Interpolation Polynomials of Entire Functions for Erdös-Type Weights

Abstract

Let ℝ=(−∞,∞), and let 𝑄∈𝐶¹(ℝ)∶ℝ→[0,∞) be an even function. In this paper, we consider some Lagrange interpolation polynomials and the Gauss-Jacobi quadrature formula of entire functions associated with Erdös-type weights 𝑤(𝑥)=𝑒^{−𝑄(𝑥)}, 𝑥∈ℝ, and we will estimate the error terms.

1. Introduction

For a weight 𝑤, we suppose that ∫^∞₀𝑥^𝑛𝑤²(𝑥)𝑑𝑥<∞ for all 𝑛=0,1,2,…. Then we define a unique sequence of orthonormal polynomials {𝑝_𝑛(𝑤;𝑥)} as follows: ^∞_−∞𝑝_𝑛(𝑤;𝑥)𝑝_𝑚(𝑤;𝑥)𝑤(𝑥)𝑑𝑥=𝛿_𝑛,𝑚theKroneckersymbol, (1) where 𝑝_𝑛(𝑤;𝑥)∶=𝛾_𝑛𝑥^𝑛+⋯,𝛾_𝑛>0 is a polynomial of degree 𝑛. Furthermore, we denote the zeros of 𝑝_𝑛(𝑤;𝑥) by 𝑥_𝑘,𝑛  (𝑘=1,2,…,𝑛),  𝑥_1,𝑛>𝑥_2,𝑛>⋯>𝑥_𝑛,𝑛. Hence, 𝑝_𝑛(𝑤;𝑥)=𝛾_𝑛𝑥−𝑥_𝑛,𝑛⋯𝑥−𝑥_2,𝑛𝑥−𝑥_1,𝑛. (2) For a given function 𝑓, the Lagrange interpolation polynomial 𝐿_𝑛(𝑤;𝑓) with the weight 𝑤 is defined to be a unique polynomial of degree at most 𝑛−1 which coincides with 𝑓 at 𝑥_𝑘,𝑛,  𝑘=1,2,…,𝑛. Then we have the representation 𝐿_𝑛𝑤;𝑓;𝜉=^𝑛_𝑘=1𝑓𝑥_𝑘,𝑛𝑙_𝑘,𝑛(𝜉), (3) where 𝑙_𝑘,𝑛(𝜉)=𝑝_𝑛(𝑤;𝜉)𝑝^′_𝑛𝑤;𝑥_𝑘,𝑛𝜉−𝑥_𝑘,𝑛,𝑘=1,2,…,𝑛. (4) The Gauss-Jacobi quadrature formula to the function 𝑓 is defined by 𝑄_𝑛𝑤;𝑓∶=^𝑛_𝑘=1𝜆_𝑛𝑤;𝑥_𝑘,𝑛𝑓𝑥_𝑘,𝑛, (5) where the coefficients 𝜆_𝑛(𝑤;𝑥_𝑘,𝑛) are the Christoffel numbers, which are given by 𝜆_𝑛(𝑤;𝑥)∶=^𝑛−1_𝑘=1𝑝_𝑘(𝑤;𝑥)²⁻¹. (6)

We denote by 𝑊 a class of all weight functions of the form 𝑤=exp(−2𝑄) such that 𝑄 is an even differentiable function on ℝ, except possibly at 𝑥=0, increasing for 𝑥>0, and there exists 𝜌<1 such that 𝑥^𝜌𝑄^′(𝑥) is increasing. In addition, the unique positive sequence {𝑞_𝑛} determined by 𝑞_𝑛𝑄^′(𝑞_𝑛)=𝑛 satisfies the condition 𝑞_2𝑛/𝑞_𝑛≥𝐶>1,𝑛=1,2,…, for some constant 𝐶.

For an entire function 𝑓 and 𝑅>0, we set 𝑀_𝑓(𝑅)∶=max_|𝑧|=𝑅𝑓(𝑧),𝑧∈ℂ. (7) In [1, 2], Al-Jarrah showed the following results (see also [3–5]).

Theorem 1. Let 𝑤∈𝑊 and 𝑓 be an entire function.(1)There exists a constant 𝐴∈(0,1), depending on 𝑄 only, such that if 𝑓 satisfies limsup_𝑅→∞log𝑀_𝑓(𝑅)2𝑄(𝑅)≤𝐴, (8) then, uniformly on compact subsets of ℂ, limsup_𝑛→∞𝑓(𝜉)−𝐿_𝑛𝑤;𝑓;𝜉^1/𝑛<1,limsup_{𝑛→∞ℝ}𝑓(𝑥)𝑤(𝑥)𝑑𝑥−𝑄_𝑛𝑤;𝑓^1/𝑛<1. (9)(2)If the entire function 𝑓 satisfies lim_𝑅→∞log𝑀_𝑓(𝑅)2𝑄(𝑅)=0, (10) then, uniformly on compact subsets of ℂ, lim_𝑛→∞𝑓(𝜉)−𝐿_𝑛𝑤;𝑓;𝜉^1/𝑛=0,lim_{𝑛→∞ℝ}𝑓(𝑥)𝑤(𝑥)𝑑𝑥−𝑄_𝑛𝑤;𝑓^1/𝑛=0. (11)

In this paper, we extend the above results to Erdös-type weights. Note that every weight in 𝑊 is not Erdös-type (see Remark 4). In Section 2, we recall the definition of ℱ(lip(1/2)+), which is the class of weights we discuss here. Our theorems are stated in Section 3.

2. Preliminaries

For any nonzero real valued functions 𝑓(𝑥) and 𝑔(𝑥), we write 𝑓(𝑥)∼𝑔(𝑥) if there exists a constant 𝐶>1 such that 𝐶⁻¹𝑔(𝑥)≤𝑓(𝑥)≤𝐶𝑔(𝑥) for all 𝑥. Throughout this paper, 𝐶 denotes a positive constant independent of 𝑛,𝑥,𝑡. The same symbol does not necessarily denote the same constant in different occurrences. We say that 𝑓∶ℝ⁺∶=[0,∞)→ℝ⁺ is quasi-increasing if there exists 𝐶>0 such that 𝑓(𝑥)≤𝐶𝑓(𝑦),0<𝑥<𝑦.

First, we recall a class of weights.

Definition 2 (see cf. [6, Definition 1.2]). Let 𝑄∶ℝ→[0,∞) be a continuous even function. We write 𝑤=exp(−𝑄)∈ℱ(lip(1/2)+) if 𝑄 satisfies the following properties.(a)𝑄^′(𝑥) is continuous in ℝ, with 𝑄(0)=0. (b)𝑄^′(𝑥) is nondecreasing in ℝ. (c)lim_𝑥→∞𝑄(𝑥)=∞. (d)The function 𝑇(𝑥)∶=𝑥𝑄^′(𝑥)/𝑄(𝑥), 𝑥≠0, is quasi-increasing in (0,∞) and 𝑇(𝑥)⩾Λ>1, 𝑥∈ℝ⧵{0}. (e)There exists 𝜀₀∈(0,1) such that for 𝑥∈ℝ⧵{0}, 𝑇(𝑥)∼𝑇(𝑥(1−(𝜀₀/𝑇(𝑥)))). (f)For any 𝜀>0, there exists 𝛿>0 such that for any 𝑥∈ℝ⧵{0}, _|𝑥|(𝑥)^{𝑥+𝛿|𝑥|/𝑇(𝑥)}_{𝑥−𝛿/𝑇}𝑄^′(𝑠)−𝑄^′(𝑥)|𝑠−𝑥|^3/2𝑑𝑠≤𝜀𝑄^′(𝑥)𝑇(𝑥)|𝑥|. (12)

Example 3. Let 𝑤=exp(−𝑄)∈ℱ(lip(1/2)+).(1)If 𝑇(𝑥) is bounded, then the weight 𝑤(𝑥) is called a Freud-type weight (see [5]). We see that 𝑤(𝑥)=exp(−|𝑥|^𝛼), 𝛼>1, is a Freud-type weight.(2)If 𝑇(𝑥) is unbounded, then 𝑤(𝑥) is called an Erdös-type weight. The following 𝑄s give Erdös-type weights 𝑤=exp(−𝑄) (see [7, Example 1.2], [8, Theorem 3.1]). Let exp_𝑙(𝑥)∶=exp(exp(exp⋯exp𝑥)⋯)(𝑙times) for 𝑙=1,2,…. Then for 𝛼+𝛽>1, 𝛼>0, and 𝛽≥0, we set 𝑄(𝑥)=𝑄_{𝑙,𝛼,𝛽}(𝑥)∶=|𝑥|^𝛽exp_𝑙|𝑥|^𝛼−exp_𝑙(0). (13)Also, for 𝛼>1, put 𝑄(𝑥)=𝑄_𝛼(𝑥)∶=(1+|𝑥|)^|𝑥|𝛼−1.

Remark 4. Every weight 𝑤 in 𝑊 is Freud type. In fact, by definition, 𝑞_2𝑘≥𝐶^𝑘𝑞₁. On the other hand, if we assume that 𝑤 is Erdös type, then for any given 𝑀>0, there exists a constant 𝐶>0 such that 𝑄^′(𝑥)/𝑄(𝑥)=𝑇(𝑥)/𝑥≥(𝑀+1)/𝑥 for 𝑥>𝐶. By integration, we see 𝑥^𝑀+1≤𝐶𝑄(𝑥) (14) for 𝑥>𝐶. Since 𝑞_𝑛𝑄(𝑞_𝑛)=𝑛, 𝑞_𝑛^𝑀+1≤𝐶𝑄(𝑞_𝑛)=𝐶𝑛/𝑞_𝑛; that is, 𝑞_𝑛≤𝐶𝑛^1/(𝑀+2). If we take 𝑀>0 sufficiently large, this contradicts the fact that 𝑞_2𝑘≥𝐶^𝑘𝑞₁ (cf. [7, Lemma 3.2]).

For 𝑢>0, we define the Mhaskar-Rakhmanov-Saff number 𝑎_𝑢 by 2𝑢=𝜋¹₀𝑎_𝑢𝑡𝑄^′𝑎_𝑢𝑡1−𝑡^21/2𝑑𝑡. (15)

Lemma 5. Let 𝑤=exp(−𝑄)∈ℱ(lip(1/2)+).(1)([6, Lemma  3.4 (3.18)]) For 𝑡>0, 𝑄(𝑎_𝑡)∼𝑡/𝑇(𝑎_𝑡).(2)([6, Theorem 1.19 (f)]) There exists 𝑛₀>0 such that for 𝑛⩾𝑛₀, 1−𝑥_1,𝑛/𝑎_𝑛∼1/(𝑛𝑇(𝑎_𝑛))^2/3. In particular 𝑥_1,𝑛≤𝑎_𝑛 holds.(3)([6, Theorem 15.2 (15.8)]) There exists 𝐶>0 such that 𝛾_𝑛−1/𝛾_𝑛≤𝐶𝑎_𝑛 for all 𝑛=1,2,….

For an entire function 𝑓 and 𝑛=1,2,…, we set 𝐸_𝑛(𝜉)∶=𝑓(𝜉)−𝐿_𝑛𝑤;𝑓;𝜉. (16) Then by the residue theorem, for 𝜉∈𝐷_𝑛, we have 𝐸_𝑛(𝜉)=𝑝_𝑛(𝑤;𝜉)2𝜋𝑖_𝐶𝑛𝑓(𝑧)𝑑𝑧𝑝_𝑛(),𝑤;𝑧)(𝑧−𝜉 (17) where 𝐷_𝑛 is a simply connected domain containing all zeros of 𝑝_𝑛(𝑤) and 𝐶_𝑛 is its boundary. Note that for any 𝜉∈ℂ, there exists 𝑛 such that 𝜉∈𝐷_𝑛. For each 𝑚=1,2,… we also set 𝐿_𝑛,𝑚𝑤;𝑓;𝜉∶=^𝑛_𝑗=1^𝑚−1_𝑠=01()!𝑓(𝑧)𝑚−1−𝑠𝛾^𝑚_𝑛∏_𝑘≠𝑗𝑧−𝑥_𝑘,𝑛^{𝑚(𝑚−1−𝑠)}_{𝑧=𝑥𝑗,𝑛}×𝑝_𝑛(𝑤;𝜉)^𝑚𝜉−𝑥_𝑗,𝑛^𝑠+1,𝐸_𝑛,𝑚(𝜉)∶=𝑓(𝜉)−𝐿_𝑛,𝑚𝑤;𝑓;𝜉. (18) Then 𝐿_𝑛,𝑚(𝑤;𝑓;𝜉) is a polynomial of degree at most 𝑛𝑚−1 such that 𝐿^(𝑗)_𝑛,𝑚𝑤;𝑓;𝑥_𝑘,𝑛=𝑓^(𝑗)𝑥_𝑘,𝑛,𝑘=1,2,…,𝑛,𝑗=1,2,…,𝑚−1 (19) holds. By the residue theorem again, we see 𝐸_𝑛,𝑚(𝜉)=𝑝_𝑛(𝑤;𝜉)^𝑚2𝜋𝑖_𝐶𝑛𝑓(𝑧)𝑑𝑧𝑝_𝑛(𝑤;𝑧)^𝑚(𝑧−𝜉) (20) for 𝜉∈𝐷_𝑛.

For the Gauss-Jacobi quadrature, we set 𝐸_𝑛,∗∶=_ℝ𝑓(𝑥)𝑤(𝑥)𝑑𝑥−𝑄_𝑛𝑤;𝑓. (21) Then, we also see 𝐸_𝑛,∗=^∞_𝑘=𝑛𝛾_𝑘+1𝛾_𝑘12𝜋𝑖_𝐶𝑘𝑓(𝑧)𝑑𝑧𝑝_𝑘(𝑤;𝑧)p_𝑘+1(),𝑤;𝑧 (22) (cf. [1–5]).

3. Theorems and Proofs

We state our theorems. Throughout this section, we assume 𝑤=exp(−𝑄)∈ℱ(lip(1/2)+), and let 𝜂∶=limsup_𝑡→∞𝑄𝑎_𝑡𝑇𝑎_𝑡𝑡. (23) Note that 𝜂 is determined finitely by Lemma 5(1).

Theorem 6. (1) If an entire function 𝑓 satisfies limsup_𝑅→∞log𝑀_𝑓(5𝑅)𝑇(𝑅)𝑄(𝑅)≤1𝜂, (24) then limsup_𝑛→∞𝑓(𝜉)−𝐿_𝑛𝑤;𝑓;𝜉^1/𝑛<1 (25) uniformly on compact subsets; that is, for any compact subset 𝐾 in ℂ, there exists a constant 0<𝑐<1 such that limsup_𝑛→∞𝑓(𝜉)−𝐿_𝑛𝑤;𝑓;𝜉^1/𝑛≤𝑐 (26) for 𝜉∈𝐾. Moreover, for any 𝑚=1,2,…, limsup_𝑛→∞𝑓(𝜉)−𝐿_𝑛,𝑚𝑤;𝑓;𝜉^1/𝑛<1 (27) holds also uniformly on compact subsets.
(2) If an entire function 𝑓 satisfies limsup_𝑅→∞log𝑀(10𝑅)𝑇(𝑅)𝑄(𝑅)≤1𝜂, (28) then for any 𝑗=1,2,3,…, limsup_𝑛→∞𝑓^(𝑗)(𝜉)−𝐿_𝑛𝑤;𝑓^(𝑗);𝜉^1/𝑛<1 (29) uniformly on compact subsets. Moreover, for any 𝑚=1,2,…, limsup_𝑛→∞𝑓^(𝑗)(𝜉)−𝐿_𝑛,𝑚𝑤;𝑓^(𝑗);𝜉^1/𝑛<1 (30) holds also uniformly on compact subsets.

As for the Gauss-Jacobi quadrature, we have the following theorem.

Theorem 7. Let 𝑓 be an entire function. If limsup_𝑅→∞log𝑀_𝑓𝜆₀𝑅𝑇(𝑅)𝑄(𝑅)≤1𝜂 (31) holds for some 𝜆₀>1 large enough, then limsup_{𝑛→∞ℝ}𝑓(𝑥)𝑤(𝑥)𝑑𝑥−𝑄_𝑛𝑤;𝑓^1/𝑛≤𝑐<1, (32) for a constant 𝑐>0. Further, if limsup_𝑅→∞log𝑀_𝑓2𝜆₀𝑅𝑇(𝑅)𝑄(𝑅)≤1𝜂 (33) holds, then for any 𝑗=1,2,3,…, limsup_{𝑛→∞ℝ}𝑓^(𝑗)(𝑥)𝑤(𝑥)𝑑𝑥−𝑄_𝑛𝑤;𝑓^(𝑗)1/𝑛≤𝑐<1. (34)

Theorem 8. Let 𝑓 be an entire function. Suppose that lim_𝑅→∞log𝑀_𝑓(𝜆𝑅)𝑇(𝑅)𝑄(𝑅)=0, (35) for every 𝜆>0 large enough. Then lim_𝑛→∞𝑓(𝜉)−𝐿_𝑛𝑤;𝑓;𝜉^1/𝑛=0 (36) holds uniformly on compact subsets of ℂ and for any 𝑗=1,2,3,…, lim_𝑛→∞𝑓^(𝑗)(𝜉)−𝐿_𝑛,𝑚𝑤;𝑓^(𝑗);𝜉^1/𝑛=0 (37) also holds uniformly on compact subsets of ℂ. Moreover for any 𝑗=1,2,3,…, lim_{𝑛→∞ℝ}𝑓^(𝑗)(𝑥)𝑤(𝑥)𝑑𝑥−𝑄_𝑛𝑤;𝑓^(𝑗)1/𝑛=0 (38) holds true.

Remark 9. The order of an entire function 𝑓 is given by 𝜇𝑓∶=limsup_𝑅→∞loglog𝑀_𝑓(𝑅).log𝑅 (39) Also the type of 𝑓 with order 𝜇<∞ is given by 𝜏𝑓∶=𝜏_𝜇𝑓=limsup_𝑅→∞log𝑀_𝑓(𝑅)𝑅^𝜇. (40) If 𝑤=exp(−𝑄) is an Erdös-type weight and 𝜇(𝑓),𝜏(𝑓)<∞, then (35) holds for every 𝜆>0. In fact, by (14), lim_𝑅→∞log𝑀_𝑓(𝜆𝑅)𝑇(𝑅)𝑄(𝑅)=lim_𝑅→∞log𝑀_𝑓(𝜆𝑅)(𝜆𝑅)^𝜇(𝜆𝑅)^𝜇𝑇(𝑅)𝑄(𝑅)=0. (41)
The following lemma was proved in [2, Lemmas 3.3 and 3.4]. Its proof is available for our case.

Lemma 10. Let 𝑤=exp(−𝑄)∈ℱ(lip(1/2)+), and let 𝑝_𝑛(𝑤;𝑧) be the 𝑛th orthonormal polynomial generated by 𝑤. Then 𝑝_𝑛(𝑤;𝑧)≤𝛾_𝑛2^𝑛/2𝑥^𝑛_1,𝑛, (42) for all 𝑧 such that |𝑧|≤𝑥_1,𝑛, and 𝑝_𝑛(𝑤;𝑧)⁻¹≤1𝛾_𝑛|𝑧|^𝑛exp𝑛𝑥²_1,𝑛2|𝑧|²−𝑥²_1,𝑛 (43) for all 𝑧 such that |𝑧|>𝑥_1,𝑛. Also we have ^[𝑛/2]_𝑗=1𝑥²_𝑗,𝑛=^𝑛−1_𝑗=1𝛾_𝑗−1𝛾_𝑗². (44)

Proof of Theorem 6. (1) We estimate the error form 𝐸_𝑛(𝜉) given by the formula (17). For a given compact subset 𝐾, choose a number 𝑛 such that |𝜉|≤𝑥_1,𝑛 for 𝜉∈𝐾. We may take the path of integration 𝐶_𝑛 to be the circle |𝑧|=5𝑎_𝑛 because 𝑥_1,𝑛≤𝑎_𝑛. By (42), we see |𝑝_𝑛(𝑤;𝜉)|≤𝛾_𝑛2^𝑛/2𝑎^𝑛_𝑛. From the definition of 𝜂 and (24), for any 𝜀>0, there exists 𝑁_𝜀>0 such that for 𝑛≥𝑁_𝜀, 𝑄𝑎_𝑛𝑇𝑎_𝑛≤𝜂+𝜀𝑛,log𝑀_𝑓5𝑎_𝑛≤1𝜂+𝜀𝑇𝑎_𝑛𝑄𝑎_𝑛. (45) Hence by (43), for 𝑛 large enough, 𝐸_𝑛(𝜉)≤𝐶𝛾_𝑛2^𝑛/2𝑎^𝑛_𝑛1𝛾_𝑛5𝑎_𝑛^𝑛×exp𝑛𝑥²_1,𝑛25𝑎_𝑛²−𝑥²_1,𝑛𝑀_𝑓5𝑎_𝑛2𝜋_𝐶𝑛|𝑑𝑧|4𝑎_𝑛≤𝐶2^𝑛/215^𝑛𝑛exp148exp𝜂+𝜀𝑄𝑎_𝑛𝑇𝑎_𝑛≤𝐶2^𝑛/215^𝑛𝑛exp148exp𝜂+𝜀𝜂+𝜀𝑛. (46) Here we select 𝜀>0 such as 1+148𝜂+𝜀𝜂+𝜀≤25.24 (47) Then we have for 𝑛≥𝑁_𝜀 and any 𝜉∈𝐾, 𝐸_𝑛(𝜉√)≤𝐶25^𝑛exp25𝑛√24=𝐶25exp2524^𝑛. (48) Since (√2/5)exp(25/24)=∶𝑐<1, we conclude (26). In the above proof, we only exchange (48) for 𝐸_𝑛,𝑚(𝜉√)≤𝐶25^𝑛𝑚exp49𝑛+𝑛𝑚√48≤𝐶25exp2524^𝑛𝑚. (49) Then (27) follows.
(2) Since 𝑓^(𝑗)(𝑧)=𝑗!2𝜋𝑖_𝐶2𝑅𝑓(𝜁)(𝜁−𝑧)^𝑗+1𝑑𝜁, (50) where 𝐶_2𝑅={𝜁;|𝜁|=2𝑅}, we see 𝑀_𝑓^(𝑗)(𝑅)≤2(𝑗!)𝑀_𝑓(2𝑅)/𝑅^𝑗. Hence, (28) gives us limsup_𝑅→∞log𝑀_𝑓^(𝑗)(5𝑅)𝑇(𝑅)𝑄(𝑅)≤limsup_𝑅→∞log2𝑗!/𝑅^𝑗+log𝑀_𝑓(10𝑅)𝑇(𝑅)𝑄(𝑅)≤limsup_𝑅→∞log𝑀_𝑓(10𝑅)𝑇(𝑅)𝑄(𝑅)≤1𝜂. (51) Thus, (29) and (30) follow from (26) and (27), respectively.

Proof of Theorem 7. First, we show the case 𝑗=0. From Lemma 5(3) and (44), we have ^[𝑛/2]_𝑘=1𝑥²_𝑘,𝑛≤𝐶^𝑛−1_𝑘=1𝑎²_𝑘≤𝐶𝑎²_𝑛−1(𝑛−1), (52) where 𝐶>0 is an absolutely constant. Also by Lemma 5(3), we see 1𝛾_𝑛𝑎≤𝐶^𝑛𝛾_𝑛−1≤⋯≤𝐶^𝑛𝑎_𝑛𝑎_𝑛−1⋯𝑎₁≤𝐶𝑎_𝑛^𝑛. (53) Now let |𝑧|>𝑥_1,𝑛. Since 𝑝_𝑛(𝑤;𝑧)⁻¹≤1𝛾_𝑛|𝑧|^𝑛1exp|𝑧|²−𝑥²_1,𝑛^[𝑛/2]_𝑗=1𝑥²_𝑗,𝑛, (54) (see [3, page 131]), (52) gives us 𝑝_𝑛(𝑤∶𝑧)⁻¹≤1𝛾_𝑛|𝑧|^𝑛exp𝐶𝑎²_𝑛−1(𝑛−1)|𝑧|²−𝑥²_1,𝑛. (55) Therefore, for |𝑧|>𝑥_1,𝑛+1, we conclude that 𝑝_𝑛(𝑤;𝑧)𝑝_𝑛+1(𝑤;𝑧)⁻¹≤1𝛾_𝑛𝛾_𝑛+1|𝑧|^2𝑛+1exp2𝐶𝑎²_𝑛𝑛|𝑧|²−𝑥²_1,𝑛+1. (56) We consider the circle 𝐶_𝑛={𝑧;|𝑧|=𝜆₀𝑎_𝑛}, where we will choose 𝜆₀>0 large enough later. Now we estimate 𝐼_𝑛𝛾∶=^𝑛+1𝛾_𝑛12𝜋𝑖_𝐶𝑛𝑓(𝑧)𝑑𝑧𝑝_𝑛(𝑤;𝑧)𝑝_𝑛+1().𝑤;𝑧 (57) From Lemma 5(1), (53), (56), and our assumption (31), we have for 𝛿>0 small enough, 𝐼_𝑛≤1𝛾²_𝑛1𝜆₀𝑎_𝑛^2𝑛+1𝑀_𝑓𝜆₀𝑎_𝑛exp2𝐶𝑎²_𝑛𝑛𝜆₀𝑎_𝑛²−𝑥²_1,𝑛𝜆₀𝑎_𝑛≤𝐶𝑎_𝑛^2𝑛exp1/𝜂+𝛿𝜂+𝛿𝑛𝜆₀𝑎_𝑛^2𝑛exp2𝐶𝑛𝜆²₀≤𝐶−1𝜆₀²1exp𝜂𝐶+𝛿𝜂+𝛿exp𝜆²₀−1^𝑛. (58) Here we select 𝜆₀ as 𝐶0<𝑐∶=𝜆₀²1exp𝜂+𝐶+𝛿𝜂+𝛿𝜆²₀−1<1. (59) Then we have 𝐸_𝑛,∗≤^∞_𝑗=𝑛𝐼_𝑗≤𝑐^𝑛,1−𝑐 (60) which shows that limsup_𝑛→∞𝐸_𝑛,∗^1/𝑛≤𝑐limsup_𝑛→∞11−𝑐^1/𝑛=𝑐<1. (61) For the case of 𝑗⩾1, as in the proof of (2) of Theorem 6, (33) gives us limsup_𝑅→∞log𝑀_𝑓^(𝑗)𝜆₀𝑅𝑇(𝑅)𝑄(𝑅)≤limsup_𝑅→∞𝑀_𝑓2𝜆₀𝑅𝑇(𝑅)𝑄(𝑅)≤1𝜂. (62) Hence, (34) follows from (32).

Proof of Theorem 8. In (17), we take the path of integration 𝐶_𝑛 to be the circle |𝑧|=𝜆𝑎_𝑛  (𝜆>0). Then we have 𝐸_𝑛(𝜉√)≤𝐶2𝜆𝜆exp²𝜆²−1^𝑛. (63) In this right-hand side, we can take 𝜆 as large as we like because of our assumption (35). Hence, we have (36). Similarly as above, (37) follows from (36). Also Theorem 7 shows (38).