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

Volume 2011, Article ID 928194,25pages doi:10.1155/2011/928194

Research Article

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

Martin Chuaqui,

Janne Gr ¨ohn,

Janne Heittokangas,

and Jouni R ¨atty ¨a

1Departamento de Matem´aticas, Pontificia Universidad Cat´olica de Chile, Casilla 306, Correo 22 Santiago, 6904411 Macual Santiago, Chile

2Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland

Correspondence should be addressed to Janne Heittokangas,janne.heittokangas@uef.fi Received 19 May 2011; Accepted 5 July 2011

Academic Editor: Jean Michel Combes

Copyrightq2011 Martin Chuaqui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 off^ka_k−1zf^k−1· · ·a1zfa0zf 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 possibleT- andM-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 forT- andM-orders are introduced to give detailed information about the growth of solutions. Finally, these findings yield sharp lower bounds for the sums ofT- andM-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^ka_k−1zf^k−1· · ·a1zfa0zf0, 1.1 where the coefficientsa₀z, . . . , a_k−1zare analytic functions in the unit discD: {z:|z|<

1}of the complex planeC. 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 in1 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 inD, it is known thatT- andM-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 Dto 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 derivatives2 . Second, if the growth of solutions is measured by using theM-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 sets3 .

The main focus of this paper is in improving the lower bounds for the growth of solutions of1.1given in 2,3 and explore some consequences, which are motivated by the following observations.

By the classical results inCmaking use of Newton-Puiseux diagram, there is a finite set containing the possible growth orders of solutions of1.1assuming 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 in1.1 4, Theorem 1 . Their proof relies on classical Wiman-Valiron theory inC.

Even though a recent unit disc counterpart of Wiman-Valiron theory5 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 areα-polynomial regular. Theseα-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 inD 6, Theorem 6.1 . Note that in the case ofC, all solutions of1.1are of finite order if and only if coefficients are polynomials7, 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 of1.1, 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 andM-orders, giving detailed information about the growth of solutions. Finally, these findings are applied to estimate the sums ofT- andM-orders of functions in the solution bases of1.1from below.

2. Results and Motivation

The results concerning T- and M-orders of solutions of 1.1 are given, respectively, in Sections2.1-2.2and2.3-2.4. Due to the similarities of the assertions, we omit the proofs of results regardingM-orders of solutions of1.1, excluding the sketched proof ofTheorem 2.5 inSection 7.

LetMDandHDdenote the sets of all meromorphic and analytic functions inD.

For simplicity, we writeα :max{α,0}for anyα∈R,|fz||gz|if there exists a constant

C > 0 independent ofzsuch that |fz| ≤ C|gz|, andfz ∼ gzif there exist constants C₁>0 andC₂>0 independent ofzsuch thatC₁|gz| ≤ |fz| ≤C₂|gz|.

2.1. Growth of Solutions with Respect to Nevanlinna Characteristic TheT-order of growth off∈ MDis defined as

σT

f

:lim sup

r→1⁻

logT r, f

−log1−r, 2.1

whereTr, fis the Nevanlinna characteristic off. Forp >0 andα >−1, the weighted Bergman spaceA^p_αconsists of thosef ∈ HDfor which

f

A^pα :

D

fz^p

1− |z|^{2 α}dmz _1/p

<∞. 2.2

Functions of maximal growth in

q<α<∞A^p_α are distinguished by denotingf ∈ A^pq, 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 of1.1with respect toT-order.

Theorem A see 2, Theorems 1 and 2 . Suppose that aj ∈ A^1/k−jαj , where αj ≥ 0 for j 0, . . . , k−1, and denoteα_k:0.

iLet 0 ≤ α < ∞. Then all solutions f of 1.1 satisfy σTf ≤ α if and only if max_j0,...,k−1{αj} ≤α.

iiAll nontrivial solutionsfof1.1satisfy

j1,...,kmin k

α₀−α_j

j α_j

≤σ_T f

≤ max

j0,...,k−1

α_j

. 2.3

iiiIf q ∈ {0, . . . , k −1}is the smallest index for which αq max_j0,...,k−1{αj}, then each solution base of 1.1 contains at leastk−qlinearly independent solutionsf such that σ_Tf α_q.

The assumptiona_j∈A^1/k−jαj in Theorem Aicannot be replaced bya_j∈A^1/k−j_αj ; see 8 . We refine Theorem A and then further underscore its consequences.

Theorem 2.1. Suppose thataj∈A^1/k−jαj , whereαj≥ −1 forj0, . . . , k−1, and letq∈ {0, . . . , k−1}

be the smallest index for whichα_q max_j0,...,k−1{αj}. Ifs ∈ {0, . . . , q}, then each solution base of 1.1contains at leastk−slinearly independent solutionsfsuch that

js1,...,kmin

k−s

α_s−α_j

j−s αj

≤σT

f

≤α_q, 2.4

whereα_k:−1.

The case s 0 clearly reduces to Theorem Aii. If s q, then the conditionαq max_j0,...,k−1{αj}implies that

jq1,...,kmin

k−q αq−αj

j−q αj

min

jq1,...,k

k−j αq−αj

j−q αq

αq, 2.5

where the minimum is attained forj k. Hence the assertion ofTheorem 2.1fors qis contained in Theorem Aiii. Our contribution is to extend the first inequality in2.4 for s∈ {1, . . . , q−1}.Theorem 2.1is proved inSection 4, and the sharpness and the special cases k2 andk3 are further discussed inSection 3.1.

Letq ∈ {0, . . . , k−1}be the smallest index for whichα_q max_j0,...,k−1{αj}. Ifα_q ≤0, then all solutions in each solution base of1.1are of zeroT-order by Theorem Aii. Suppose thatαq >0. In order to state the following corollaries ofTheorem 2.1, we denote

βTs: min

js1,...,k

k−s

α_s−α_j

j−s αj

, s0, . . . , q, 2.6

whereαk:−1. Moreover, we define s:min

s∈

0, . . . , q

:β_Ts>0

. 2.7

Remark thatβTq>0, since2.5impliesαq βTq.

Corollary 2.2. Suppose thataj∈A^1/k−j_αj , whereαj≥ −1 forj0, . . . , k−1, and letq∈ {0, . . . , k− 1}be the smallest index for whichα_qmax_j0,...,k−1{αj}>0. Then each solution base of1.1admits at mosts≤qsolutionsfsatisfyingσTf< βTs. In particular, there are at mosts≤qsolutions fsatisfyingσTf 0.

To estimate the quantity_k

j1σ_Tfjby usingTheorem 2.1, we set γ_T

j

:max

β_T0, . . . , βT

j

, j0, . . . , q. 2.8

EvidentlyγTj>0 forj∈ {s, . . . , q}, andγTj≤0 forj∈ {0, . . . , s−1}.

Corollary 2.3. Suppose thataj∈A^1/k−jαj , whereαj≥ −1 forj0, . . . , k−1, and letq∈ {0, . . . , k− 1}be the smallest index for whichα_q max_j0,...,k−1{αj} >0. Let{f1, . . . , f_k}be a solution base of 1.1. Ifq0, then_k

j1σTfj kα0, while ifq≥1, then k−q

αq^q−1

js

γT

j

≤^k

j1

σT

fj

≤kαq. 2.9

Note that the sum in2.9is considered to be empty, if sq.

2.2. Gundersen-Steinbart-Wang Method forT-Order

We proceed to give an alternative statement ofTheorem 2.1and its corollaries by modifying the key steps in4 . This yields a natural way to define possible intervals forT-orders of solutions of1.1. As a consequence, we get a useful estimate following fromCorollary 2.3.

Setδj : αj1k−jfor allj0, . . . , k−1. Lets1∈ {0, . . . , k−1}be the smallest index satisfyingα_s1 max_j0,...,k−1{αj}>0, which is equivalent to

δs1

k−s₁ max

j0,...,k−1

δ_j k−j

>1. 2.10

If s1 cannot be found, then all solutions of 1.1 are of zero T-order by Theorem Aii.

Otherwise, for a givens_m,m∈N, lets_m1∈ {0, . . . , sm−1}be the smallest index satisfying δs_m1−δsm

sm−s_m1 max

j0,...,sm−1

δj−δsm

sm−j

>1. 2.11

Eventually this process will stop, yielding a finite list of indicess₁, . . . , s_psuch thatp≤kand s1> s2>· · ·> sp≥0. Further, set

BTt: δst−δst−1

s_t−1−s_t −1, t1, . . . , p, 2.12 wheres₀:kandδ_k:0. Due to resemblance between2.12and4, Equation2.4 , it seems plausible that the possible nonzeroT-orders of solutions of1.1in the unit disc case could be found among the numbersBTt, wheret1, . . . , p. However,Example 3.1shows that this is not the case.

The following lemma allows us to view the results inSection 2.1in a new perspective.

In particular,Lemma 2.4emphasizes the connection betweenBTandγ_T. Lemma 2.4. One has the following:

iBT1>BT2>· · ·>BTp>0;

iiβTst BTtfor allt∈ {1, . . . , p};

iiiγ_Tq BT1,γ_Tj BTtfor alls_t≤j < s_t−1andt∈ {2, . . . , p}, andγ_Tj≤0 for all j < s_p. In particular,s_ps.

By relying onLemma 2.4,Theorem 2.1, andCorollary 2.2, we proceed to state possible intervals for T-orders of functions in solution bases of 1.1in the caseaj ∈ A^1/k−jαj , where α_j≥ −1 forj0, . . . , k−1. In fact, each solution base of1.1contains the following:

iat leastk−s1solutionsfsatisfyingσTf BT1;

iiat leastk−s_tsolutionsfsatisfyingσ_Tf∈BTt,BT1 fort2, . . . , p;

iiiat mosts_psolutionsfsatisfyingσ_Tf∈0,BTsp.

For the following application, let{f1, . . . , fk}be a solution base of1.1. Knowing the possible intervals forT-orders, we get

k j1

σT

fj

≥k−s1BT1 · · ·

s_p−1−sp

BT

p

sp·0δspsp−k. 2.13

In view ofLemma 2.4, the lower estimates in2.9and2.13are equal.

Finally, we point out a useful consequence of2.13. Ifsp 0, thenδsp sp δ0. If s_p >0, thenδ0−δ_sp/sp ≤1 by2.11, andδ_sp s_p ≥δ₀. Hence, in both cases we can state that

k j1

σ_T f_j

≥δ_sps_p−k≥δ₀−k≥α₀k, 2.14

where the equalities hold, ifα₀max_j0,...,k−1{αj}>0.

2.3. Growth of Solutions with Respect to Maximum Modulus

Alongside of theT-order, we may also define theM-order of growth off∈ HDby

σM

f

:lim sup

r→1⁻

loglogM r, f

−log1−r , 2.15

where Mr, f : max_|z|r|fz|is the maximum modulus of f. It is well known that the inequalities

σ_T f

≤σ_M f

≤σ_T f

1 2.16

are satisfied for all f ∈ HD, and all possibilities allowed by 2.16 can be assumed 9, Theorems 3.5–3.7 . A functionf ∈ HDis said to belong to the weighted Hardy spaceH_α^∞, if there existsα≥0 such that

sup

z∈D

1− |z|^{2 α} fz<∞. 2.17

Functions of maximal growth in

α>pH_α^∞ are distinguished by denotingf ∈ H^∞_p, if p inf{α≥ 0 :f ∈H_α^∞}. Remark thatH₀^∞ H^∞is the space of all bounded analytic functions inD. The union

α>0H_α^∞is also known as the Korenblum spaceA^−∞10 , and since11 H^∞_p is also known asG_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 toM-order.

Theorem Bsee3, Theorem 1.4 . Suppose thata_j∈H^∞_pj1k−j, wherep_j≥ −1 forj0, . . . , k− 1, and denotep_k:−1.

iSuppose that

j1,...,kmin k

p0−pj

j pj

>1, 2.18

and let 1 ≤ α < ∞. Then all solutions f of 1.1 satisfyσ_Mf ≤ α if and only if max_j0,...,k−1{pj} ≤α.

iiAll nontrivial solutionsfof1.1satisfyσ_Mf≤max_j0,...,k−1{p_j}, and

j1,...,kmin k

p0−pj

j pj

≤max σM

f ,1

. 2.19

iiiSuppose that 2.18 holds. If q ∈ {0, . . . , k − 1} is the smallest index for which p_q max_j0,...,k−1{pj}, then each solution base of 1.1contains at leastk−qlinearly independent solutionsfsuch thatσMf pq.

To conclude 3, Equation 4.17 in the proof of Theorem B, the inequality 3, Equation1.9 , corresponding to2.18, must be strict. By a simple modification of the proof of Theorem B, the assumption2.18can be relaxed to

j0,...,k−1max pj

>1, 2.20

which allows us to apply Theorem Biiialso in the case that there are solutionsf satisfying σ_Mf ≤ 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 inSection 3.2. In this case

j1,...,kmin k

p₀−p_j

j pj

−4, max

j0,...,k−1

pj

>1. 2.21

Note that by takingjkin2.18, we obtainp₀>1. Hence2.18implies2.20.

Theorem 2.5corresponds toTheorem 2.1.

Theorem 2.5. Suppose that aj ∈ H^∞_pj1k−j, where pj ≥ −1 forj 0, . . . , k −1, and let q ∈ {0, . . . , k−1}be the smallest index for whichp_q max_j0,...,k−1{pj}. If s ∈ {0, . . . , q}, then each solution base of 1.1contains at leastk−slinearly independent solutionsfsuch that

js1,...,kmin

k−s p_s−p_j

j−s pj

≤max σM

f ,1

. 2.22

Note that2.22gives information onσMfonly in the case when the minimum in 2.22is strictly greater than 1. The cases0 inTheorem 2.5reduces to Theorem Bii, and the casesqreduces to Theorem Biiiwith the assumption2.20, since now

jq1,...,kmin

k−q pq−pj

j−q pj

pq max

j0,...,k−1

pj

, 2.23

where the minimum is attained for j k. For a similar argumentation, see 2.5. Our contribution is to extend2.22fors ∈ {1, . . . , q−1}. The proof ofTheorem 2.5is sketched inSection 7, and the sharpness and the the special casesk2 andk3 are further discussed inSection 3.2.

Letq∈ {0, . . . , k−1}be the smallest index for whichpq max_j0,...,k−1{pj}. Ifpq ≤1, then all solutionsfin each solution base of1.1satisfyσMf≤1 by Theorem Bii. Suppose thatp_q>1. In order to state the following corollaries ofTheorem 2.5, we denote

βMs: min

js1,...,k

k−s p_s−p_j

j−s pj

, s0, . . . , q, 2.24

wherep_k:−1. Moreover, we define s:min

s∈

0, . . . , q

:βMs>1

. 2.25

Remark thatβMq>1, since2.23impliespq βMq.

Corollary 2.6. Suppose that a_j ∈ H^∞_p

j1k−j, wherep_j ≥ −1 for j 0, . . . , k −1, and letq ∈ {0, . . . , k−1}be the smallest index for whichp_q max_j0,...,k−1{pj}>1. Then each solution base of 1.1admits at mosts ≤qsolutionsf satisfyingσMf < βMs. In particular, there are at most s≤qsolutionsfsatisfyingσ_Mf≤1.

To estimate the quantity_k

j1σ_Mfjby usingTheorem 2.5, we set γ_M

j

:max

β_M0, . . . , βM

j

, j 0, . . . , q. 2.26

EvidentlyγMj>1 forj∈ {s, . . . , q}, andγMj≤1 forj ∈ {0, . . . , s−1}.

Corollary 2.7. Suppose that a_j ∈ H^∞_p

j1k−j, wherep_j ≥ −1 for j 0, . . . , k −1, and letq ∈ {0, . . . , k−1} be the smallest index for which pq max_j0,...,k−1{pj} > 1. Let {f1, . . . , fk} be a solution base of 1.1. Ifq0, then_k

j1σMfj kp0, while ifq≥1, then k−q

p_q^q−1

js

γ_M j

≤^k

j1

σ_M f_j

≤kp_q. 2.27

Note that the sum in2.27is considered to be empty, ifsq.

2.4. Gundersen-Steinbart-Wang Method forM-Order

We proceed to give an alternative statement ofTheorem 2.5and its corollaries by modifying the key steps in4 . This yields a natural way to define the possible intervals forM-orders of solutions of1.1. As a consequence, we get a useful estimate following fromCorollary 2.7.

Setδj : pj1k−jfor allj 0, . . . , k−1. Lets1∈ {0, . . . , k−1}be the smallest index satisfyingp_s1max_j0,...,k−1{pj}>1, which is equivalent to

δ_s1

k−s1 max

j0,...,k−1

δj

k−j

>2. 2.28

If s1 cannot be found, then all solutions f of 1.1 satisfy σMf ≤ 1 by Theorem Bii.

Otherwise, for a givens_m,m∈N, lets_m1∈ {0, . . . , sm−1}be the smallest index satisfying δ_sm1−δ_sm

s_m−s_m1 max

j0,...,sm−1

δ_j−δ_sm s_m−j

>2. 2.29

Eventually this process will stop, yielding a finite list of indicess1, . . . , spsuch thatp≤kand s1> s2>· · ·> sp≥0. Further, set

BMt: δst−δs_t−1

s_t−1−st −1, t1, . . . , p, 2.30 wheres0 : kandδk : 0. ByExample 3.1, it is possible that1.1possesses a solutionf of M-order strictly greater than one such thatσ_Mf/BMtfor allt1, . . . , p.

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

Lemma 2.8. One has the following:

iBM1>BM2>· · ·>BMp>1;

iiβ_Mst BMtfor allt∈ {1, . . . , p};

iiiγ_Mq BM1,γ_Mj BMtfor alls_t≤j < s_t−1andt∈ {2, . . . , p}, andγ_Mj≤1 for allj < sp. In particular,sps.

By relying onLemma 2.8,Theorem 2.5, andCorollary 2.6, we proceed to state possible intervals forM-orders of functions in solution bases of1.1in the casea_j ∈H^∞_p

j1k−j, where pj≥ −1 forj0, . . . , k−1. In fact, each solution base of1.1contains the following:

iat leastk−s₁solutionsfsatisfyingσ_Mf BM1;

iiat leastk−s_tsolutionsfsatisfyingσ_Mf∈BMt,BM1 fort2, . . . , p;

iiiat mosts_psolutionsfsatisfyingσ_Mf∈0,BMsp.

For results of the same type, we refer to12, Theorem 1 and 13, Corollary 1 . To compareiandiito the estimates given in13, Corollary 1 , note that there is−1 in2.30 instead of−2 in13, Equation1.3 . Evidently, assertionsiandiiimprove the estimates

given for theM-orders of solutions in13, Corollary 1 . Moreover, by means of2.16we see thatiandiireduce to13, Corollary 1 , if we consider the growth of solutions of1.1 with respect toT-order.

For the following application, let{f1, . . . , fk}be a solution base of1.1. Knowing the possible intervals forM-orders, we get

k j1

σ_M f_j

≥k−s₁BM1 · · ·

s_p−1−s_p BM

p

s_p·0δ_sps_p−k. 2.31

Corresponding to the case in Section 2.2, by means of Lemma 2.8 we see that the lower estimates in2.27and2.31are equal.

Finally, we point out a practical estimate, which is a consequence of2.31. Ifsp 0, thenδspspδ0−sp. Ifsp>0, thenδ0−δsp/sp≤2 by2.29, andδspsp≥δ0−sp. Hence, in both cases we can state that

k j1

σ_M f_j

≥δ₀−s_p−k≥p₀k−s_p. 2.32

We conclude that ifs₁ 0, then the equalities hold in2.32, since in this cases_p s₁ 0.

Note that if2.18holds, then we can conclude thats_p0.

3. Sharpness Discussion

We may assume that max_j0,...,k−1{αj} > 0, for otherwise all solutions are of zeroT-order. If k2, then the statement ofTheorem 2.1is contained in Theorem A, and all the assertions are sharp2, Examples 3 and 6 .

Ifk3, then we have three different cases to consider.

A1Ifα1, α2 ≤α0, then all nontrivial solutionsfof1.1satisfyσTf α0. In this case s0q.

A2Ifα₀ < α₁ andα₂ ≤ α₁, then in every solution base{f1, f₂, f₃}of1.1there are at least two solutionsf1andf2such thatσTfj α1for bothj 1,2, and all solutions fjsatisfy

σT

fj

≥min

3α0−2α1,3 2α0−1

2α2, α0

, j1,2,3. 3.1

In this cases0 ors1q.

A3If α0, α1 < α2, then in every solution base{f1, f2, f3} of1.1there is at least one solutionf₁such thatσ_Tf1 α₂, two solutionsf₁andf₂such that

σ_T f_j

≥min{2α1−α₂, α₁}, j1,2, 3.2 and all solutionsf_jsatisfy3.1. In this cases0,s1, ors2q.

It is clear that the assertion inA1is sharp, and so are the ones inA2by2, Example 10 . Moreover,2, Example 9 shows that the assertions inA3are sharp fors0 ands2.

Example 3.2shows the sharpness of the assertions inA3fors 0,1,2. That is, in all cases there exists a solution for which the lower bound for theT-order of growth is attained.

3.2. Sharpness ofTheorem 2.5

We may assume that max_j0,...,k−1{pj} > 1, for otherwise all solutions f of 1.1 satisfy max{σMf,1}1, and we cannot conclude anything from2.22. Ifk2, then the statement ofTheorem 2.5is contained in Theorem B, and all the assertions are sharp by2, Examples 3 and 6 . In the case of2, Examples 3 and 6 , forβ >1, linearly independent solutionsf1and f₂satisfyσ_Mf1 βandσ_Mf2 β2. Moreover,a_j ∈ H^∞_p

j12−j, wherep₀ β1 and p1 β2. Note that max{p0, p1} p1 β2 > 1, and henceq 1. An easy computation shows the sharpness fors 0 and fors q 1. In the case of2, Example 6 , forβ > 1, linearly independent solutions f₁ and f₂ satisfy σ_Mf1 β and σ_Mf2 β. Moreover, aj ∈ H^∞_pj12−j, where p0 β and p1 −1/2. Note now that max{p0, p1} p0 β > 1, and henceq 0. This example shows the sharpness fors q0. For another example, see 3, Example 2 .

Ifk3, then we have three different cases to consider.

B1Ifp₁, p₂≤p₀, thens0q, and all nontrivial solutionsfof1.1satisfyσ_Mf p₀ by2.23.

B2Ifp₀ < p₁, andp₂ ≤ p₁, then in every solution base{f1, f₂, f₃}of1.1there are at least two solutionsf1andf2such thatσMfj p1for bothj 1,2, and all solutions f_jsatisfy

max σM

fj

,1

≥min

3p0−2p1,3 2p0−1

2p2, p0

, j1,2,3. 3.3

In this cases0 ors1q.

B3If p₀, p₁ < p₂, then in every solution base {f1, f₂, f₃} of 1.1 there is at least one solutionf1such thatσMf1 p2, two solutionsf1andf2such that

max σ_M

f_j ,1

≥min

2p₁−p₂, p₁

, j1,2, 3.4

and all solutionsf_jsatisfy3.3. In this cases0,s1, ors2q.

It is clear that the assertion inB1is sharp. By2, Example 10 , we see that the asser- tion inB2corresponding tos 1 is sharp. In this case, for β > 1, linearly independent solutionsf1, f2, andf3 satisfyσMf1 σMf2 βandσMf3 0. Nowaj ∈ H^∞_pj13−j, wherep₀ 2/3β,p₁ β2, andp₂ 0. Moreover, by2, Example 9 , we see that the assertions in B3 are sharp fors 0 and s 2. In this case for β > 1, linearly indepen- dent solutionsf1,f2, andf3 satisfyσMf1 σMf2 βandσMf3 2β. Moreover,aj ∈ H^∞_p

j13−j, wherep₀ 4/3β,p₁ β, andp₂ 2β.Example 3.2shows the sharpness of the assertions inB3fors0,1,2. That is, in all cases there exists a solution for which equality holds in2.22.

3.3. Examples

Example 3.1shows that a unit disc counterpart of the finite set constructed by Gundersen- Steinbart-Wang does not contain growth orders of solutions of

fa1zfa0zf 0, 3.5

if coefficients belong either to weighted Bergman spaces or to weighted Hardy spaces.

Example 3.1. Letα, β∈Rbe constants satisfying 1< β < α <2β−1. Then the functions

f₁z 1−z^αβexp 1

1−z _α

1

1z _β

,

f2z 1−z^αβexp 1

1z

_β 3.6

are linearly independent analytic solutions of3.5, where

a₀z β²

1z^22β β

α3β γz 1−z1z^2β − α

αβ 1−z^2α

− αβ

1−z^1α1z^1ββ αβ 1−z² ,

a₁z 2β

1z^1β− α

1−z^1α α2β−1 1−z

3.7

belong toHD, andγ αβ−2/α3β∈0,1/2.

It is clear thata_j ∈A^1/2−jαj , whereα₀ β−1 andα₁ α−1. We calculate thats₁ 1, s2 0,BT1 α−1, andBT2 2β−α−1. Hence2β−α−1, α−1 is the only possible interval forT-orders of solutions of3.5. Sinceσ_Tf2 β−1, we conclude that theT-order of a solution does not have to be one of the endpoints.

On the other hand, it is also clear thataj ∈ H^∞_pj12−j, wherep0 βandp1 α. We calculate thats₁ 1,s₂ 0,BM1 α, andBM2 2β−α. Hence2β−α, α is the only possible interval forM-orders of solutions of3.5. Sinceσ_Mf2 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 Theorems2.1and2.5in the case that they do not reduce to known results.

Example 3.2. Letβ >1, and denotegz 5/1−z^β. Then the functions

f_jz 1−z^βexp

gz^j , j 1,2,3, 3.8

are linearly independent solutions offa2zfa1zfa0zf 0, where

a2z P₂

gz

1−zQ

gz, a1z P₁

gz

1−z²Q

gz, a0z β³P₀

gz

1−z³Q

gz 3.9

are such that

P₂ζ 54βζ⁸−27βζ⁷−24βζ⁶

108β54 ζ⁵−

82β63

ζ⁴3βζ³

22β39 ζ²−

6β24 ζ6, P₁ζ −108β²ζ¹⁰72β²ζ⁹

27β²−54β ζ⁸

27β−135β² ζ⁷

24β15β² ζ⁶

64β²−108β−18 ζ⁵

2182β−51β² ζ⁴

−

3β15β² ζ³

31β²−22β−13 ζ²

86β−14β² ζ2β²−2, P₀ζ 108ζ¹¹−234ζ¹⁰126ζ⁹123ζ⁸−276ζ⁷183ζ⁶−104ζ⁵40ζ⁴

−6ζ³−4ζ²,

Qζ −18ζ⁵21ζ⁴−13ζ²8ζ−2.

3.10

The zeros ofQζlie in the open disc of radius 121/18 centered at the origin by14, Lem- ma 1.3.2 . Since|gz|>|5/1−z|>5/2>39/18 for allz∈D, we conclude thata0, a1, a2 ∈ HD. In fact, the coefficientsa₀, a₁, and a₂satisfy

a2z∼ 1

1−z _3β1

, a1z∼

1 1−z

_5β2

, a0z∼

1 1−z

_6β3

, 3.11

in a neighborhood ofz1, while they are bounded in a neighborhood of any boundary point in∂D\ {1}.

Note thataj ∈A^1/3−jαj , whereα2 3β−1,α1 5/2β−1, andα0 2β−1. Evidently σ_Tfj βj−1 forj 1,2,3. We deduce that there is one solutionf₃such thatσ_Tf3 α₂ 3β−1, two solutionsf2andf3such that

σT

f3

> σT

f2

min{2α1−α2, α1}2β−1, 3.12

and three solutionsf₁,f₂, andf₃such that

σT

f3

> σT

f2

> σT

f1

min

3α0−2α1,3 2α0− 1

2α2, α0

β−1. 3.13

That is, in all casess 0,1,2 there exists a solution for which the lower bound in2.4is attained. Further, this example is in line withCorollary 2.2, since all solutionsf1,f2, andf3

are of strictly positiveT-order, and in this cases^∗0.

NowγT0 βT0 β−1,γT1 βT1 2β−1, andγT2 βT2 3β−1. It follows that for the solution base{f1, f₂, f₃}equality holds in the first inequality in2.9, and for the solution base{f1f₃, f₂f₃, f₃}equality holds in the last inequality in2.9. This shows the sharpness ofCorollary 2.3.

On the other hand,aj ∈H^∞_p

j13−j, wherep2 3β,p1 5β/2, andp0 2β. Evidently σ_Mfj βj forj 1,2,3. We deduce that there is one solutionf₃such thatσ_Mf3 p₂ 3β >1, two solutionsf2andf3such that

σM

f3

> σM

f2

min

2p1−p2, p1

2β >1, 3.14

and three solutionsf₁,f₂, andf₃such that

σM

f3

> σM

f2

> σM

f1

min

3p0−2p1,3 2p0− 1

2p2, p0

β. 3.15

That is, in all casess 0,1,2 there exists a solution for which the lower bound in2.22is attained. Further, this example is in line withCorollary 2.6, since all solutionsf1,f2, andf3

are ofM-order strictly greater than 1, and in this cases0.

Nowγ_M0 β_M0 β,γ_M1 β_M1 2β, andγ_M2 β_M2 3β. It follows that for the solution base{f1, f2, f3}equality holds in2.27, and for the solution base{f1 f3, f2f3, f3}upper bound for the sum ofM-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 fromC.

Lemma Csee4, Lemma 6.4 . Letf0,1, f0,2, . . . , f0,mbe m≥2 linearly independent meromorphic solutions of

y^ka_0,k−1zy^k−1· · ·a_0,0zy0, k≥m, 4.1

wherea0,0z, . . . , a0,k−1zare meromorphic functions inD. For 1≤p≤m−1, set

f_p,j

f_p−1,j1 f_p−1,1

, j1, . . . , m−p. 4.2

Thenf_p,1, f_p,2, . . . , f_p,m−pare linearly independent meromorphic solutions of

y^k−pa_p,k−p−1zy^k−p−1· · ·ap,0zy0, 4.3

where

ap,jz ^k−p1

nj1

n j1

a_p−1,nzf_p−1,1^n−j−1z

f_p−1,1z 4.4

forj 0, . . . , k−p−1. Herea_n,k−nz≡1 for alln0, . . . , p.

Lemma Dsee15, Lemma Eb . Letkandjbe integers satisfyingk > j≥0, and letε >0. Iff is meromorphic inDsuch thatσ_Tf<∞, andf^j/≡0, then

D

f^kz f^jz

1/k−j

1− |z|^σTfεdmz<∞. 4.5

4.1. Cases1

Let k ≥ 3, q ≥ 2, s 1, and β_T1 > 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 βT1≤α1≤0. Let{f0,1, f0,2, . . . , f0,k}be a solution base of1.1, and assume on the contrary to the assertion that there exists12 linearly independent solutionsf_0,1andf_0,2such that max{σTf0,1, σTf0,2}:σ < βT1. Then the meromorphic functiong: f0,1/f0,2satisfies σTg≤σ. Moreover, Lemma C implies thatgsatisfies

g^k−1a_1,k−2zg^k−2· · ·a1,0zg0, 4.6

where

a_1,jz a_0,j1z ^k

nj2

n j1

a_0,nzf_0,1^n−j−1z

f0,1z 4.7

forj0,1, . . . , k−2, anda_0,kz≡1. Therefore

|a0,1z| ≤ |a1,0z|^k

n2

n 1

|a0,nz|

f_0,1ⁿ⁻¹z f0,1z

, 4.8

where

|a1,0z| ≤

g^k−1z gz

|a1,k−2z|

g^k−2z gz

· · ·|a1,1z|

gz gz

, 4.9

sincegsatisfies4.6. Putting the last two inequalities together, we obtain

|a0,1z|^k−1

j1

a1,jz

g^jz gz

^k

n2

|a0,nz|

f_0,1ⁿ⁻¹z f0,1z

. 4.10

Letε >0. Raising both sides to the power 1/k−1and integrating over the discD0, rof radiusr∈0,1with respect to1− |z|^2α1−εdmz, we obtain

D0,r|a0,1z|^1/k−1

1− |z|^{2 α1−ε}dmz

^k−1

j1

D

a1,jz^1/k−1

g^jz gz

1/k−1

1− |z|^{2 α1−ε}dmz

^k

n2

D|a0,nz|^1/k−1

f_0,1ⁿ⁻¹z f0,1z

1/k−1

1− |z|^{2 α1−ε}dmz.

4.11

To deal with the second sum in4.11, consider

I_n:

D|a0,nz|^1/k−1

f_0,1ⁿ⁻¹z f0,1z

1/k−1

1− |z|^{2 α1−ε}dmz, n2, . . . , k. 4.12

By Lemma D,

I_k

D

f_0,1^k−1z f0,1z

1/k−1

1− |z|^{2 α1−ε}dmz<∞ 4.13

for ε > 0 being small enough since σTf0,1 ≤ σ < βT1 ≤ α1. Moreover, by H ¨older’s inequality, with indicesk−1/k−nandk−1/n−1, we have

I_n≤

D|a0,nz|^1/k−n

1− |z|^{2 αnε}dmz

_k−n/k−1

·

⎛

⎜⎝

D

f_0,1ⁿ⁻¹z f_0,1z

1/n−1

1− |z|^{2 ω1n}dmz

⎞

⎟⎠

n−1/k−1 4.14

for alln2, . . . , k−1, where

ω_sn: k−sαs−α_n

n−s α_n−2k−n−s

n−s ε. 4.15

The first member in the product is finite sincea0,n ∈ A^1/k−nαn for alln 2, . . . , k−1 by the assumption, and so is the second one forε >0 small enough since

σT

f0,1

≤σ < βT1≤ k−1α1−αn

n−1 αn, n2, . . . , k−1, 4.16 by the antithesis. Thus_k

n2I_nis finite forε >0 being small enough.