following inequality: 𝑚1𝑟,𝜓≤𝑁1𝑟,𝜓1−𝑁𝑟,𝜓+𝑁1𝑟,+𝜓−𝑐𝑁1𝑟,𝑓+log𝑇11−𝑟+𝑜𝑟,𝑓+𝑂loglog1−𝑟as𝑟⟶1. (5.11) Since 𝑁(𝑟,1/𝜓)≤𝑁(𝑟,1/𝜓), the 𝑁(𝑟,1/𝜓) terms cancel, and so we can say that 𝑚1𝑟,𝜓≤𝑁1𝑟,+𝜓−𝑐𝑁1𝑟,𝑓+log𝑇11−𝑟+𝑜𝑟,𝑓+𝑂loglog1−𝑟as𝑟⟶1. (5.12) So, using the First Fundamental theorem, properties of the proximity function and (5.12) give us the following: 𝑇1𝑟,𝑓=𝑚𝑟,𝑓1+𝑁𝑟,𝑓(1)1+𝑂≤𝑚1,𝜓𝜓𝑓1+𝑁𝑟,𝑓(1)1+𝑂≤𝑚𝑟,𝜓𝜓+𝑚𝑟,𝑓1+𝑁𝑟,𝑓(1)≤+𝑂𝑁1𝑟,+𝜓−𝑐𝑁1𝑟,𝑓+log𝜓1−𝑟+𝑚𝑟,𝑓1+𝑁𝑟,𝑓𝑇1+𝑜𝑟,𝑓+𝑂loglog1−𝑟as𝑟⟶1. (5.13) Noticing the fact that 𝑁(𝑟,𝑓)≤𝑁(𝑟,𝑓) and using the hypothesis that 𝑁1𝑟,𝑓+𝑁𝑟,𝑓=𝑜𝑇𝑟,𝑓as𝑟⟶1, (5.14) we can say that 𝑇𝑟,𝑓≤𝑁1𝑟,𝜓𝜓−𝑐+𝑚𝑟,𝑓1+log𝑇11−𝑟+𝑜𝑟,𝑓+𝑂loglog1−𝑟as𝑟⟶1. (5.15)
We now estimate 𝑚(𝑟,𝜓/𝑓). By using properties of the proximity function, we get 𝑚𝜓𝑟,𝑓∑=𝑚𝑟,^𝑛𝑘=0𝑎_𝑘𝑓^(𝑘)𝑓=𝑚𝑟,^𝑛_𝑘=0𝑎_𝑘𝑓^(𝑘)𝑓≤^𝑛_𝑘=0𝑚𝑟,𝑎_𝑘𝑓^(𝑘)𝑓()≤+log𝑛+1_𝑛𝑘=0𝑚𝑟,𝑎_𝑘+^𝑛_𝑘=1𝑚𝑓𝑟,^(𝑘)𝑓+log(𝑛+1). (5.16) Recall the set 𝐸={𝑘∶𝑚(𝑟,𝑓^(𝑘)/𝑓)=𝑜(𝑇(𝑟,𝑓))as𝑟→1}. Notice that since 𝑓∈𝒫, 𝐸 is not empty. The set 𝐸 also allows us to split the following sum into two pieces. Indeed, ^𝑛_𝑘=1𝑚𝑓𝑟,^(𝑘)𝑓=_𝑘∈𝐸𝑚𝑓𝑟,^(𝑘)𝑓+_𝑘∉𝐸𝑚𝑓𝑟,^(𝑘)𝑓. (5.17) But now we can say that _𝑘∈𝐸𝑚𝑓𝑟,^(𝑘)𝑓=𝑜𝑇𝑟,𝑓as𝑟⟶1, (5.18) since this is true for each 𝑘∈𝐸. Therefore, using (5.18) and the hypothesis 𝑇𝑟,𝑎_𝑘=𝑜𝑇𝑟,𝑓,as𝑟⟶1, (5.19) we can update (5.16) to say that 𝑚𝜓𝑟,𝑓≤_𝑘∉𝐸𝑚𝑓𝑟,^(𝑘)𝑓+𝑜𝑇𝑟,𝑓as𝑟⟶1. (5.20) We use (3.26) to say that 𝑚𝑓𝑟,^(𝑘)𝑓1≤𝑘log𝑇1−𝑟+𝑜𝑟,𝑓as𝑟⟶1, (5.21) and, thus, (5.20) becomes 𝑚𝜓𝑟,𝑓≤_𝑘∉𝐸1𝑘log𝑇1−𝑟+𝑜𝑟,𝑓as𝑟⟶1. (5.22) We can calculate ∑_𝑘∉𝐸𝑘log(1/(1−𝑟)) by noting that _𝑘∉𝐸1𝑘log11−𝑟=log1−𝑟_𝑘∉𝐸𝑛()𝑘=𝑛+121−𝐸log,1−𝑟 (5.23) where ∑𝐸 is the sum of the elements in 𝐸. Note that 𝑛(𝑛+1)/2−∑𝐸=𝛽≥0. Thus, (5.20) becomes 𝑚𝜓𝑟,𝑓≤𝑛(𝑛+1)21−𝐸log𝑇1−𝑟+𝑜𝑟,𝑓as𝑟⟶1. (5.24) Therefore, we can now update (5.15) to say 𝑇𝑟,𝑓≤𝑁1𝑟,+𝑛()𝜓−𝑐𝑛+121−∑𝐸+1log𝑇11−𝑟+𝑜𝑟,𝑓+𝑂loglog1−𝑟as𝑟⟶1. (5.25)
Since the index of 𝑓 is equal to 𝛼>𝑛(𝑛+1)/2−∑𝐸+1, we have that 𝛾𝑇𝑟,𝑓≤𝑁1𝑟,𝑇𝜓−𝑐+𝑜𝑟,𝑓as𝑟⟶1, (5.26) where 𝛾>0. Since 𝑇(𝑟,𝑓) is unbounded, we have proved the claim that 𝜓 assumes every complex number except possibly zero infinitely often.