Possible Intervals for 𝑇- and 𝑀-Orders of Solutions of Linear Differential Equations in the Unit Disc

Abstract

In the case of the complex plane, it is known that there exists a finite set of rational numbers containing all possible growth orders of solutions of 𝑓^(𝑘)+𝑎_𝑘−1(𝑧)𝑓^(𝑘−1)+⋯+𝑎₁(𝑧)𝑓^′+𝑎₀(𝑧)𝑓=0 with polynomial coefficients. In the present paper, it is shown by an example that a unit disc counterpart of such finite set does not contain all possible 𝑇- and 𝑀-orders of solutions, with respect to Nevanlinna characteristic and maximum modulus, if the coefficients are analytic functions belonging either to weighted Bergman spaces or to weighted Hardy spaces. In contrast to a finite set, possible intervals for 𝑇- and 𝑀-orders are introduced to give detailed information about the growth of solutions. Finally, these findings yield sharp lower bounds for the sums of 𝑇- and 𝑀-orders of functions in the solution bases.

1. Introduction

This research is a continuation of recent activity in the field of complex differential equations. In particular, the present paper concerns linear differential equations of the type 𝑓^(𝑘)+𝑎_𝑘−1(𝑧)𝑓^(𝑘−1)+⋯+𝑎₁(𝑧)𝑓^′+𝑎₀(𝑧)𝑓=0,(1.1) where the coefficients 𝑎₀(𝑧),…,𝑎_𝑘−1(𝑧) are analytic functions in the unit disc 𝔻∶={𝑧∶|𝑧|<1} of the complex plane ℂ. A variety of publications in the existing literature illustrate that the connection between the growth of coefficient functions and the growth of solutions is relatively well understood. On the one hand, the growth estimates in [1] have been proven to be instrumental tools in estimating the growth of solutions when the growth of coefficients is known. On the other hand, proofs of the converse direction have taken advantage of the method of order reduction as well as different types of logarithmic derivative estimates.

For an analytic function in 𝔻, it is known that 𝑇- and 𝑀-orders of growth, with respect to Nevanlinna characteristic and maximum modulus, are not equal in general. This is in contrast to the corresponding case in ℂ. Hence, there are two distinct cases in 𝔻 to work with. First, if the growth of solutions is measured by using the 𝑇-order, then it is natural to express the other growth aspects by means of integration as well. In particular, it is reasonable to consider coefficient functions belonging to some weighted Bergman spaces and use integrated estimates for logarithmic derivatives [2]. Second, if the growth of solutions is measured by using the 𝑀-order, then it is natural to express the other growth aspects by means of the maximum modulus function. In particular, it is sensible to restrict the growth of the maximum modulus of coefficient functions, which leads to weighted Hardy spaces, and work with estimates for the maximum modulus of logarithmic derivatives involving exceptional sets [3].

The main focus of this paper is in improving the lower bounds for the growth of solutions of (1.1) given in [2, 3] and explore some consequences, which are motivated by the following observations.

By the classical results in ℂ making use of Newton-Puiseux diagram, there is a finite set containing the possible growth orders of solutions of (1.1) assuming that coefficients are polynomials. In particular, Gundersen-Steinbart-Wang showed that this finite set consists of rational numbers obtained from simple arithmetic with the degrees of the polynomial coefficients in (1.1) [4, Theorem  1]. Their proof relies on classical Wiman-Valiron theory in ℂ. Even though a recent unit disc counterpart of Wiman-Valiron theory [5] has been successfully applied to differential equations, the possible orders of solutions of (1.1) in 𝔻 have been obtained only by assuming that coefficients are 𝛼-polynomial regular. These 𝛼-polynomial regular functions have similar growth properties than polynomials in the sense that maximal growth is attained in every direction. However, they appear to be only a relatively small subset of the Korenblum space, which characterizes finite-order solutions of (1.1) in 𝔻 [6, Theorem  6.1]. Note that in the case of ℂ, all solutions of (1.1) are of finite order if and only if coefficients are polynomials [7, Satz  1].

In the present paper, it is shown by an example that a unit disc counterpart of the finite set constructed by Gundersen-Steinbart-Wang does not contain all possible orders of solutions of (1.1), provided that the coefficients belong either to weighted Bergman spaces or to weighted Hardy spaces. In contrast to a finite set, we introduce possible intervals for 𝑇-orders and 𝑀-orders, giving detailed information about the growth of solutions. Finally, these findings are applied to estimate the sums of 𝑇- and 𝑀-orders of functions in the solution bases of (1.1) from below.

2. Results and Motivation

The results concerning 𝑇- and 𝑀-orders of solutions of (1.1) are given, respectively, in Sections 2.1-2.2 and 2.3-2.4. Due to the similarities of the assertions, we omit the proofs of results regarding 𝑀-orders of solutions of (1.1), excluding the sketched proof of Theorem 2.5 in Section 7.

Let ℳ(𝔻) and ℋ(𝔻) denote the sets of all meromorphic and analytic functions in 𝔻. For simplicity, we write 𝛼⁺∶=max{𝛼,0} for any 𝛼∈ℝ, |𝑓(𝑧)|≲|𝑔(𝑧)| if there exists a constant 𝐶>0 independent of 𝑧 such that |𝑓(𝑧)|≤𝐶|𝑔(𝑧)|, and 𝑓(𝑧)∼𝑔(𝑧) if there exist constants 𝐶₁>0 and 𝐶₂>0 independent of 𝑧 such that 𝐶₁|𝑔(𝑧)|≤|𝑓(𝑧)|≤𝐶₂|𝑔(𝑧)|.

2.1. Growth of Solutions with Respect to Nevanlinna Characteristic

The 𝑇-order of growth of 𝑓∈ℳ(𝔻) is defined as 𝜎_𝑇𝑓∶=limsup_𝑟→1−log⁺𝑇𝑟,𝑓(),−log1−𝑟(2.1) where 𝑇(𝑟,𝑓) is the Nevanlinna characteristic of 𝑓. For 𝑝>0 and 𝛼>−1, the weighted Bergman space 𝐴^𝑝_𝛼 consists of those 𝑓∈ℋ(𝔻) for which 𝑓_𝐴^𝑝_𝛼∶=_𝔻𝑓(𝑧)^𝑝1−|𝑧|^2𝛼𝑑𝑚(𝑧)^1/𝑝<∞.(2.2) Functions of maximal growth in ⋂_{𝑞<𝛼<∞}𝐴^𝑝_𝛼 are distinguished by denoting 𝑓∈𝔸^𝑝_𝑞, if 𝑞=inf{𝛼>−1∶𝑓∈𝐴^𝑝_𝛼}.

If the growth of the coefficients is expressed by means of integration, then it is natural to consider the growth of solutions of (1.1) with respect to 𝑇-order.

Theorem A (see [2, Theorems  1 and  2]). Suppose that 𝑎_𝑗∈𝔸_𝛼^{1/(𝑘−𝑗)}_𝑗, where 𝛼_𝑗≥0 for 𝑗=0,…,𝑘−1, and denote 𝛼_𝑘∶=0. (i)Let 0≤𝛼<∞. Then all solutions 𝑓 of (1.1) satisfy 𝜎_𝑇(𝑓)≤𝛼 if and only if max_{𝑗=0,…,𝑘−1}{𝛼_𝑗}≤𝛼.(ii)All nontrivial solutions 𝑓 of (1.1) satisfy min_{𝑗=1,…,𝑘}𝑘𝛼₀−𝛼_𝑗𝑗+𝛼_𝑗≤𝜎_𝑇𝑓≤max_{𝑗=0,…,𝑘−1}𝛼_𝑗.(2.3)(iii)If 𝑞∈{0,…,𝑘−1} is the smallest index for which 𝛼_𝑞=max_{𝑗=0,…,𝑘−1}{𝛼_𝑗}, then each solution base of (1.1) contains at least 𝑘−𝑞 linearly independent solutions 𝑓 such that 𝜎_𝑇(𝑓)=𝛼_𝑞.

The assumption 𝑎_𝑗∈𝔸_𝛼^{1/(𝑘−𝑗)}_𝑗 in Theorem A(i) cannot be replaced by 𝑎_𝑗∈𝐴_𝛼^{1/(𝑘−𝑗)}_𝑗; see [8]. We refine Theorem A and then further underscore its consequences.

Theorem 2.1. Suppose that 𝑎_𝑗∈𝔸_𝛼^{1/(𝑘−𝑗)}_𝑗, where 𝛼_𝑗≥−1 for 𝑗=0,…,𝑘−1, and let 𝑞∈{0,…,𝑘−1} be the smallest index for which 𝛼_𝑞=max_{𝑗=0,…,𝑘−1}{𝛼_𝑗}. If 𝑠∈{0,…,𝑞}, then each solution base of (1.1) contains at least 𝑘−𝑠 linearly independent solutions 𝑓 such that min_{𝑗=𝑠+1,…,𝑘}(𝑘−𝑠)𝛼_𝑠−𝛼_𝑗𝑗−𝑠+𝛼_𝑗≤𝜎_𝑇𝑓≤𝛼⁺_𝑞,(2.4) where 𝛼_𝑘∶=−1.

The case 𝑠=0 clearly reduces to Theorem A(ii). If 𝑠=𝑞, then the condition 𝛼_𝑞=max_{𝑗=0,…,𝑘−1}{𝛼_𝑗} implies that min_{𝑗=𝑞+1,…,𝑘}𝑘−𝑞𝛼_𝑞−𝛼_𝑗𝑗−𝑞+𝛼_𝑗=min_{𝑗=𝑞+1,…,𝑘}𝑘−𝑗𝛼_𝑞−𝛼_𝑗𝑗−𝑞+𝛼_𝑞=𝛼_𝑞,(2.5) where the minimum is attained for 𝑗=𝑘. Hence the assertion of Theorem 2.1 for 𝑠=𝑞 is contained in Theorem A(iii). Our contribution is to extend the first inequality in (2.4) for 𝑠∈{1,…,𝑞−1}. Theorem 2.1 is proved in Section 4, and the sharpness and the special cases 𝑘=2 and 𝑘=3 are further discussed in Section 3.1.

Let 𝑞∈{0,…,𝑘−1} be the smallest index for which 𝛼_𝑞=max_{𝑗=0,…,𝑘−1}{𝛼_𝑗}. If 𝛼_𝑞≤0, then all solutions in each solution base of (1.1) are of zero 𝑇-order by Theorem A(ii). Suppose that 𝛼_𝑞>0. In order to state the following corollaries of Theorem 2.1, we denote 𝛽_𝑇(𝑠)∶=min_{𝑗=𝑠+1,…,𝑘}(𝑘−𝑠)𝛼_𝑠−𝛼_𝑗𝑗−𝑠+𝛼_𝑗,𝑠=0,…,𝑞,(2.6) where 𝛼_𝑘∶=−1. Moreover, we define 𝑠^⋆∶=min𝑠∈0,…,𝑞∶𝛽_𝑇(𝑠)>0.(2.7) Remark that 𝛽_𝑇(𝑞)>0, since (2.5) implies 𝛼_𝑞=𝛽_𝑇(𝑞).

Corollary 2.2. Suppose that 𝑎_𝑗∈𝔸_𝛼^{1/(𝑘−𝑗)}_𝑗, where 𝛼_𝑗≥−1 for 𝑗=0,…,𝑘−1, and let 𝑞∈{0,…,𝑘−1} be the smallest index for which 𝛼_𝑞=max_{𝑗=0,…,𝑘−1}{𝛼_𝑗}>0. Then each solution base of (1.1) admits at most 𝑠^⋆≤𝑞 solutions 𝑓 satisfying 𝜎_𝑇(𝑓)<𝛽_𝑇(𝑠^⋆). In particular, there are at most 𝑠^⋆≤𝑞 solutions 𝑓 satisfying 𝜎_𝑇(𝑓)=0.

To estimate the quantity ∑^𝑘_𝑗=1𝜎_𝑇(𝑓_𝑗) by using Theorem 2.1, we set 𝛾_𝑇𝑗∶=max𝛽_𝑇(0),…,𝛽_𝑇𝑗,𝑗=0,…,𝑞.(2.8) Evidently 𝛾_𝑇(𝑗)>0 for 𝑗∈{𝑠^⋆,…,𝑞}, and 𝛾_𝑇(𝑗)≤0 for 𝑗∈{0,…,𝑠^⋆−1}.

Corollary 2.3. Suppose that 𝑎_𝑗∈𝔸_𝛼^{1/(𝑘−𝑗)}_𝑗, where 𝛼_𝑗≥−1 for 𝑗=0,…,𝑘−1, and let 𝑞∈{0,…,𝑘−1} be the smallest index for which 𝛼_𝑞=max_{𝑗=0,…,𝑘−1}{𝛼_𝑗}>0. Let {𝑓₁,…,𝑓_𝑘} be a solution base of (1.1). If 𝑞=0, then ∑^𝑘_𝑗=1𝜎_𝑇(𝑓_𝑗)=𝑘𝛼₀, while if 𝑞≥1, then 𝑘−𝑞𝛼_𝑞+^𝑞−1_𝑗=𝑠⋆𝛾_𝑇𝑗≤^𝑘_𝑗=1𝜎_𝑇𝑓_𝑗≤𝑘𝛼_𝑞.(2.9)

Note that the sum in (2.9) is considered to be empty, if 𝑠^⋆=𝑞.

2.2. Gundersen-Steinbart-Wang Method for 𝑇-Order

We proceed to give an alternative statement of Theorem 2.1 and its corollaries by modifying the key steps in [4]. This yields a natural way to define possible intervals for 𝑇-orders of solutions of (1.1). As a consequence, we get a useful estimate following from Corollary 2.3.

Set 𝛿_𝑗∶=(𝛼_𝑗+1)(𝑘−𝑗) for all 𝑗=0,…,𝑘−1. Let 𝑠₁∈{0,…,𝑘−1} be the smallest index satisfying 𝛼_𝑠1=max_{𝑗=0,…,𝑘−1}{𝛼_𝑗}>0, which is equivalent to 𝛿_𝑠1𝑘−𝑠₁=max_{𝑗=0,…,𝑘−1}𝛿_𝑗𝑘−𝑗>1.(2.10) If 𝑠₁ cannot be found, then all solutions of (1.1) are of zero 𝑇-order by Theorem A(ii). Otherwise, for a given 𝑠_𝑚, 𝑚∈ℕ, let 𝑠_𝑚+1∈{0,…,𝑠_𝑚−1} be the smallest index satisfying 𝛿_𝑠𝑚+1−𝛿_𝑠𝑚𝑠_𝑚−𝑠_𝑚+1=max_{𝑗=0,…,𝑠𝑚−1}𝛿_𝑗−𝛿_𝑠𝑚𝑠_𝑚−𝑗>1.(2.11) Eventually this process will stop, yielding a finite list of indices 𝑠₁,…,𝑠_𝑝 such that 𝑝≤𝑘 and 𝑠₁>𝑠₂>⋯>𝑠_𝑝≥0. Further, set ℬ_𝑇(𝑡)𝛿∶=^𝑠𝑡−𝛿_𝑠𝑡−1𝑠_𝑡−1−𝑠_𝑡−1,𝑡=1,…,𝑝,(2.12) where 𝑠₀∶=𝑘 and 𝛿_𝑘∶=0. Due to resemblance between (2.12) and [4, Equation (2.4)], it seems plausible that the possible nonzero 𝑇-orders of solutions of (1.1) in the unit disc case could be found among the numbers ℬ_𝑇(𝑡), where 𝑡=1,…,𝑝. However, Example 3.1 shows that this is not the case.

The following lemma allows us to view the results in Section 2.1 in a new perspective. In particular, Lemma 2.4 emphasizes the connection between ℬ_𝑇 and 𝛾_𝑇.

Lemma 2.4. One has the following: (i)ℬ_𝑇(1)>ℬ_𝑇(2)>⋯>ℬ_𝑇(𝑝)>0; (ii)𝛽_𝑇(𝑠_𝑡)=ℬ_𝑇(𝑡) for all 𝑡∈{1,…,𝑝}; (iii)𝛾_𝑇(𝑞)=ℬ_𝑇(1), 𝛾_𝑇(𝑗)=ℬ_𝑇(𝑡) for all 𝑠_𝑡≤𝑗<𝑠_𝑡−1 and 𝑡∈{2,…,𝑝}, and 𝛾_𝑇(𝑗)≤0 for all 𝑗<𝑠_𝑝. In particular, 𝑠_𝑝=𝑠^⋆.

By relying on Lemma 2.4, Theorem 2.1, and Corollary 2.2, we proceed to state possible intervals for 𝑇-orders of functions in solution bases of (1.1) in the case 𝑎_𝑗∈𝔸_𝛼^{1/(𝑘−𝑗)}_𝑗, where 𝛼_𝑗≥−1 for 𝑗=0,…,𝑘−1. In fact, each solution base of (1.1) contains the following: (i)at least 𝑘−𝑠₁ solutions 𝑓 satisfying 𝜎_𝑇(𝑓)=ℬ_𝑇(1); (ii)at least 𝑘−𝑠_𝑡 solutions 𝑓 satisfying 𝜎_𝑇(𝑓)∈[ℬ_𝑇(𝑡),ℬ_𝑇(1)] for 𝑡=2,…,𝑝; (iii)at most 𝑠_𝑝 solutions 𝑓 satisfying 𝜎_𝑇(𝑓)∈[0,ℬ_𝑇(𝑠_𝑝)). For the following application, let {𝑓₁,…,𝑓_𝑘} be a solution base of (1.1). Knowing the possible intervals for 𝑇-orders, we get ^𝑘_𝑗=1𝜎_𝑇𝑓_𝑗≥𝑘−𝑠₁ℬ_𝑇(1)+⋯+𝑠_𝑝−1−𝑠_𝑝ℬ_𝑇𝑝+𝑠_𝑝⋅0=𝛿_𝑠𝑝+𝑠_𝑝−𝑘.(2.13) In view of Lemma 2.4, the lower estimates in (2.9) and (2.13) are equal.

Finally, we point out a useful consequence of (2.13). If 𝑠_𝑝=0, then 𝛿_𝑠𝑝+𝑠_𝑝=𝛿₀. If 𝑠_𝑝>0, then (𝛿₀−𝛿_𝑠𝑝)/𝑠_𝑝≤1 by (2.11), and 𝛿_𝑠𝑝+𝑠_𝑝≥𝛿₀. Hence, in both cases we can state that ^𝑘_𝑗=1𝜎_𝑇𝑓_𝑗≥𝛿_𝑠𝑝+𝑠_𝑝−𝑘≥𝛿₀−𝑘≥𝛼₀𝑘,(2.14) where the equalities hold, if 𝛼₀=max_{𝑗=0,…,𝑘−1}{𝛼_𝑗}>0.

2.3. Growth of Solutions with Respect to Maximum Modulus

Alongside of the 𝑇-order, we may also define the 𝑀-order of growth of 𝑓∈ℋ(𝔻) by 𝜎_𝑀𝑓∶=limsup_𝑟→1−log⁺log⁺𝑀𝑟,𝑓(),−log1−𝑟(2.15) where 𝑀(𝑟,𝑓)∶=max_|𝑧|=𝑟|𝑓(𝑧)| is the maximum modulus of 𝑓. It is well known that the inequalities 𝜎_𝑇𝑓≤𝜎_𝑀𝑓≤𝜎_𝑇𝑓+1(2.16) are satisfied for all 𝑓∈ℋ(𝔻), and all possibilities allowed by (2.16) can be assumed [9, Theorems  3.5–3.7]. A function 𝑓∈ℋ(𝔻) is said to belong to the weighted Hardy space 𝐻^∞_𝛼, if there exists 𝛼≥0 such that sup_𝑧∈𝔻1−|𝑧|^2𝛼𝑓(𝑧)<∞.(2.17) Functions of maximal growth in ⋂_𝛼>𝑝𝐻^∞_𝛼 are distinguished by denoting 𝑓∈ℍ^∞_𝑝, if 𝑝=inf{𝛼≥0∶𝑓∈𝐻^∞_𝛼}. Remark that 𝐻^∞₀=𝐻^∞ is the space of all bounded analytic functions in 𝔻. The union ⋃_𝛼>0𝐻^∞_𝛼 is also known as the Korenblum space 𝒜^−∞ [10], and since [11] ℍ^∞_𝑝 is also known as 𝐺_𝑝.

If the growth of coefficients is measured by means of maximum modulus estimates, then it is natural to consider the growth of solutions with respect to 𝑀-order.

Theorem B (see [3, Theorem  1.4]). Suppose that 𝑎_𝑗∈ℍ^∞_{(𝑝𝑗+1)(𝑘−𝑗)}, where 𝑝_𝑗≥−1 for 𝑗=0,…,𝑘−1, and denote 𝑝_𝑘∶=−1. (i)Suppose that min_{𝑗=1,…,𝑘}𝑘𝑝₀−𝑝_𝑗𝑗+𝑝_𝑗>1,(2.18) and let 1≤𝛼<∞. Then all solutions 𝑓 of (1.1) satisfy𝜎_𝑀(𝑓)≤𝛼 if and only if max_{𝑗=0,…,𝑘−1}{𝑝_𝑗}≤𝛼.(ii)All nontrivial solutions 𝑓 of (1.1) satisfy 𝜎_𝑀(𝑓)≤max_{𝑗=0,…,𝑘−1}{𝑝⁺_𝑗}, and min_{𝑗=1,…,𝑘}𝑘𝑝₀−𝑝_𝑗𝑗+𝑝_𝑗≤max𝜎_𝑀𝑓,1.(2.19)(iii)Suppose that (2.18) holds. If 𝑞∈{0,…,𝑘−1} is the smallest index for which 𝑝_𝑞=max_{𝑗=0,…,𝑘−1}{𝑝_𝑗}, then each solution base of (1.1) contains at least 𝑘−𝑞 linearly independent solutions 𝑓 such that 𝜎_𝑀(𝑓)=𝑝_𝑞.

To conclude [3, Equation  (4.17)] in the proof of Theorem B, the inequality [3, Equation  (1.9)], corresponding to (2.18), must be strict. By a simple modification of the proof of Theorem B, the assumption (2.18) can be relaxed to max_{𝑗=0,…,𝑘−1}𝑝_𝑗>1,(2.20) which allows us to apply Theorem B(iii) also in the case that there are solutions 𝑓 satisfying 𝜎_𝑀(𝑓)≤1. To see that (2.20) is in fact a weaker assumption than (2.18), we refer to [2, Example  10], which is further considered in Section 3.2. In this case min_{𝑗=1,…,𝑘}𝑘𝑝₀−𝑝_𝑗𝑗+𝑝_𝑗=−4,max_{𝑗=0,…,𝑘−1}𝑝_𝑗>1.(2.21) Note that by taking 𝑗=𝑘 in (2.18), we obtain 𝑝₀>1. Hence (2.18) implies (2.20).

Theorem 2.5 corresponds to Theorem 2.1.

Theorem 2.5. Suppose that 𝑎_𝑗∈ℍ^∞_{(𝑝𝑗+1)(𝑘−𝑗)}, where 𝑝_𝑗≥−1 for 𝑗=0,…,𝑘−1, and let 𝑞∈{0,…,𝑘−1} be the smallest index for which 𝑝_𝑞=max_{𝑗=0,…,𝑘−1}{𝑝_𝑗}. If 𝑠∈{0,…,𝑞}, then each solution base of (1.1) contains at least 𝑘−𝑠 linearly independent solutions 𝑓 such that min_{𝑗=𝑠+1,…,𝑘}(𝑘−𝑠)𝑝_𝑠−𝑝_𝑗𝑗−𝑠+𝑝_𝑗≤max𝜎_𝑀𝑓,1.(2.22)

Note that (2.22) gives information on 𝜎_𝑀(𝑓) only in the case when the minimum in (2.22) is strictly greater than 1. The case 𝑠=0 in Theorem 2.5 reduces to Theorem B(ii), and the case 𝑠=𝑞 reduces to Theorem B(iii) with the assumption (2.20), since now min_{𝑗=𝑞+1,…,𝑘}𝑘−𝑞𝑝_𝑞−𝑝_𝑗𝑗−𝑞+𝑝_𝑗=𝑝_𝑞=max_{𝑗=0,…,𝑘−1}𝑝_𝑗,(2.23) where the minimum is attained for 𝑗=𝑘. For a similar argumentation, see (2.5). Our contribution is to extend (2.22) for 𝑠∈{1,…,𝑞−1}. The proof of Theorem 2.5 is sketched in Section 7, and the sharpness and the the special cases 𝑘=2 and 𝑘=3 are further discussed in Section 3.2.

Let 𝑞∈{0,…,𝑘−1} be the smallest index for which 𝑝_𝑞=max_{𝑗=0,…,𝑘−1}{𝑝_𝑗}. If 𝑝_𝑞≤1, then all solutions 𝑓 in each solution base of (1.1) satisfy 𝜎_𝑀(𝑓)≤1 by Theorem B(ii). Suppose that 𝑝_𝑞>1. In order to state the following corollaries of Theorem 2.5, we denote 𝛽_𝑀(𝑠)∶=min_{𝑗=𝑠+1,…,𝑘}(𝑘−𝑠)𝑝_𝑠−𝑝_𝑗𝑗−𝑠+𝑝_𝑗,𝑠=0,…,𝑞,(2.24) where 𝑝_𝑘∶=−1. Moreover, we define 𝑠^⋆∶=min𝑠∈0,…,𝑞∶𝛽_𝑀(𝑠)>1.(2.25) Remark that 𝛽_𝑀(𝑞)>1, since (2.23) implies 𝑝_𝑞=𝛽_𝑀(𝑞).

Corollary 2.6. Suppose that 𝑎_𝑗∈ℍ^∞_{(𝑝𝑗+1)(𝑘−𝑗)}, where 𝑝_𝑗≥−1 for 𝑗=0,…,𝑘−1, and let 𝑞∈{0,…,𝑘−1} be the smallest index for which 𝑝_𝑞=max_{𝑗=0,…,𝑘−1}{𝑝_𝑗}>1. Then each solution base of (1.1) admits at most 𝑠^⋆≤𝑞 solutions 𝑓 satisfying 𝜎_𝑀(𝑓)<𝛽_𝑀(𝑠^⋆). In particular, there are at most 𝑠^⋆≤𝑞 solutions 𝑓 satisfying 𝜎_𝑀(𝑓)≤1.

To estimate the quantity ∑^𝑘_𝑗=1𝜎_𝑀(𝑓_𝑗) by using Theorem 2.5, we set 𝛾_𝑀𝑗∶=max𝛽_𝑀(0),…,𝛽_𝑀𝑗,𝑗=0,…,𝑞.(2.26) Evidently 𝛾_𝑀(𝑗)>1 for 𝑗∈{𝑠^⋆,…,𝑞}, and 𝛾_𝑀(𝑗)≤1 for 𝑗∈{0,…,𝑠^⋆−1}.

Corollary 2.7. Suppose that 𝑎_𝑗∈ℍ^∞_{(𝑝𝑗+1)(𝑘−𝑗)}, where 𝑝_𝑗≥−1 for 𝑗=0,…,𝑘−1, and let 𝑞∈{0,…,𝑘−1} be the smallest index for which 𝑝_𝑞=max_{𝑗=0,…,𝑘−1}{𝑝_𝑗}>1. Let {𝑓₁,…,𝑓_𝑘} be a solution base of (1.1). If 𝑞=0, then ∑^𝑘_𝑗=1𝜎_𝑀(𝑓_𝑗)=𝑘𝑝₀, while if 𝑞≥1, then 𝑘−𝑞𝑝_𝑞+^𝑞−1_𝑗=𝑠⋆𝛾_𝑀𝑗≤^𝑘_𝑗=1𝜎_𝑀𝑓_𝑗≤𝑘𝑝_𝑞.(2.27)

Note that the sum in (2.27) is considered to be empty, if 𝑠^⋆=𝑞.

2.4. Gundersen-Steinbart-Wang Method for 𝑀-Order

We proceed to give an alternative statement of Theorem 2.5 and its corollaries by modifying the key steps in [4]. This yields a natural way to define the possible intervals for 𝑀-orders of solutions of (1.1). As a consequence, we get a useful estimate following from Corollary 2.7.

Set 𝛿_𝑗∶=(𝑝_𝑗+1)(𝑘−𝑗) for all 𝑗=0,…,𝑘−1. Let 𝑠₁∈{0,…,𝑘−1} be the smallest index satisfying 𝑝_𝑠1=max_{𝑗=0,…,𝑘−1}{𝑝_𝑗}>1, which is equivalent to 𝛿_𝑠1𝑘−𝑠₁=max_{𝑗=0,…,𝑘−1}𝛿_𝑗𝑘−𝑗>2.(2.28) If 𝑠₁ cannot be found, then all solutions 𝑓 of (1.1) satisfy 𝜎_𝑀(𝑓)≤1 by Theorem B(ii). Otherwise, for a given 𝑠_𝑚, 𝑚∈ℕ, let 𝑠_𝑚+1∈{0,…,𝑠_𝑚−1} be the smallest index satisfying 𝛿_𝑠𝑚+1−𝛿_𝑠𝑚𝑠_𝑚−𝑠_𝑚+1=max_{𝑗=0,…,𝑠𝑚−1}𝛿_𝑗−𝛿_𝑠𝑚𝑠_𝑚−𝑗>2.(2.29) Eventually this process will stop, yielding a finite list of indices 𝑠₁,…,𝑠_𝑝 such that 𝑝≤𝑘 and 𝑠₁>𝑠₂>⋯>𝑠_𝑝≥0. Further, set ℬ_𝑀(𝑡)𝛿∶=^𝑠𝑡−𝛿_𝑠𝑡−1𝑠_𝑡−1−𝑠_𝑡−1,𝑡=1,…,𝑝,(2.30) where 𝑠₀∶=𝑘 and 𝛿_𝑘∶=0. By Example 3.1, it is possible that (1.1) possesses a solution 𝑓 of 𝑀-order strictly greater than one such that 𝜎_𝑀(𝑓)≠ℬ_𝑀(𝑡) for all 𝑡=1,…,𝑝.

The following lemma, which can be proved similarly than Lemma 2.4, allows us to view the results in Section 2.3 in a new perspective.

Lemma 2.8. One has the following: (i)ℬ_𝑀(1)>ℬ_𝑀(2)>⋯>ℬ_𝑀(𝑝)>1; (ii)𝛽_𝑀(𝑠_𝑡)=ℬ_𝑀(𝑡) for all 𝑡∈{1,…,𝑝}; (iii)𝛾_𝑀(𝑞)=ℬ_𝑀(1), 𝛾_𝑀(𝑗)=ℬ_𝑀(𝑡) for all 𝑠_𝑡≤𝑗<𝑠_𝑡−1 and 𝑡∈{2,…,𝑝}, and 𝛾_𝑀(𝑗)≤1 for all 𝑗<𝑠_𝑝. In particular, 𝑠_𝑝=𝑠^⋆.

By relying on Lemma 2.8, Theorem 2.5, and Corollary 2.6, we proceed to state possible intervals for 𝑀-orders of functions in solution bases of (1.1) in the case 𝑎_𝑗∈ℍ^∞_{(𝑝𝑗+1)(𝑘−𝑗)}, where 𝑝_𝑗≥−1 for 𝑗=0,…,𝑘−1. In fact, each solution base of (1.1) contains the following: (i)at least 𝑘−𝑠₁ solutions 𝑓 satisfying 𝜎_𝑀(𝑓)=ℬ_𝑀(1); (ii)at least 𝑘−𝑠_𝑡 solutions 𝑓 satisfying 𝜎_𝑀(𝑓)∈[ℬ_𝑀(𝑡),ℬ_𝑀(1)] for 𝑡=2,…,𝑝; (iii)at most 𝑠_𝑝 solutions 𝑓 satisfying 𝜎_𝑀(𝑓)∈[0,ℬ_𝑀(𝑠_𝑝)).

For results of the same type, we refer to [12, Theorem  1] and [13,  Corollary   1]. To compare (i) and (ii) to the estimates given in [13, Corollary  1], note that there is −1 in (2.30) instead of −2 in [13, Equation  (1.3)]. Evidently, assertions (i) and (ii) improve the estimates given for the 𝑀-orders of solutions in [13, Corollary    1]. Moreover, by means of (2.16) we see that (i) and (ii) reduce to [13, Corollary    1], if we consider the growth of solutions of (1.1) with respect to 𝑇-order.

For the following application, let {𝑓₁,…,𝑓_𝑘} be a solution base of (1.1). Knowing the possible intervals for 𝑀-orders, we get ^𝑘_𝑗=1𝜎_𝑀𝑓_𝑗≥𝑘−𝑠₁ℬ_𝑀(1)+⋯+𝑠_𝑝−1−𝑠_𝑝ℬ_𝑀𝑝+𝑠_𝑝⋅0=𝛿_𝑠𝑝+𝑠_𝑝−𝑘.(2.31) Corresponding to the case in Section 2.2, by means of Lemma 2.8 we see that the lower estimates in (2.27) and (2.31) are equal.

Finally, we point out a practical estimate, which is a consequence of (2.31). If 𝑠_𝑝=0, then 𝛿_𝑠𝑝+𝑠_𝑝=𝛿₀−𝑠_𝑝. If 𝑠_𝑝>0, then (𝛿₀−𝛿_𝑠𝑝)/𝑠_𝑝≤2 by (2.29), and 𝛿_𝑠𝑝+𝑠_𝑝≥𝛿₀−𝑠_𝑝. Hence, in both cases we can state that ^𝑘_𝑗=1𝜎_𝑀𝑓_𝑗≥𝛿₀−𝑠_𝑝−𝑘≥𝑝₀𝑘−𝑠_𝑝.(2.32) We conclude that if 𝑠₁=0, then the equalities hold in (2.32), since in this case 𝑠_𝑝=𝑠₁=0. Note that if (2.18) holds, then we can conclude that 𝑠_𝑝=0.

3. Sharpness Discussion

3.1. Sharpness of Theorem 2.1

We may assume that max_{𝑗=0,…,𝑘−1}{𝛼_𝑗}>0, for otherwise all solutions are of zero 𝑇-order. If 𝑘=2, then the statement of Theorem 2.1 is contained in Theorem A, and all the assertions are sharp [2, Examples  3 and 6].

If 𝑘=3, then we have three different cases to consider. (A1) If 𝛼₁,𝛼₂≤𝛼₀, then all nontrivial solutions 𝑓 of (1.1) satisfy 𝜎_𝑇(𝑓)=𝛼₀. In this case 𝑠=0=𝑞.(A2) If 𝛼₀<𝛼₁ and 𝛼₂≤𝛼₁, then in every solution base {𝑓₁,𝑓₂,𝑓₃} of (1.1) there are at least two solutions 𝑓₁ and 𝑓₂ such that 𝜎_𝑇(𝑓_𝑗)=𝛼₁ for both 𝑗=1,2, and all solutions 𝑓_𝑗 satisfy 𝜎_𝑇𝑓_𝑗≥min3𝛼₀−2𝛼₁,32𝛼₀−12𝛼₂,𝛼₀,𝑗=1,2,3.(3.1) In this case 𝑠=0 or 𝑠=1=𝑞.(A3) If 𝛼₀,𝛼₁<𝛼₂, then in every solution base {𝑓₁,𝑓₂,𝑓₃} of (1.1) there is at least one solution 𝑓₁ such that 𝜎_𝑇(𝑓₁)=𝛼₂, two solutions 𝑓₁ and 𝑓₂ such that 𝜎_𝑇𝑓_𝑗≥min2𝛼₁−𝛼₂,𝛼₁,𝑗=1,2,(3.2) and all solutions 𝑓_𝑗 satisfy (3.1). In this case 𝑠=0, 𝑠=1, or 𝑠=2=𝑞.

It is clear that the assertion in (A1) is sharp, and so are the ones in (A2) by [2, Example   10]. Moreover, [2,  Example   9] shows that the assertions in (A3) are sharp for 𝑠=0 and 𝑠=2. Example 3.2 shows the sharpness of the assertions in (A3) for 𝑠=0,1,2. That is, in all cases there exists a solution for which the lower bound for the 𝑇-order of growth is attained.

3.2. Sharpness of Theorem 2.5

We may assume that max_{𝑗=0,…,𝑘−1}{𝑝_𝑗}>1, for otherwise all solutions 𝑓 of (1.1) satisfy max{𝜎_𝑀(𝑓),1}=1, and we cannot conclude anything from (2.22). If 𝑘=2, then the statement of Theorem 2.5 is contained in Theorem B, and all the assertions are sharp by [2, Examples  3 and 6]. In the case of [2, Examples  3 and 6], for 𝛽>1, linearly independent solutions 𝑓₁ and 𝑓₂ satisfy 𝜎_𝑀(𝑓₁)=𝛽 and 𝜎_𝑀(𝑓₂)=𝛽+2. Moreover, 𝑎_𝑗∈ℍ^∞_{(𝑝𝑗+1)(2−𝑗)}, where 𝑝₀=𝛽+1 and 𝑝₁=𝛽+2. Note that max{𝑝₀,𝑝₁}=𝑝₁=𝛽+2>1, and hence 𝑞=1. An easy computation shows the sharpness for 𝑠=0 and for 𝑠=𝑞=1. In the case of [2, Example  6], for 𝛽>1, linearly independent solutions 𝑓₁ and 𝑓₂ satisfy 𝜎_𝑀(𝑓₁)=𝛽 and 𝜎_𝑀(𝑓₂)=𝛽. Moreover, 𝑎_𝑗∈ℍ^∞_{(𝑝𝑗+1)(2−𝑗)}, where 𝑝₀=𝛽 and 𝑝₁=−1/2. Note now that max{𝑝₀,𝑝₁}=𝑝₀=𝛽>1, and hence 𝑞=0. This example shows the sharpness for 𝑠=𝑞=0. For another example, see [3, Example  2].

If 𝑘=3, then we have three different cases to consider. (B1) If 𝑝₁,𝑝₂≤𝑝₀, then 𝑠=0=𝑞, and all nontrivial solutions 𝑓 of (1.1) satisfy 𝜎_𝑀(𝑓)=𝑝₀ by (2.23).(B2) If 𝑝₀<𝑝₁, and 𝑝₂≤𝑝₁, then in every solution base {𝑓₁,𝑓₂,𝑓₃} of (1.1) there are at least two solutions 𝑓₁ and 𝑓₂ such that 𝜎_𝑀(𝑓_𝑗)=𝑝₁ for both 𝑗=1,2, and all solutions 𝑓_𝑗 satisfy max𝜎_𝑀𝑓_𝑗,1≥min3𝑝₀−2𝑝₁,32𝑝₀−12𝑝₂,𝑝₀,𝑗=1,2,3.(3.3) In this case 𝑠=0 or 𝑠=1=𝑞.(B3)If 𝑝₀,𝑝₁<𝑝₂, then in every solution base {𝑓₁,𝑓₂,𝑓₃} of (1.1) there is at least one solution 𝑓₁ such that 𝜎_𝑀(𝑓₁)=𝑝₂, two solutions 𝑓₁ and 𝑓₂ such that max𝜎_𝑀𝑓_𝑗,1≥min2𝑝₁−𝑝₂,𝑝₁,𝑗=1,2,(3.4) and all solutions 𝑓_𝑗 satisfy (3.3). In this case 𝑠=0, 𝑠=1, or 𝑠=2=𝑞.

It is clear that the assertion in (B1) is sharp. By [2, Example  10], we see that the assertion in (B2) corresponding to 𝑠=1 is sharp. In this case, for 𝛽>1, linearly independent solutions 𝑓₁,𝑓₂, and 𝑓₃ satisfy 𝜎_𝑀(𝑓₁)=𝜎_𝑀(𝑓₂)=𝛽 and 𝜎_𝑀(𝑓₃)=0. Now 𝑎_𝑗∈ℍ^∞_{(𝑝𝑗+1)(3−𝑗)}, where 𝑝₀=(2/3)𝛽, 𝑝₁=𝛽+2, and 𝑝₂=0. Moreover, by [2, Example   9], we see that the assertions in (B3) are sharp for 𝑠=0 and 𝑠=2. In this case for 𝛽>1, linearly independent solutions 𝑓₁, 𝑓₂, and 𝑓₃ satisfy 𝜎_𝑀(𝑓₁)=𝜎_𝑀(𝑓₂)=𝛽 and 𝜎_𝑀(𝑓₃)=2𝛽. Moreover, 𝑎_𝑗∈ℍ^∞_{(𝑝𝑗+1)(3−𝑗)}, where 𝑝₀=(4/3)𝛽, 𝑝₁=𝛽, and 𝑝₂=2𝛽. Example 3.2 shows the sharpness of the assertions in (B3) for 𝑠=0,1,2. That is, in all cases there exists a solution for which equality holds in (2.22).

3.3. Examples

Example 3.1 shows that a unit disc counterpart of the finite set constructed by Gundersen-Steinbart-Wang does not contain growth orders of solutions of 𝑓^′′+𝑎₁(𝑧)𝑓^′+𝑎₀(𝑧)𝑓=0,(3.5) if coefficients belong either to weighted Bergman spaces or to weighted Hardy spaces.

Example 3.1. Let 𝛼,𝛽∈ℝ be constants satisfying 1<𝛽<𝛼<2𝛽−1. Then the functions 𝑓₁(𝑧)=(1−𝑧)^𝛼+𝛽1exp1−𝑧^𝛼+11+𝑧^𝛽,𝑓₂(𝑧)=(1−𝑧)^𝛼+𝛽1exp1+𝑧^𝛽(3.6) are linearly independent analytic solutions of (3.5), where 𝑎₀(𝑧)=𝛽²(1+𝑧)^2+2𝛽+𝛽𝛼+3𝛽𝛾+𝑧(1−𝑧)(1+𝑧)^2+𝛽−𝛼𝛼+𝛽(1−𝑧)^2+𝛼−𝛼𝛽(1−𝑧)^1+𝛼(1+𝑧)^1+𝛽+𝛽𝛼+𝛽(1−𝑧)²,𝑎₁(𝑧)=2𝛽(1+𝑧)^1+𝛽−𝛼(1−𝑧)^1+𝛼+𝛼+2𝛽−11−𝑧(3.7) belong to ℋ(𝔻), and 𝛾=(𝛼+𝛽−2)/(𝛼+3𝛽)∈(0,1/2).
It is clear that 𝑎_𝑗∈𝔸_𝛼^1/(2−𝑗)_𝑗, where 𝛼₀=𝛽−1 and 𝛼₁=𝛼−1. We calculate that 𝑠₁=1, 𝑠₂=0, ℬ_𝑇(1)=𝛼−1, and ℬ_𝑇(2)=2𝛽−𝛼−1. Hence [2𝛽−𝛼−1,𝛼−1] is the only possible interval for 𝑇-orders of solutions of (3.5). Since 𝜎_𝑇(𝑓₂)=𝛽−1, we conclude that the 𝑇-order of a solution does not have to be one of the endpoints.
On the other hand, it is also clear that 𝑎_𝑗∈ℍ^∞_{(𝑝𝑗+1)(2−𝑗)}, where 𝑝₀=𝛽 and 𝑝₁=𝛼. We calculate that 𝑠₁=1, 𝑠₂=0, ℬ_𝑀(1)=𝛼, and ℬ_𝑀(2)=2𝛽−𝛼. Hence [2𝛽−𝛼,𝛼] is the only possible interval for 𝑀-orders of solutions of (3.5). Since 𝜎_𝑀(𝑓₂)=𝛽, we conclude that the 𝑀-order of a solution does not have to be one of the endpoints.
The following example demonstrates the sharpness of Theorems 2.1 and 2.5 in the case that they do not reduce to known results.

Example 3.2   2. Let 𝛽>1, and denote 𝑔(𝑧)=(5/(1−𝑧))^𝛽. Then the functions 𝑓_𝑗(𝑧)=(1−𝑧)^𝛽exp𝑔(𝑧)^𝑗,𝑗=1,2,3,(3.8) are linearly independent solutions of 𝑓^′′′+𝑎₂(𝑧)𝑓^′′+𝑎₁(𝑧)𝑓′+𝑎₀(𝑧)𝑓=0, where 𝑎₂(𝑧)=𝑃₂𝑔(𝑧)(1−𝑧)𝑄𝑔(𝑧),𝑎₁(𝑧)=𝑃₁𝑔(𝑧)(1−𝑧)²𝑄𝑔(𝑧),𝑎₀(𝑧)=𝛽³𝑃₀𝑔(𝑧)(1−𝑧)³𝑄𝑔(𝑧)(3.9) are such that 𝑃₂(𝜁)=54𝛽𝜁⁸−27𝛽𝜁⁷−24𝛽𝜁⁶+108𝛽+54𝜁⁵−82𝛽+63𝜁⁴+3𝛽𝜁³+22𝛽+39𝜁²−𝑃6𝛽+24𝜁+6,₁(𝜁)=−108𝛽²𝜁¹⁰+72𝛽²𝜁⁹+27𝛽²−54𝛽𝜁⁸+27𝛽−135𝛽²𝜁⁷+24𝛽+15𝛽²𝜁⁶+64𝛽²−108𝛽−18𝜁⁵+21+82𝛽−51𝛽²𝜁⁴−3𝛽+15𝛽²𝜁³+31𝛽²−22𝛽−13𝜁²+8+6𝛽−14𝛽²𝜁+2𝛽²𝑃−2,₀(𝜁)=108𝜁¹¹−234𝜁¹⁰+126𝜁⁹+123𝜁⁸−276𝜁⁷+183𝜁⁶−104𝜁⁵+40𝜁⁴−6𝜁³−4𝜁²,𝑄(𝜁)=−18𝜁⁵+21𝜁⁴−13𝜁²+8𝜁−2.(3.10) The zeros of 𝑄(𝜁) lie in the open disc of radius 1+21/18 centered at the origin by [14, Lemma  1.3.2]. Since |𝑔(𝑧)|>|5/(1−𝑧)|>5/2>39/18 for all 𝑧∈𝔻, we conclude that 𝑎₀,𝑎₁,𝑎₂∈ℋ(𝔻). In fact, the coefficients 𝑎₀,𝑎₁,and𝑎₂ satisfy 𝑎₂(𝑧)∼11−𝑧^3𝛽+1,𝑎₁(𝑧)∼11−𝑧^5𝛽+2,𝑎₀(𝑧)∼11−𝑧^6𝛽+3,(3.11) in a neighborhood of 𝑧=1, while they are bounded in a neighborhood of any boundary point in 𝜕𝔻⧵{1}.
Note that 𝑎_𝑗∈𝔸_𝛼^1/(3−𝑗)_𝑗, where 𝛼₂=3𝛽−1, 𝛼₁=(5/2)𝛽−1, and 𝛼₀=2𝛽−1. Evidently 𝜎_𝑇(𝑓_𝑗)=𝛽𝑗−1 for 𝑗=1,2,3. We deduce that there is one solution 𝑓₃ such that 𝜎_𝑇(𝑓₃)=𝛼₂=3𝛽−1, two solutions 𝑓₂ and 𝑓₃ such that 𝜎_𝑇𝑓₃>𝜎_𝑇𝑓₂=min2𝛼₁−𝛼₂,𝛼₁=2𝛽−1,(3.12) and three solutions 𝑓₁, 𝑓₂, and 𝑓₃ such that 𝜎_𝑇𝑓₃>𝜎_𝑇𝑓₂>𝜎_𝑇𝑓₁=min3𝛼₀−2𝛼₁,32𝛼₀−12𝛼₂,𝛼₀=𝛽−1.(3.13) That is, in all cases 𝑠=0,1,2 there exists a solution for which the lower bound in (2.4) is attained. Further, this example is in line with Corollary 2.2, since all solutions 𝑓₁, 𝑓₂, and 𝑓₃ are of strictly positive 𝑇-order, and in this case 𝑠^∗=0.
Now 𝛾_𝑇(0)=𝛽_𝑇(0)=𝛽−1, 𝛾_𝑇(1)=𝛽_𝑇(1)=2𝛽−1, and 𝛾_𝑇(2)=𝛽_𝑇(2)=3𝛽−1. It follows that for the solution base {𝑓₁,𝑓₂,𝑓₃} equality holds in the first inequality in (2.9), and for the solution base {𝑓₁+𝑓₃,𝑓₂+𝑓₃,𝑓₃} equality holds in the last inequality in (2.9). This shows the sharpness of Corollary 2.3.
On the other hand, 𝑎_𝑗∈ℍ^∞_{(𝑝𝑗+1)(3−𝑗)}, where 𝑝₂=3𝛽, 𝑝₁=5𝛽/2, and 𝑝₀=2𝛽. Evidently 𝜎_𝑀(𝑓_𝑗)=𝛽𝑗 for 𝑗=1,2,3. We deduce that there is one solution 𝑓₃ such that 𝜎_𝑀(𝑓₃)=𝑝₂=3𝛽>1, two solutions 𝑓₂ and 𝑓₃ such that 𝜎_𝑀𝑓₃>𝜎_𝑀𝑓₂=min2𝑝₁−𝑝₂,𝑝₁=2𝛽>1,(3.14) and three solutions 𝑓₁, 𝑓₂, and 𝑓₃ such that 𝜎_𝑀𝑓₃>𝜎_𝑀𝑓₂>𝜎_𝑀𝑓₁=min3𝑝₀−2𝑝₁,32𝑝₀−12𝑝₂,𝑝₀=𝛽.(3.15) That is, in all cases 𝑠=0,1,2 there exists a solution for which the lower bound in (2.22) is attained. Further, this example is in line with Corollary 2.6, since all solutions 𝑓₁, 𝑓₂, and 𝑓₃ are of 𝑀-order strictly greater than 1, and in this case 𝑠^⋆=0.
Now 𝛾_𝑀(0)=𝛽_𝑀(0)=𝛽, 𝛾_𝑀(1)=𝛽_𝑀(1)=2𝛽, and 𝛾_𝑀(2)=𝛽_𝑀(2)=3𝛽. It follows that for the solution base {𝑓₁,𝑓₂,𝑓₃} equality holds in (2.27), and for the solution base {𝑓₁+𝑓₃,𝑓₂+𝑓₃,𝑓₃} upper bound for the sum of 𝑀-orders is attained. This shows the sharpness of Corollary 2.7.

4. Proof of Theorem 2.1

The following lemma on the order reduction procedure originates from ℂ.

Lemma C (see [4, Lemma  6.4]). Let 𝑓_0,1,𝑓_0,2,…,𝑓_0,𝑚 be 𝑚≥2 linearly independent meromorphic solutions of 𝑦^(𝑘)+𝑎_0,𝑘−1(𝑧)𝑦^(𝑘−1)+⋯+𝑎_0,0(𝑧)𝑦=0,𝑘≥𝑚,(4.1) where 𝑎_0,0(𝑧),…,𝑎_0,𝑘−1(𝑧) are meromorphic functions in 𝔻. For 1≤𝑝≤𝑚−1, set 𝑓_𝑝,𝑗=𝑓_{𝑝−1,𝑗+1}𝑓_𝑝−1,1^′,𝑗=1,…,𝑚−𝑝.(4.2) Then 𝑓_𝑝,1,𝑓_𝑝,2,…,𝑓_{𝑝,𝑚−𝑝} are linearly independent meromorphic solutions of 𝑦^{(𝑘−𝑝)}+𝑎_{𝑝,𝑘−𝑝−1}(𝑧)𝑦^{(𝑘−𝑝−1)}+⋯+𝑎_𝑝,0(𝑧)𝑦=0,(4.3) where 𝑎_𝑝,𝑗(𝑧)=^{𝑘−𝑝+1}_𝑛=𝑗+1𝑛𝑎𝑗+1^{𝑝−1,𝑛}(𝑧)𝑓^{(𝑛−𝑗−1)}_𝑝−1,1(𝑧)𝑓_𝑝−1,1(𝑧)(4.4) for 𝑗=0,…,𝑘−𝑝−1. Here 𝑎_{𝑛,𝑘−𝑛}(𝑧)≡1 for all 𝑛=0,…,𝑝.

Lemma D (see [15, Theorem  E(b)]). Let 𝑘 and 𝑗 be integers satisfying 𝑘>𝑗≥0, and let 𝜀>0. If 𝑓 is meromorphic in 𝔻 such that 𝜎_𝑇(𝑓)<∞, and 𝑓^(𝑗)≢0, then _𝔻𝑓^(𝑘)(𝑧)𝑓^(𝑗)(𝑧)^{1/(𝑘−𝑗)}(1−|𝑧|)^{𝜎𝑇(𝑓)+𝜀}𝑑𝑚(𝑧)<∞.(4.5)

4.1. Case 𝑠=1

Let 𝑘≥3, 𝑞≥2, 𝑠=1, and 𝛽_𝑇(1)>0, since otherwise there is nothing to prove. In particular, if 𝛼₁≤0, then (2.4) is trivial, since by taking 𝑗=𝑘 in (2.6), we obtain 𝛽_𝑇(1)≤𝛼₁≤0. Let {𝑓_0,1,𝑓_0,2,…,𝑓_0,𝑘} be a solution base of (1.1), and assume on the contrary to the assertion that there exist 𝑠+1=2 linearly independent solutions 𝑓_0,1 and 𝑓_0,2 such that max{𝜎_𝑇(𝑓_0,1),𝜎_𝑇(𝑓_0,2)}=∶𝜎<𝛽_𝑇(1). Then the meromorphic function 𝑔∶=(𝑓_0,1/𝑓_0,2)′ satisfies 𝜎_𝑇(𝑔)≤𝜎. Moreover, Lemma C implies that 𝑔 satisfies 𝑔^(𝑘−1)+𝑎_1,𝑘−2(𝑧)𝑔^(𝑘−2)+⋯+𝑎_1,0(𝑧)𝑔=0,(4.6) where 𝑎_1,𝑗(𝑧)=𝑎_0,𝑗+1(𝑧)+^𝑘_𝑛=𝑗+2𝑛𝑎𝑗+1^0,𝑛(𝑧)𝑓^{(𝑛−𝑗−1)}_0,1(𝑧)𝑓_0,1(𝑧)(4.7) for 𝑗=0,1,…,𝑘−2, and 𝑎_0,𝑘(𝑧)≡1. Therefore 𝑎_0,1(𝑧)≤𝑎_1,0(𝑧)+^𝑘_𝑛=2𝑛1𝑎_0,𝑛(𝑧𝑓)^{(𝑛−1)0,1}(𝑧)𝑓_0,1(𝑧),(4.8) where 𝑎_1,0(𝑧≤𝑔)^(𝑘−1)(𝑧)𝑔(𝑧)+𝑎_1,𝑘−2(𝑧𝑔)^(𝑘−2)(𝑧)𝑔(𝑧)+⋯+𝑎_1,1(𝑧𝑔)^′(𝑧)𝑔(𝑧),(4.9) since 𝑔 satisfies (4.6). Putting the last two inequalities together, we obtain 𝑎_0,1(𝑧)≲^𝑘−1_𝑗=1𝑎_1,𝑗(𝑧𝑔)^(𝑗)(𝑧)𝑔(𝑧)+^𝑘_𝑛=2𝑎_0,𝑛(𝑧𝑓)^{(𝑛−1)0,1}(𝑧)𝑓_0,1(𝑧).(4.10) Let 𝜀>0. Raising both sides to the power 1/(𝑘−1) and integrating over the disc 𝐷(0,𝑟) of radius 𝑟∈(0,1) with respect to (1−|𝑧|²)^𝛼1−𝜀𝑑𝑚(𝑧), we obtain _𝐷(0,𝑟)𝑎_0,1(𝑧)^1/(𝑘−1)1−|𝑧|^{2𝛼1−𝜀}𝑑𝑚(𝑧)≲^𝑘−1_𝑗=1𝔻𝑎_1,𝑗(𝑧)^1/(𝑘−1)𝑔^(𝑗)(𝑧)𝑔(𝑧)^1/(𝑘−1)1−|𝑧|^{2𝛼1−𝜀}𝑑𝑚(𝑧)+^𝑘_𝑛=2𝔻𝑎_0,𝑛(𝑧)^1/(𝑘−1)𝑓^(𝑛−1)_0,1(𝑧)𝑓_0,1(𝑧)^1/(𝑘−1)1−|𝑧|^{2𝛼1−𝜀}𝑑𝑚(𝑧).(4.11)

To deal with the second sum in (4.11), consider 𝐼_𝑛∶=_𝔻𝑎_0,𝑛(𝑧)^1/(𝑘−1)𝑓^(𝑛−1)_0,1(𝑧)𝑓_0,1(𝑧)^1/(𝑘−1)1−|𝑧|^{2𝛼1−𝜀}𝑑𝑚(𝑧),𝑛=2,…,𝑘.(4.12) By Lemma D, 𝐼_𝑘=_𝔻𝑓^(𝑘−1)_0,1(𝑧)𝑓_0,1(𝑧)^1/(𝑘−1)1−|𝑧|^{2𝛼1−𝜀}𝑑𝑚(𝑧)<∞(4.13) for 𝜀>0 being small enough since 𝜎_𝑇(𝑓_0,1)≤𝜎<𝛽_𝑇(1)≤𝛼₁. Moreover, by Hölder’s inequality, with indices (𝑘−1)/(𝑘−𝑛) and (𝑘−1)/(𝑛−1), we have 𝐼_𝑛≤_𝔻𝑎_0,𝑛(𝑧)^{1/(𝑘−𝑛)}1−|𝑧|^{2𝛼𝑛+𝜀}𝑑𝑚(𝑧)^{(𝑘−𝑛)/(𝑘−1)}⋅_𝔻𝑓^(𝑛−1)_0,1(𝑧)𝑓_0,1(𝑧)^1/(𝑛−1)1−|𝑧|^2𝜔1(𝑛)𝑑𝑚(𝑧)^{(𝑛−1)/(𝑘−1)}(4.14) for all 𝑛=2,…,𝑘−1, where 𝜔_𝑠(𝑛)()𝛼∶=𝑘−𝑠_𝑠−𝛼_𝑛𝑛−𝑠+𝛼_𝑛−2𝑘−𝑛−𝑠𝑛−𝑠𝜀.(4.15) The first member in the product is finite since 𝑎_0,𝑛∈𝔸_𝛼^{1/(𝑘−𝑛)}_𝑛 for all 𝑛=2,…,𝑘−1 by the assumption, and so is the second one for 𝜀>0 small enough since 𝜎_𝑇𝑓_0,1≤𝜎<𝛽_𝑇(1)≤(𝑘−1)𝛼₁−𝛼_𝑛𝑛−1+𝛼_𝑛,𝑛=2,…,𝑘−1,(4.16) by the antithesis. Thus ∑^𝑘_𝑛=2𝐼_𝑛 is finite for 𝜀>0 being small enough.

To deal with the first sum in (4.11), denote 𝐽_𝑗∶=_𝔻𝑎_1,𝑗(𝑧)^1/(𝑘−1)𝑔^(𝑗)(𝑧)𝑔(𝑧)^1/(𝑘−1)1−|𝑧|^{2𝛼1−𝜀}𝑑𝑚(𝑧),𝑗=1,…,𝑘−1.(4.17) Lemma D implies that 𝐽_𝑘−1=_𝔻𝑔^(𝑘−1)(𝑧)𝑔(𝑧)^1/(𝑘−1)1−|𝑧|^{2𝛼1−𝜀}𝑑𝑚(𝑧)<∞(4.18) for 𝜀>0 being small enough since 𝜎_𝑇(𝑔)≤𝜎<𝛽_𝑇(1)≤𝛼₁. Moreover, by (4.7) we have 𝐽_𝑗≲_𝔻𝑎_0,𝑗+1(𝑧)^1/(𝑘−1)𝑔^(𝑗)(𝑧)𝑔(𝑧)^1/(𝑘−1)1−|𝑧|^{2𝛼1−𝜀}(𝑧)+𝑑𝑚_{𝑘𝑛=𝑗+2𝔻}𝑎_0,𝑛(𝑧)^1/(𝑘−1)𝑓^{(𝑛−𝑗−1)}_0,1(𝑧)𝑓_0,1(𝑧)^1/(𝑘−1)𝑔^(𝑗)(𝑧)𝑔(𝑧)^1/(𝑘−1)1−|𝑧|^{2𝛼1−𝜀}𝑑𝑚(𝑧)=∶𝐾_𝑗+𝐿_𝑗,𝑘+^𝑘−1_𝑛=𝑗+2𝐿_𝑗,𝑛(4.19) for all 𝑗=1,…,𝑘−2. Since max{𝜎_𝑇(𝑔),𝜎_𝑇(𝑓_0,1)}≤𝜎<𝛽_𝑇(1), we deduce that 𝐾_𝑗 behaves like 𝐼_𝑗+1 and hence ∑^𝑘−2_𝑗=1𝐾_𝑗<∞ for 𝜀>0 being small enough. Moreover, by Hölder’s inequality, with indices (𝑘−1)/(𝑘−𝑗−1) and (𝑘−1)/𝑗, and Lemma D we have 𝐿_𝑗,𝑘≤_𝔻𝑓^{(𝑘−𝑗−1)}_0,1(𝑧)𝑓_0,1(𝑧)^{1/(𝑘−𝑗−1)}1−|𝑧|^{2𝛼1−𝜀}𝑑𝑚(𝑧)^{(𝑘−𝑗−1)/(𝑘−1)}⋅_𝔻𝑔^(𝑗)(𝑧)𝑔(𝑧)^1/𝑗1−|𝑧|^{2𝛼1−𝜀}𝑑𝑚(𝑧)^{𝑗/(𝑘−1)}<∞(4.20) for all 𝑗=1,…,𝑘−2 when 𝜀>0 is sufficiently small. It remains to consider the double sum ∑^𝑘−3_𝑗=1∑^𝑘−1_𝑛=𝑗+2𝐿_𝑗,𝑛. By Hölder’s inequality, with indices (𝑘−1)/(𝑘−𝑛) and (𝑘−1)/(𝑛−1), we have 𝐿_𝑗,𝑛≤_𝔻𝑎_0,𝑛(𝑧)^{1/(𝑘−𝑛)}1−|𝑧|^{2𝛼𝑛+𝜀}𝑑𝑚(𝑧)^{(𝑘−𝑛)/(𝑘−1)}⋅_𝔻𝑓^{(𝑛−𝑗−1)}_0,1(𝑧)𝑓_0,1(𝑧)^1/(𝑛−1)𝑔^(𝑗)(𝑧)𝑔(𝑧)^1/(𝑛−1)1−|𝑧|^2𝜔1(𝑛)𝑑𝑚(𝑧)^{(𝑛−1)/(𝑘−1)},(4.21) where 𝜔₁(𝑛) is defined in (4.15). The first term in the product is bounded for all 𝜀>0 since 𝑎_0,𝑛∈𝔸_𝛼^{1/(𝑘−𝑛)}_𝑛 for all 𝑛=3,…,𝑘−1 by the assumption. One more application of Hölder's inequality, with indices (𝑛−1)/(𝑛−𝑗−1) and (𝑛−1)/𝑗, together with Lemma D and the antithesis shows that also the second term in the product is bounded for 𝜀>0 being small enough, and thus ∑^𝑘−3_𝑗=1∑^𝑘−1_𝑛=𝑗+2𝐿_𝑗,𝑛<∞ for 𝜀>0 being small enough.

We have proved that the right-hand side of (4.11) is uniformly bounded for all 𝑟∈(0,1), if 𝜀>0 is small enough. However, 𝑎_0,1∈𝔸_𝛼^1/(𝑘−1)₁ by the assumption, and hence the left-hand side of (4.11) diverges as 𝑟→1⁻. Contradiction follows.

4.2. Case 𝑠>1

Let 𝑘≥3, 𝑞≥2, 𝑠>1, and 𝛽_𝑇(𝑠)>0, since otherwise there is nothing to prove. In particular, it follows that 𝛼_𝑠>0. Let {𝑓_0,1,𝑓_0,2,…,𝑓_0,𝑘} be a solution base of (1.1), and assume on the contrary to the assertion that there exist 𝑠+1 linearly independent solutions 𝑓_0,1,…,𝑓_0,𝑠+1 such that 𝜎∶=max𝜎_𝑇𝑓_0,1,…,𝜎_𝑇𝑓_0,𝑠+1<𝛽_𝑇(𝑠).(4.22) Then the meromorphic functions 𝑓_1,𝑗=(𝑓_0,𝑗+1/𝑓_0,1)′ satisfy 𝜎_𝑇(𝑓_1,𝑗)≤𝜎 for all 𝑗=1,…,𝑠. This in turn implies that 𝑓_2,𝑗=(𝑓_1,𝑗+1/𝑓_1,1)′ satisfy 𝜎_𝑇(𝑓_2,𝑗)≤𝜎 for all 𝑗=1,…,𝑠−1. In general, 𝜎_𝑇(𝑓_𝑛,𝑗)≤𝜎 for all 𝑗=1,…,𝑠−𝑛+1 and 𝑛=1,…,𝑠. Moreover, as in the case 𝑠=1, Lemma C implies 𝑎_0,𝑠(𝑧)≤𝑎_1,𝑠−1(𝑧)+^𝑘_𝑛=𝑠+1𝑛𝑠𝑎_0,𝑛(𝑧𝑓)^{(𝑛−𝑠)0,1}(𝑧)𝑓_0,1(𝑧)≤𝑎_2,𝑠−2(𝑧)+^𝑘−1_𝑛=𝑠𝑛𝑎𝑠−1^1,𝑛(𝑧𝑓)^{(𝑛−𝑠+1)1,1}(𝑧)𝑓_1,1(𝑧)+^𝑘_𝑛=𝑠+1𝑛𝑠𝑎_0,𝑛(𝑧𝑓)^{(𝑛−𝑠)0,1}(𝑧)𝑓_0,1(𝑧)≤𝑎≤⋯_𝑠,0(𝑧)+^𝑠−1_𝑚=0^𝑘−𝑚_{𝑛=𝑠+1−𝑚}𝑛𝑎𝑠−𝑚^𝑚,𝑛(𝑧𝑓)^{(𝑛−𝑠+𝑚)𝑚,1}(𝑧)𝑓_𝑚,1(𝑧),(4.23) where 𝑎_𝑠,0(𝑧≤𝑓)^{(𝑘−𝑠)𝑠,1}(𝑧)𝑓_𝑠,1(𝑧)+^{𝑘−𝑠−1}_𝑚=1𝑎_𝑠,𝑚(𝑧𝑓)^(𝑚)𝑠,1(𝑧)𝑓_𝑠,1(𝑧).(4.24) Putting these inequalities together, we obtain 𝑎_0,𝑠(𝑧)≲^𝑠_𝑚=0^{𝑘−𝑚−1}_{𝑛=𝑠+1−𝑚}𝑎_𝑚,𝑛(𝑧𝑓)^{(𝑛−𝑠+𝑚)𝑚,1}(𝑧)𝑓_𝑚,1(𝑧)+^𝑠_𝑚=0𝑓^{(𝑘−𝑠)}_𝑚,1(𝑧)𝑓_𝑚,1(𝑧).(4.25) Let 𝜀>0. Raising both sides to the power 1/(𝑘−𝑠) and integrating over the disc 𝐷(0,𝑟) with respect to (1−|𝑧|²)^{𝛼𝑠−𝜀}𝑑𝑚(𝑧), we obtain _𝐷(0,𝑟)𝑎_0,𝑠(𝑧)^{1/(𝑘−𝑠)}1−|𝑧|^{2𝛼𝑠−𝜀}𝑑𝑚(𝑧)≲^𝑠_𝑚=0^{𝑘−𝑚−1}_{𝑛=𝑠+1−𝑚𝔻}𝑎_𝑚,𝑛(𝑧)^{1/(𝑘−𝑠)}𝑓^{(𝑛−𝑠+𝑚)}_𝑚,1(𝑧)𝑓_𝑚,1(𝑧)^{1/(𝑘−𝑠)}1−|𝑧|^{2𝛼𝑠−𝜀}𝑑𝑚(𝑧)+^𝑠_𝑚=0𝔻𝑓^{(𝑘−𝑠)}_𝑚,1(𝑧)𝑓_𝑚,1(𝑧)^{1/(𝑘−𝑠)}1−|𝑧|^{2𝛼𝑠−𝜀}𝑑𝑚(𝑧)=∶^𝑠_𝑚=0^{𝑘−𝑚−1}_{𝑛=𝑠+1−𝑚}𝐼_𝑚,𝑛+^𝑠_𝑚=0𝐽_𝑚.(4.26) Lemma D and the antithesis imply that ∑^𝑠_𝑚=0𝐽_𝑚<∞ for 𝜀>0 being small enough. Hence, in order to obtain a contradiction with (4.26) and the assumption 𝑎_0,𝑠∈𝔸_𝛼^{1/(𝑘−𝑠)}_𝑠, it suffices to show that 𝐼_𝑚,𝑛<∞ for all 𝑚=0,…,𝑠 and 𝑛=𝑠+1−𝑚,…,𝑘−𝑚−1 when 𝜀>0 is sufficiently small.

By Hölder's inequality, with indices (𝑘−𝑠)/(𝑘−𝑛) and (𝑘−𝑠)/(𝑛−𝑠), we have 𝐼_0,𝑛≤_𝔻𝑎_0,𝑛(𝑧)^{1/(𝑘−𝑛)}1−|𝑧|^{2𝛼𝑛+𝜀}𝑑𝑚(𝑧)^{(𝑘−𝑛)/(𝑘−𝑠)}⋅_𝔻𝑓^{(𝑛−𝑠)}_0,1(𝑧)𝑓_0,1(𝑧)^{1/(𝑛−𝑠)}1−|𝑧|^{2𝜔𝑠(𝑛)}𝑑𝑚(𝑧)^{(𝑛−𝑠)/(𝑘−𝑠)}(4.27) for all 𝑛=𝑠+1,…,𝑘−1, where 𝜔_𝑠(𝑛) is defined in (4.15). The first member in the product is finite since 𝑎_0,𝑛∈𝔸_𝛼^{1/(𝑘−𝑛)}_𝑛 for all 𝑛=𝑠+1,…,𝑘−1 by the assumption, and so is the second one for 𝜀>0 being small enough, since 𝜎_𝑇𝑓_0,1≤𝜎<𝛽_𝑇(𝑠)≤(𝑘−𝑠)𝛼_𝑠−𝛼_𝑛𝑛−𝑠+𝛼_𝑛,𝑛=𝑠+1,…,𝑘−1,(4.28) by the antithesis. In the general case Lemma C gives 𝐼_𝑚,𝑛=_𝔻𝑎_𝑚,𝑛(𝑧)^{1/(𝑘−𝑠)}𝑓^{(𝑛−𝑠+𝑚)}_𝑚,1(𝑧)𝑓_𝑚,1(𝑧)^{1/(𝑘−𝑠)}1−|𝑧|^{2𝛼𝑠−𝜀}(𝑧)≲𝑑𝑚_{𝑛𝑘−𝑚+11=𝑛+1𝔻}𝑎_{𝑚−1,𝑛1}(𝑧)^{1/(𝑘−𝑠)}𝑓^{(𝑛1−𝑛−1)}_𝑚−1,1(𝑧)𝑓_𝑚−1,1(𝑧)^{1/(𝑘−𝑠)}⋅𝑓^{(𝑛−𝑠+𝑚)}_𝑚,1(𝑧)𝑓_𝑚,1(𝑧)^{1/(𝑘−𝑠)}1−|𝑧|^{2𝛼𝑠−𝜀}𝑑𝑚(𝑧)≲_𝑛^{𝑘−𝑚+1}_1=𝑛+1𝑛^{𝑘−𝑚+2}_{2=𝑛1+1𝔻}𝑎_{𝑚−2,𝑛2}(𝑧)^{1/(𝑘−𝑠)}𝑓^{(𝑛2−𝑛1−1)}_𝑚−2,1(𝑧)𝑓_𝑚−2,1(𝑧)^{1/(𝑘−𝑠)}⋅𝑓^{(𝑛1−𝑛−1)}_𝑚−1,1(𝑧)𝑓_𝑚−1,1(𝑧)^{1/(𝑘−𝑠)}𝑓^{(𝑛−𝑠+𝑚)}_𝑚,1(𝑧)𝑓_𝑚,1(𝑧)^{1/(𝑘−𝑠)}1−|𝑧|^{2𝛼𝑠−𝜀}𝑑𝑚(𝑧),(4.29) and finally 𝐼_𝑚,𝑛≲_𝑛^{𝑘−𝑚+1}_1=𝑛+1𝑛^{𝑘−𝑚+2}_2=𝑛1+1⋯^𝑘_{𝑛𝑚=𝑛𝑚−1+1}𝐾𝑛,𝑛₁,…,𝑛_𝑚,(4.30) where 𝐾𝑛,𝑛₁,…,𝑛_𝑚∶=_𝔻𝑎_0,𝑛𝑚(𝑧)^{1/(𝑘−𝑠)}𝑓^{(𝑛𝑚−𝑛𝑚−1−1)}_0,1(𝑧)𝑓_0,1(𝑧)^{1/(𝑘−𝑠)}⋯𝑓^{(𝑛2−𝑛1−1)}_𝑚−2,1(𝑧)𝑓_𝑚−2,1(𝑧)^{1/(𝑘−𝑠)}⋅𝑓^{(𝑛1−𝑛−1)}_𝑚−1,1(𝑧)𝑓_𝑚−1,1(𝑧)^{1/(𝑘−𝑠)}𝑓^{(𝑛−𝑠+𝑚)}_𝑚,1(𝑧)𝑓_𝑚,1(𝑧)^{1/(𝑘−𝑠)}1−|𝑧|^{2𝛼𝑠−𝜀}𝑑𝑚(𝑧).(4.31)

If 𝑛_𝑚=𝑘, then 𝑎_0,𝑛𝑚(𝑧)≡1, and general form of Hölder's inequality with indices 𝑛_𝑚−𝑠𝑛_𝑚−𝑛_𝑚−1,𝑛−1^𝑚−𝑠𝑛_𝑚−1−𝑛_𝑚−2𝑛−1,…,^𝑚−𝑠𝑛₁,𝑛−𝑛−1^𝑚−𝑠𝑛−𝑠+𝑚(4.32) together with Lemma D shows that 𝐾(𝑛,𝑛₁,…,𝑛_𝑚)<∞ for 𝜀>0 being small enough. If 𝑛_𝑚<𝑘, then an appropriate application of Hölder’s inequality with indices (𝑘−𝑠)/(𝑘−𝑛_𝑚) and (𝑘−𝑠)/(𝑛_𝑚−𝑠) separates the coefficient from the solutions. The first term is finite by the assumption, and the second term can seen to be finite by another application of general form of Hölder's inequality with indices (4.32). This gives the desired contradiction, since the left-hand side of (4.26) diverges as 𝑟→1⁻ and the right-hand side of (4.26) is uniformly bounded for all 𝑟∈(0,1).

5. Proof of Corollary 2.3

The upper bound in (2.9) follows directly from Theorem 2.1. To conclude the lower bound in (2.9), assume that solutions 𝑓₁,…,𝑓_𝑘 are given in increasing order with respect to 𝑇-order of growth; that is, 𝜎_𝑇(𝑓₁)≤⋯≤𝜎_𝑇(𝑓_𝑘). By applying Theorem 2.1 with 𝑠=0,…,𝑞, we get the following sequence of successive statements. For all solutions 𝑓 in the solution base, we have 𝛽_𝑇(0)≤𝜎_𝑇(𝑓); for 𝑘−1 solutions 𝑓 in the solution base, we have 𝛽_𝑇(1)≤𝜎_𝑇(𝑓), ending up with the fact that 𝑘−𝑞 solutions 𝑓 in the solution base satisfy 𝛼_𝑞=𝛽_𝑇(𝑞)=𝜎_𝑇(𝑓). Hence we have 𝛾_𝑇(0)=𝛽_𝑇(0)≤𝜎_𝑇(𝑓₁), 𝛾_𝑇(1)=max{𝛽_𝑇(0),𝛽_𝑇(1)}≤𝜎_𝑇(𝑓₂) continuing to 𝛾_𝑇𝑞−1=max𝛽_𝑇(0),…,𝛽_𝑇𝑞−1≤𝜎_𝑇𝑓_𝑞.(5.1)

Note that 𝛼_𝑞=𝛾_𝑇(𝑞). To see this, note that 𝛽_𝑇(𝑠)≤𝛼_𝑠 for every 𝑠=0,…,𝑞, which follows by taking 𝑗=𝑘 in (2.6), and hence 𝛼_𝑞=𝛽_𝑇𝑞≤𝛾_𝑇𝑞=max𝛽_𝑇(0),…,𝛽_𝑇𝑞≤max𝛼₀,…,𝛼_𝑞=𝛼_𝑞.(5.2) The assertion follows by noting that if 𝑗∈{0,…,𝑠^⋆−1}, then 𝛾_𝑇(𝑗)≤0, and we only have the trivial estimate 𝜎_𝑇(𝑓_𝑗)≥0.

6. Proof of Lemma 2.4

Let 𝑚∈{1,…,𝑝}. By (2.11), we obtain 𝛿_𝑗−𝛿_𝑠𝑚−1𝑠_𝑚−1≤𝛿−𝑗^𝑠𝑚−𝛿_𝑠𝑚−1𝑠_𝑚−1−𝑠_𝑚(6.1) for all 0≤𝑗<𝑠_𝑚−1, and the inequality (6.1) is strict for all 0≤𝑗<𝑠_𝑚. This estimate will be repeatedly needed later on.

6.1. Proof of Lemma 2.4(i)

Let 1≤𝑡≤𝑝−1. Note that ℬ_𝑇(𝑡)>0 by definition. Since 𝑠_𝑡+1<𝑠_𝑡, ℬ_𝑇(𝑡)−ℬ_𝑇()=𝛿𝑡+1^𝑠𝑡−𝛿_𝑠𝑡−1𝑠_𝑡−1−𝑠_𝑡−𝛿_𝑠𝑡+1−𝛿_𝑠𝑡𝑠_𝑡−𝑠_𝑡+1=𝑠_𝑡−1−𝑠_𝑡+1𝑠_𝑡−𝑠_𝑡+1𝛿_𝑠𝑡−𝛿_𝑠𝑡−1𝑠_𝑡−1−𝑠_𝑡−𝛿_𝑠𝑡+1−𝛿_𝑠𝑡−1𝑠_𝑡−1−𝑠_𝑡+1>0(6.2) by (6.1), which proves the assertion of Lemma 2.4(i).

6.2. Proof of Lemma 2.4(ii)

Since max_{𝑗=0,…,𝑘−1}{𝛼_𝑗}=𝛼_𝑠1>0, we get 𝛽_𝑇𝑠₁=min_{𝑗=𝑠1+1,…,𝑘}𝑘−𝑗𝑗−𝑠₁𝛼_𝑠1−𝛼_𝑗+𝛼_𝑠1=𝛼_𝑠1=𝛿_𝑠1𝑘−𝑠₁−1=ℬ_𝑇(1),(6.3) where the minimum is obtained with 𝑗=𝑘. This proves the claim for 𝑡=1.

Assume that 𝑡∈{2,…,𝑝}. To prove the claim, we need the following observations. If 𝑚∈{1,…,𝑡−1}, then by (6.1) we get 𝛿_𝑠𝑡−𝛿_𝑠𝑚−1𝑠_𝑚−1−𝑠_𝑡−𝛿_𝑠𝑡−𝛿_𝑠𝑚𝑠_𝑚−𝑠_𝑡=𝑠_𝑚−1−𝑠_𝑚𝑠_𝑚−𝑠_𝑡𝛿_𝑠𝑚−𝛿_𝑠𝑚−1𝑠_𝑚−1−𝑠_𝑚−𝛿_𝑠𝑡−𝛿_𝑠𝑚−1𝑠_𝑚−1−𝑠_𝑡>0.(6.4) On the other hand, if 𝑚∈{1,…,𝑡}, then 𝛿_𝑠𝑚−𝛿_𝑗𝑗−𝑠_𝑚≥𝛿_𝑠𝑚−𝛿_𝑠𝑚−1𝑠_𝑚−1−𝑠_𝑚(6.5) for all 𝑠_𝑚<𝑗≤𝑠_𝑚−1. To verify (6.5), we consider the following two cases. If 𝑗=𝑠_𝑚−1, then the equality in (6.5) holds. If 𝑗<𝑠_𝑚−1, then by using (6.1), we obtain 𝛿_𝑠𝑚−𝛿_𝑗𝑗−𝑠_𝑚=𝛿_𝑠𝑚−𝛿_𝑠𝑚−1𝑗−𝑠_𝑚−𝛿_𝑗−𝛿_𝑠𝑚−1𝑠_𝑚−1𝑠−𝑗^𝑚−1−𝑗𝑗−𝑠_𝑚≥𝛿_𝑠𝑚−𝛿_𝑠𝑚−1𝑗−𝑠_𝑚−𝛿_𝑠𝑚−𝛿_𝑠𝑚−1𝑠_𝑚−1−𝑠_𝑚𝑠_𝑚−1−𝑗𝑗−𝑠_𝑚=𝛿_𝑠𝑚−𝛿_𝑠𝑚−1𝑠_𝑚−1−𝑠_𝑚,(6.6) which proves (6.5).

To complete the proof of 𝛽_𝑇𝑠_𝑡=min_{𝑗=𝑠𝑡+1,…,𝑘}𝛿_𝑠𝑡−𝛿_𝑗𝑗−𝑠_𝑡=𝛿−1^𝑠𝑡−𝛿_𝑠𝑡−1𝑠_𝑡−1−𝑠_𝑡−1=ℬ_𝑇(𝑡),(6.7) we argue as follows. First, we show that 𝛽_𝑇(𝑠_𝑡)≥ℬ_𝑇(𝑡). If 𝑠_𝑡<𝑗≤𝑠_𝑡−1, then (6.5) holds for 𝑚=𝑡. If 𝑗>𝑠_𝑡−1, then let 𝑚∈{1,…,𝑡−1} be the smallest index such that 𝑠_𝑚<𝑗. From (6.1), (6.4), and (6.5), we obtain 𝛿_𝑠𝑡−𝛿_𝑗𝑗−𝑠_𝑡−𝛿_𝑠𝑡−𝛿_𝑠𝑚𝑠_𝑚−𝑠_𝑡=𝑗−𝑠_𝑚𝑗−𝑠_𝑡𝛿_𝑠𝑚−𝛿_𝑗𝑗−𝑠_𝑚−𝛿_𝑠𝑡−𝛿_𝑠𝑚𝑠_𝑚−𝑠_𝑡>𝑗−𝑠_𝑚𝑗−𝑠_𝑡𝛿_𝑠𝑚−𝛿_𝑠𝑚−1𝑠_𝑚−1−𝑠_𝑚−𝛿_𝑠𝑡−𝛿_𝑠𝑚−1𝑠_𝑚−1−𝑠_𝑡>0,(6.8) which together with (6.4) shows that 𝛿_𝑠𝑡−𝛿_𝑗𝑗−𝑠_𝑡>𝛿_𝑠𝑡−𝛿_𝑠𝑚𝑠_𝑚−𝑠_𝑡𝛿>⋯>^𝑠𝑡−𝛿_𝑠𝑡−1𝑠_𝑡−1−𝑠_𝑡.(6.9) Second, we note that equality in (6.7) follows by taking 𝑗=𝑠_𝑡−1.

6.3. Proof of Lemma 2.4(iii)

Since 𝛼_𝑠1=max_{𝑗=0,…,𝑘−1}{𝛼_𝑗}>0 and 𝑠₁=𝑞, we get by means of (5.2) that ℬ_𝑇(1)=𝛿_𝑞𝑘−𝑞−1=𝛼_𝑞=𝛾_𝑇𝑞.(6.10) Let 𝑡∈{2,…,𝑝}. We proceed to prove that 𝛾_𝑇(𝑗)=ℬ_𝑇(𝑡) for all 𝑠_𝑡≤𝑗<𝑠_𝑡−1. Evidently, 𝛾_𝑇(𝑗)=max{𝛽_𝑇(0),…,𝛽_𝑇(𝑗)}≥𝛽_𝑇(𝑠_𝑡). By Lemma 2.4(ii), we conclude that 𝛾_𝑇(𝑗)≥ℬ_𝑇(𝑡). To prove that 𝛾_𝑇(𝑗)≤ℬ_𝑇(𝑡), it is enough to show that 𝛽_𝑇(𝑚)≤ℬ_𝑇(𝑡) for all 𝑚∈{0,…,𝑗}. Since 𝑚+1≤𝑠_𝑡−1, we obtain by appealing to (6.1) that 𝛽_𝑇(𝑚)=min_{𝑗=𝑚+1,…,𝑘}𝛿_𝑚−𝛿_𝑗≤𝛿𝑗−𝑚−1^𝑚−𝛿_𝑠𝑡−1𝑠_𝑡−1𝛿−𝑚−1≤^𝑠𝑡−𝛿_𝑠𝑡−1𝑠_𝑡−1−𝑠_𝑡−1=ℬ_𝑇(𝑡).(6.11)

If 𝑠_𝑝>0, then for all 𝑚∈{0,…,𝑠_𝑝−1} we have 𝛽_𝑇(𝑚)=min_{𝑗=𝑚+1,…,𝑘}𝛿_𝑚−𝛿_𝑗≤𝛿𝑗−𝑚−1^𝑚−𝛿_𝑠𝑝𝑠_𝑝−𝑚−1≤0(6.12) by (2.11). Hence 𝛾_𝑇(𝑗)=max{𝛽_𝑇(0),…,𝛽_𝑇(𝑗)}≤0 for all 𝑗<𝑠_𝑝. As a consequence we see that 𝑠_𝑝=𝑠^⋆.

7. Proof of Theorem 2.5

Our proof of Theorem 2.5 is parallel to the proof of [3,  Theorem  1.4], and hence we only outline the argumentation. We may assume that 𝑘≥3, 𝑞≥2, 𝑠∈{1,…,𝑞−1}, and 𝛽_𝑀(𝑠)>1 for otherwise there is nothing to prove by Theorem B; see the discussion after Theorem 2.5. In particular, if 𝑝_𝑠≤1, then (2.22) is trivial, since by taking 𝑗=𝑘 in (2.24), we obtain 𝛽_𝑀(𝑠)≤𝑝_𝑠≤1. On the contrary to the claim, assume that (1.1) admits 𝑠+1 linearly independent solutions 𝑓_0,1,…,𝑓_0,𝑠+1 such that 𝜎_𝑀𝑓_0,𝑡<𝛼∶=𝛽_𝑀(𝑠),𝑡=1,…,𝑠+1.(7.1) Remark that if 𝛼≤1, then there is nothing to prove in (2.22); so we may assume that 𝛼>1. Now max{𝛽,1}<𝛼, where 𝛽∶=max_{𝑡=1,…,𝑠+1}{𝜎_𝑀(𝑓_0,𝑡)}<∞.

Let 𝜀,𝛿∈(0,1). Now [3, Lemma  4.3] for 𝑚=𝑠+1 implies that there exists a solution 𝑓_𝑠,1≢0 of 𝑓^{(𝑘−𝑠)}+𝑎_{𝑠,𝑘−𝑠−1}(𝑧)𝑓^{(𝑘−𝑠−1)}+⋯+𝑎_𝑠,1(𝑧)𝑓^′+𝑎_𝑠,0(𝑧)𝑓=0(7.2) of the from 𝑓_𝑠,1=𝑔_𝑠,1/ℎ_𝑠,1, where 𝑔_𝑠,1,ℎ_𝑠,1∈ℋ(𝔻) and max𝜎_𝑀𝑔_𝑠,1,𝜎_𝑀ℎ_𝑠,1≤max𝛽,1<𝛼.(7.3) It is easy to see that 𝛼=𝛽_𝑀(𝑠) yields (𝑘−𝑙)𝑝_𝑙≤(𝑘−𝑠)𝑝_𝑠−(𝑙−𝑠)𝛼 for all 𝑙∈{𝑠+1,…,𝑘}. Hence, by [3, Lemma  4.3] and the assumption 𝑎_0,𝑗∈ℍ^∞_{(𝑝𝑗+1)(𝑘−𝑗)}, we get 𝑀𝑟,𝑎_𝑠,𝑗≤11−𝑟^{(𝑝𝑠+1)(𝑘−𝑠)−𝑗(𝛼+1)+𝜀},𝑗=1,…,𝑘−𝑠−1,(7.4) for all 𝑟∈[0,1)⧵𝐸, where the set 𝐸 satisfies the upper density condition: 𝐷(𝐸)∶=limsup_𝑟→1−𝑚(𝐸∩[𝑟,1))1−𝑟≤𝛿<1.(7.5) Here 𝑚(Ω) is the Lebesque measure of the set Ω. We note that set 𝐸 may not be the same at each occurrence; however, it always satisfies (7.5).

Let 𝜂∈(𝛿,1). If we apply [3, Lemma  4.4] and use [3, Lemma  4.1] for the coefficient 𝑎_0,𝑠∈ℍ^∞_{(𝑝𝑠+1)(𝑘−𝑠)}, we conclude that for 𝜀>0 being small enough, we have 𝑀𝑟,𝑎_𝑠,0≳11−𝑟^{(𝑝𝑠+1)(𝑘−𝑠)−𝜀}(7.6) for all 𝑟∈𝐹⧵𝐸, where the set 𝐹⊂[0,1) satisfies 𝐷(𝐹)≥𝜂. In particular, we have 𝐷(𝐹⧵𝐸)≥𝜂−𝛿>0.

On the other hand, by substituting 𝑓=𝑓_𝑠,1 in (7.2) and by applying [3,  Corollary  4.2] to 𝑓_𝑠,1, it follows that 𝑎_𝑠,0(𝑧≤𝑓)^{(𝑘−𝑠)𝑠,1}(𝑧)𝑓_𝑠,1(𝑧)+^{𝑘−𝑠−1}_𝑗=1𝑎_𝑠,𝑗(𝑧𝑓)^(𝑗)𝑠,1(𝑧)𝑓_𝑠,1(𝑧)≲11−|𝑧|^{(𝑝𝑠+1)(𝑘−𝑠)−2𝜀}(7.7) for all 𝑧∈𝔻, |𝑧|∉𝐸. By comparing (7.6) to (7.7), we get a contradictory inequality: 𝑝_𝑠+1(𝑘−𝑠)−𝜀≤𝑝_𝑠+1(𝑘−𝑠)−2𝜀.(7.8) This shows that each solution base of (1.1) contains at least 𝑘−𝑠 solutions 𝑓 satisfying 𝜎_𝑀(𝑓)≥𝛽_𝑀(𝑠).

Acknowledgments

The first author was supported by the Fondecyt Grant no. 1110321 and the Mecesup Grant PUC 0711. The second author was supported by the Academy of Finland no. 121281 and the Finnish Cultural Foundation, North Karelia Regional fund. The last author was supported by the Academy of Finland no. 121281; Väisälä Fund of Finnish Science and Letters; MTM2007-30904-E, MTM2008-05891, and MTM2008-02829-E (MICINN, Spain); FQM-210 (Junta de Andalucía, Spain); and the European Science Foundation RNP HCAA.