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,**

^{1}**Janne Gr ¨ohn,**

^{2}**Janne Heittokangas,**

^{2}**and Jouni R ¨atty ¨a**

^{2}*1**Departamento de Matem´aticas, Pontificia Universidad Cat´olica de Chile, Casilla 306,*
*Correo 22 Santiago, 6904411 Macual Santiago, Chile*

*2**Department 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 of*f*^{k}*a** _{k−1}*zf

^{k−1}· · ·

*a*1zf

^{}

*a*0zf 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- 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 for

*T- andM-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}*zf

^{k−1}· · ·

*a*1zf

^{}

*a*0zf0, 1.1 where the coeﬃcients

*a*

_{0}z, . . . , a

*zare analytic functions in the unit discD: {z:|z|*

_{k−1}*<*

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 that*T- 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 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 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 are*α-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 and*M-orders, giving detailed information about the growth of solutions. Finally,*
these findings are applied to estimate the sums of*T- 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 regarding*M-orders of solutions of*1.1, excluding the sketched proof ofTheorem 2.5
inSection 7.

LetMDandHDdenote the sets of all meromorphic and analytic functions inD.

For simplicity, we write*α*^{} :max{α,0}for any*α*∈R,|fz||gz|if there exists a constant

*C >* 0 independent of*z*such that |fz| ≤ *C|g*z|, and*fz* ∼ *gz*if there exist constants
*C*_{1}*>*0 and*C*_{2}*>*0 independent of*z*such that*C*_{1}|gz| ≤ |fz| ≤*C*_{2}|gz|.

* 2.1. Growth of Solutions with Respect to Nevanlinna Characteristic*
The

*T-order of growth off*∈ MDis defined as

*σ**T*

*f*

:lim sup

*r*→1^{−}

log^{}*T*
*r, f*

−log1−*r,* 2.1

where*T*r, f*is the Nevanlinna characteristic off. Forp >*0 and*α >*−1, the weighted Bergman
*spaceA*^{p}* _{α}*consists of those

*f*∈ HDfor which

*f*

*A*^{p}*α* :

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 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 of1.1with respect to*T*-order.

**Theorem A** see 2, Theorems 1 and 2 . Suppose that *a**j* ∈ A^{1/k−j}*α**j* *, where* *α**j* ≥ *0 for* *j*
0, . . . , k−*1, and denoteα** _{k}*:

*0.*

i*Let 0* ≤ *α <* ∞. Then all solutions *f* *of* 1.1 *satisfy* *σ**T*f ≤ *α* *if and only if*
max* _{j0,...,k−1}*{α

*j*} ≤

*α.*

ii*All nontrivial solutionsfof*1.1*satisfy*

*j1,...,k*min
*k*

*α*_{0}−*α*_{j}

*j* *α*_{j}

≤*σ*_{T}*f*

≤ max

*j0,...,k−1*

*α*_{j}

*.* 2.3

iii*If* *q* ∈ {0, . . . , k −1}*is the smallest index for which* *α**q* max* _{j0,...,k−1}*{α

*j*}, then each

*solution base of*1.1

*contains at leastk*−

*qlinearly independent solutionsf*

*such that*

*σ*

*f*

_{T}*α*

_{q}*.*

The assumption*a** _{j}*∈A

^{1/k−j}

*α*

*j*in Theorem Aicannot be replaced by

*a*

*∈*

_{j}*A*

^{1/k−j}

_{α}*; see 8 . We refine Theorem A and then further underscore its consequences.*

_{j}**Theorem 2.1. Suppose that**a*j*∈A^{1/k−j}*α**j* *, whereα**j*≥ −1 for*j*0, . . . , k−1, and let*q*∈ {0, . . . , k−1}

*be the smallest index for whichα** _{q}* max

*{α*

_{j0,...,k−1}*j*}. If

*s*∈ {0, . . . , q}, then each solution base of 1.1

*contains at leastk*−

*slinearly independent solutionsfsuch that*

*js1,...,k*min

k−*s*

*α** _{s}*−

*α*

_{j}*j*−*s* *α**j*

≤*σ**T*

*f*

≤*α*^{}_{q}*,* 2.4

*whereα** _{k}*:−1.

The case *s* 0 clearly reduces to Theorem Aii. If *s* *q, then the conditionα**q*
max* _{j0,...,k−1}*{α

*j*}implies that

*jq1,...,k*min

*k*−*q*
*α**q*−*α**j*

*j*−*q* *α**j*

min

*jq1,...,k*

*k*−*j*
*α**q*−*α**j*

*j*−*q* *α**q*

*α**q**,* 2.5

where the minimum is attained for*j* *k. Hence the assertion of*Theorem 2.1for*s* *q*is
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
*k*2 and*k*3 are further discussed inSection 3.1.

Let*q* ∈ {0, . . . , k−1}be the smallest index for which*α** _{q}* max

*{α*

_{j0,...,k−1}*j*}. If

*α*

*≤0, then all solutions in each solution base of1.1are of zero*

_{q}*T-order by Theorem Aii. Suppose*that

*α*

*q*

*>*0. In order to state the following corollaries ofTheorem 2.1, we denote

*β**T*s: min

*js1,...,k*

k−*s*

*α** _{s}*−

*α*

_{j}*j*−*s* *α**j*

*,* *s*0, . . . , q, 2.6

where*α**k*:−1. Moreover, we define
*s** ^{}*:min

*s*∈

0, . . . , q

:*β** _{T}*s

*>*0

*.* 2.7

Remark that*β**T*q*>*0, since2.5implies*α**q* *β**T*q.

**Corollary 2.2. Suppose that**a*j*∈A^{1/k−j}_{α}_{j}*, whereα**j*≥ −1 for*j*0, . . . , k−*1, and letq*∈ {0, . . . , k−
1}*be the smallest index for whichα** _{q}*max

*{α*

_{j0,...,k−1}*j*}

*>0. Then each solution base of*1.1

*admits*

*at mosts*

*≤*

^{}*qsolutionsfsatisfyingσ*

*T*f

*< β*

*T*s

*. In particular, there are at most*

^{}*s*

*≤*

^{}*qsolutions*

*fsatisfyingσ*

*T*f

*0.*

To estimate the quantity_{k}

*j1**σ** _{T}*f

*j*by usingTheorem 2.1, we set

*γ*

_{T}*j*

:max

*β** _{T}*0, . . . , β

*T*

*j*

*,* *j*0, . . . , q. 2.8

Evidently*γ**T*j*>*0 for*j*∈ {s^{}*, . . . , q}, andγ**T*j≤0 for*j*∈ {0, . . . , s* ^{}*−1}.

**Corollary 2.3. Suppose that**a*j*∈A^{1/k−j}*α**j* *, whereα**j*≥ −1 for*j*0, . . . , k−*1, and letq*∈ {0, . . . , k−
1}*be the smallest index for whichα** _{q}* max

*{α*

_{j0,...,k−1}*j*}

*>0. Let*{f1

*, . . . , f*

*}*

_{k}*be a solution base of*1.1. If

*q0, then*

_{k}*j1**σ**T*f*j* *kα*0*, while ifq*≥*1, then*
*k*−*q*

*α**q*^{q−1}

*js*^{}

*γ**T*

*j*

≤^{k}

*j1*

*σ**T*

*f**j*

≤*kα**q**.* 2.9

Note that the sum in2.9is considered to be empty, if *s*^{}*q.*

**2.2. Gundersen-Steinbart-Wang Method for**T**-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 for*T*-orders of
solutions of1.1. As a consequence, we get a useful estimate following fromCorollary 2.3.

Set*δ**j* : α*j*1k−*j*for all*j*0, . . . , k−1. Let*s*1∈ {0, . . . , k−1}be the smallest index
satisfying*α*_{s}_{1} max* _{j0,...,k−1}*{α

*j*}

*>*0, which is equivalent to

*δ**s*1

*k*−*s*_{1} max

*j0,...,k−1*

*δ*_{j}*k*−*j*

*>*1. 2.10

If *s*1 cannot be found, then all solutions of 1.1 are of zero *T*-order by Theorem Aii.

Otherwise, for a given*s** _{m}*,

*m*∈N, let

*s*

*∈ {0, . . . , s*

_{m1}*m*−1}be the smallest index satisfying

*δ*

*s*

*−*

_{m1}*δ*

*s*

*m*

*s**m*−*s** _{m1}* max

*j0,...,s**m*−1

*δ**j*−*δ**s**m*

*s**m*−*j*

*>*1. 2.11

Eventually this process will stop, yielding a finite list of indices*s*_{1}*, . . . , s** _{p}*such that

*p*≤

*k*and

*s*1

*> s*2

*>*· · ·

*> s*

*p*≥0. Further, set

B*T*t: *δ**s**t*−*δ**s**t−1*

*s** _{t−1}*−

*s*

*−1,*

_{t}*t*1, . . . , p, 2.12 where

*s*

_{0}:

*k*and

*δ*

*:0. Due to resemblance between2.12and4, Equation2.4 , it seems plausible that the possible nonzero*

_{k}*T*-orders of solutions of1.1in the unit disc case could be found among the numbersB

*T*t, where

*t*1, . . . , 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 betweenB*T*and*γ** _{T}*.

**Lemma 2.4. One has the following:**iB*T*1*>*B*T*2*>*· · ·*>*B*T*p*>0;*

ii*β**T*s*t* B*T*t*for allt*∈ {1, . . . , p};

iii*γ** _{T}*q B

*T*1,

*γ*

*j B*

_{T}*T*t

*for alls*

*≤*

_{t}*j < s*

_{t−1}*andt*∈ {2, . . . , p}, and

*γ*

*j≤*

_{T}*0 for all*

*j < s*

_{p}*. In particular,s*

_{p}*s*

^{}*.*

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 case*a**j* ∈ A^{1/k−j}*α**j* , where
*α** _{j}*≥ −1 for

*j*0, . . . , k−1. In fact, each solution base of1.1contains the following:

iat least*k*−*s*1solutions*f*satisfying*σ**T*f B*T*1;

iiat least*k*−*s** _{t}*solutions

*f*satisfying

*σ*

*f∈B*

_{T}*T*t,B

*T*1 for

*t*2, . . . , p;

iiiat most*s** _{p}*solutions

*f*satisfying

*σ*

*f∈0,B*

_{T}*T*s

*p*.

For the following application, let{f1*, . . . , f**k*}be a solution base of1.1. Knowing the possible
intervals for*T*-orders, we get

*k*
*j1*

*σ**T*

*f**j*

≥k−*s*1B*T*1 · · ·

*s** _{p−1}*−

*s*

*p*

B*T*

*p*

*s**p*·0*δ**s**p**s**p*−*k.* 2.13

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

Finally, we point out a useful consequence of2.13. If*s**p* 0, then*δ**s**p* *s**p* *δ*0. If
*s*_{p}*>*0, thenδ0−*δ*_{s}* _{p}*/s

*p*≤1 by2.11, and

*δ*

_{s}

_{p}*s*

*≥*

_{p}*δ*

_{0}. Hence, in both cases we can state that

*k*
*j1*

*σ*_{T}*f*_{j}

≥*δ*_{s}_{p}*s** _{p}*−

*k*≥

*δ*

_{0}−

*k*≥

*α*

_{0}

*k,*2.14

where the equalities hold, if*α*_{0}max* _{j0,...,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 off*∈ HDby

*σ**M*

*f*

:lim sup

*r*→1^{−}

log^{}log^{}*M*
*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 function*f* ∈ HD*is said to belong to the weighted Hardy spaceH*_{α}^{∞}, if
there exists*α*≥0 such that

sup

*z∈D*

1− |*z|*^{2} ^{α}*f*z*<*∞. 2.17

Functions of maximal growth in

*α>p**H*_{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 inD. The union

*α>0**H*_{α}^{∞}*is also known as the Korenblum space*A^{−∞}10 , and since11 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**see3, Theorem 1.4 . Suppose that*a** _{j}*∈H

^{∞}

_{p}

_{j}_{1k−j}

*, wherep*

*≥ −1 for*

_{j}*j*0, . . . , k−

*1, and denotep*

*:−1.*

_{k}i*Suppose that*

*j1,...,k*min
*k*

*p*0−*p**j*

*j* *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

*{p*

_{j0,...,k−1}*j*} ≤

*α.*

ii*All nontrivial solutionsfof*1.1*satisfyσ** _{M}*f≤max

*{p*

_{j0,...,k−1}

_{j}^{}}, and

*j1,...,k*min
*k*

*p*0−*p**j*

*j* *p**j*

≤max
*σ**M*

*f*
*,*1

*.* 2.19

iii*Suppose that* 2.18 *holds. If* *q* ∈ {0, . . . , k − 1} *is the smallest index for which*
*p** _{q}* max

*{p*

_{j0,...,k−1}*j*}, then each solution base of 1.1

*contains at leastk*−

*qlinearly*

*independent solutionsfsuch thatσ*

*M*f

*p*

*q*

*.*

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−1*max
*p**j*

*>*1, 2.20

which allows us to apply Theorem Biiialso 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 inSection 3.2. In this case

*j1,...,k*min
*k*

*p*_{0}−*p*_{j}

*j* *p**j*

−4, max

*j0,...,k−1*

*p**j*

*>*1. 2.21

Note that by taking*jk*in2.18, we obtain*p*_{0}*>*1. Hence2.18implies2.20.

Theorem 2.5corresponds toTheorem 2.1.

**Theorem 2.5. Suppose that***a**j* ∈ H^{∞}_{p}_{j}_{1k−j}*, where* *p**j* ≥ −1 for*j* 0, . . . , k −*1, and let* *q* ∈
{0, . . . , k−1}*be the smallest index for whichp** _{q}* max

*{p*

_{j0,...,k−1}*j*}. If

*s*∈ {0, . . . , q}, then each

*solution base of*1.1

*contains at leastk*−

*slinearly independent solutionsfsuch that*

*js1,...,k*min

k−*s*
*p** _{s}*−

*p*

_{j}*j*−*s* *p**j*

≤max
*σ**M*

*f*
*,*1

*.* 2.22

Note that2.22gives information on*σ**M*fonly in the case when the minimum in
2.22is strictly greater than 1. The case*s*0 inTheorem 2.5reduces to Theorem Bii, and
the case*sq*reduces to Theorem Biiiwith the assumption2.20, since now

*jq1,...,k*min

*k*−*q*
*p**q*−*p**j*

*j*−*q* *p**j*

*p**q* max

*j0,...,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 extend2.22for*s* ∈ {1, . . . , q−1}. The proof ofTheorem 2.5is sketched
inSection 7, and the sharpness and the the special cases*k*2 and*k*3 are further discussed
inSection 3.2.

Let*q*∈ {0, . . . , k−1}be the smallest index for which*p**q* max* _{j0,...,k−1}*{p

*j*}. If

*p*

*q*≤1, then all solutions

*f*in each solution base of1.1satisfy

*σ*

*M*f≤1 by Theorem Bii. Suppose that

*p*

_{q}*>*1. In order to state the following corollaries ofTheorem 2.5, we denote

*β**M*s: min

*js1,...,k*

k−*s*
*p** _{s}*−

*p*

_{j}*j*−*s* *p**j*

*,* *s*0, . . . , q, 2.24

where*p** _{k}*:−1. Moreover, we define

*s*

*:min*

^{}*s*∈

0, . . . , q

:*β**M*s*>*1

*.* 2.25

Remark that*β**M*q*>*1, since2.23implies*p**q* *β**M*q.

**Corollary 2.6. Suppose that***a** _{j}* ∈ H

^{∞}

_{p}

*j*1k−j*, wherep** _{j}* ≥ −1 for

*j*0, . . . , k −

*1, and letq*∈ {0, . . . , k−1}

*be the smallest index for whichp*

*max*

_{q}*{p*

_{j0,...,k−1}*j*}

*>1. Then each solution base of*1.1

*admits at mosts*

*≤*

^{}*qsolutionsf*

*satisfyingσ*

*M*f

*< β*

*M*s

*. In particular, there are at most*

^{}*s*

*≤*

^{}*qsolutionsfsatisfyingσ*

*f≤*

_{M}*1.*

To estimate the quantity_{k}

*j1**σ** _{M}*f

*j*by usingTheorem 2.5, we set

*γ*

_{M}*j*

:max

*β** _{M}*0, . . . , β

*M*

*j*

*,* *j* 0, . . . , q. 2.26

Evidently*γ**M*j*>*1 for*j*∈ {s^{}*, . . . , q}, andγ**M*j≤1 for*j* ∈ {0, . . . , s* ^{}*−1}.

**Corollary 2.7. Suppose that***a** _{j}* ∈ H

^{∞}

_{p}

*j*1k−j*, wherep** _{j}* ≥ −1 for

*j*0, . . . , k −

*1, and letq*∈ {0, . . . , k−1}

*be the smallest index for which*

*p*

*q*max

*{p*

_{j0,...,k−1}*j*}

*>*

*1. Let*{f1

*, . . . , f*

*k*}

*be a*

*solution base of*1.1. If

*q0, then*

_{k}*j1**σ**M*f*j* *kp*0*, 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, if*s*^{}*q.*

**2.4. Gundersen-Steinbart-Wang Method for**M-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 for*M-orders of*
solutions of1.1. As a consequence, we get a useful estimate following fromCorollary 2.7.

Set*δ**j* : p*j*1k−*j*for all*j* 0, . . . , k−1. Let*s*1∈ {0, . . . , k−1}be the smallest index
satisfying*p*_{s}_{1}max* _{j0,...,k−1}*{p

*j*}

*>*1, which is equivalent to

*δ*_{s}_{1}

*k*−*s*1 max

*j0,...,k−1*

*δ**j*

*k*−*j*

*>*2. 2.28

If *s*1 cannot be found, then all solutions *f* of 1.1 satisfy *σ**M*f ≤ 1 by Theorem Bii.

Otherwise, for a given*s** _{m}*,

*m*∈N, let

*s*

*∈ {0, . . . , s*

_{m1}*m*−1}be the smallest index satisfying

*δ*

_{s}*−*

_{m1}*δ*

_{s}

_{m}*s** _{m}*−

*s*

*max*

_{m1}*j0,...,s**m*−1

*δ** _{j}*−

*δ*

_{s}

_{m}*s*

*−*

_{m}*j*

*>*2. 2.29

Eventually this process will stop, yielding a finite list of indices*s*1*, . . . , s**p*such that*p*≤*k*and
*s*1*> s*2*>*· · ·*> s**p*≥0. Further, set

B*M*t: *δ**s**t*−*δ**s*_{t−1}

*s** _{t−1}*−

*s*

*t*−1,

*t*1, . . . , p, 2.30 where

*s*0 :

*k*and

*δ*

*k*: 0. ByExample 3.1, it is possible that1.1possesses a solution

*f*of

*M-order strictly greater than one such thatσ*

*f*

_{M}*/*B

*M*tfor all

*t*1, . . . , 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:**

iB*M*1*>*B*M*2*>*· · ·*>*B*M*p*>1;*

ii*β** _{M}*s

*t*B

*M*t

*for allt*∈ {1, . . . , p};

iii*γ** _{M}*q B

*M*1,

*γ*

*j B*

_{M}*M*t

*for alls*

*≤*

_{t}*j < s*

_{t−1}*andt*∈ {2, . . . , p}, and

*γ*

*j≤*

_{M}*1 for*

*allj < s*

*p*

*. In particular,s*

*p*

*s*

^{}*.*

By relying onLemma 2.8,Theorem 2.5, andCorollary 2.6, we proceed to state possible
*intervals forM-orders of functions in solution bases of*1.1in the case*a** _{j}* ∈H

^{∞}

_{p}

*j*1k−j, where
*p**j*≥ −1 for*j*0, . . . , k−1. In fact, each solution base of1.1contains the following:

iat least*k*−*s*_{1}solutions*f*satisfying*σ** _{M}*f B

*M*1;

iiat least*k*−*s** _{t}*solutions

*f*satisfying

*σ*

*f∈B*

_{M}*M*t,B

*M*1 for

*t*2, . . . , p;

iiiat most*s** _{p}*solutions

*f*satisfying

*σ*

*f∈0,B*

_{M}*M*s

*p*.

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 the*M-orders of solutions in*13, Corollary 1 . Moreover, by means of2.16we
see thatiandiireduce to13, Corollary 1 , if we consider the growth of solutions of1.1
with respect to*T*-order.

For the following application, let{f1*, . . . , f**k*}be a solution base of1.1. Knowing the
possible intervals for*M-orders, we get*

*k*
*j1*

*σ*_{M}*f*_{j}

≥k−*s*_{1}B*M*1 · · ·

*s** _{p−1}*−

*s*

*B*

_{p}*M*

*p*

*s** _{p}*·0

*δ*

_{s}

_{p}*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 in2.27and2.31are equal.

Finally, we point out a practical estimate, which is a consequence of2.31. If*s**p* 0,
then*δ**s**p**s**p**δ*0−*s**p*. If*s**p**>*0, thenδ0−*δ**s**p*/s*p*≤2 by2.29, and*δ**s**p**s**p*≥*δ*0−*s**p*. Hence,
in both cases we can state that

*k*
*j1*

*σ*_{M}*f*_{j}

≥*δ*_{0}−*s** _{p}*−

*k*≥

*p*

_{0}

*k*−

*s*

_{p}*.*2.32

We conclude that if*s*_{1} 0, then the equalities hold in2.32, since in this case*s*_{p}*s*_{1} 0.

Note that if2.18holds, then we can conclude that*s** _{p}*0.

**3. Sharpness Discussion**

**3.1. Sharpness of****Theorem 2.1**We may assume that max* _{j0,...,k−1}*{α

*j*}

*>*0, for otherwise all solutions are of zero

*T-order. If*

*k*2, then the statement ofTheorem 2.1is contained in Theorem A, and all the assertions are sharp2, Examples 3 and 6 .

If*k*3, then we have three diﬀerent cases to consider.

A1If*α*1*, α*2 ≤*α*0, then all nontrivial solutions*f*of1.1satisfy*σ**T*f *α*0. In this case
*s*0*q.*

A2If*α*_{0} *< α*_{1} and*α*_{2} ≤ *α*_{1}, then in every solution base{f1*, f*_{2}*, f*_{3}}of1.1there are at
least two solutions*f*1and*f*2such that*σ**T*f*j* *α*1for both*j* 1,2, and all solutions
*f**j*satisfy

*σ**T*

*f**j*

≥min

3α0−2α1*,*3
2*α*0−1

2*α*2*, α*0

*,* *j*1,2,3. 3.1

In this case*s*0 or*s*1*q.*

A3If *α*0*, α*1 *< α*2, then in every solution base{f1*, f*2*, f*3} of1.1there is at least one
solution*f*_{1}such that*σ** _{T}*f1

*α*

_{2}, two solutions

*f*

_{1}and

*f*

_{2}such that

*σ*_{T}*f*_{j}

≥min{2α1−*α*_{2}*, α*_{1}}, *j*1,2, 3.2
and all solutions*f** _{j}*satisfy3.1. In this case

*s*0,

*s*1, or

*s*2

*q.*

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 for*s*0 and*s*2.

Example 3.2shows the sharpness of the assertions inA3for*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* _{j0,...,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 from2.22. If

*k*2, 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 , for

*β >*1, linearly independent solutions

*f*1and

*f*

_{2}satisfy

*σ*

*f1*

_{M}*β*and

*σ*

*f2*

_{M}*β*2. Moreover,

*a*

*∈ H*

_{j}^{∞}

_{p}

*j*12−j, where*p*_{0} *β*1 and
*p*1 *β*2. Note that max{p0*, p*1} *p*1 *β*2 *>* 1, and hence*q* 1. An easy computation
shows the sharpness for*s* 0 and for*s* *q* 1. In the case of2, Example 6 , for*β >* 1,
linearly independent solutions *f*_{1} and *f*_{2} satisfy *σ** _{M}*f1

*β*and

*σ*

*f2*

_{M}*β. Moreover,*

*a*

*j*∈ H

^{∞}

_{p}

_{j}_{12−j}, where

*p*0

*β*and

*p*1 −1/2. Note now that max{p0

*, p*1}

*p*0

*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.

B1If*p*_{1}*, p*_{2}≤*p*_{0}, then*s*0*q, and all nontrivial solutionsf*of1.1satisfy*σ** _{M}*f

*p*

_{0}by2.23.

B2If*p*_{0} *< p*_{1}, and*p*_{2} ≤ *p*_{1}, then in every solution base{f1*, f*_{2}*, f*_{3}}of1.1there are at
least two solutions*f*1and*f*2such that*σ**M*f*j* *p*1for both*j* 1,2, and all solutions
*f** _{j}*satisfy

max
*σ**M*

*f**j*

*,*1

≥min

3p0−2p1*,*3
2*p*0−1

2*p*2*, p*0

*,* *j*1,2,3. 3.3

In this case*s*0 or*s*1*q.*

B3If *p*_{0}*, p*_{1} *< p*_{2}, then in every solution base {f1*, f*_{2}*, f*_{3}} of 1.1 there is at least one
solution*f*1such that*σ**M*f1 *p*2, two solutions*f*1and*f*2such that

max
*σ*_{M}

*f*_{j}*,*1

≥min

2p_{1}−*p*_{2}*, p*_{1}

*,* *j*1,2, 3.4

and all solutions*f** _{j}*satisfy3.3. In this case

*s*0,

*s*1, or

*s*2

*q.*

It is clear that the assertion inB1is sharp. By2, Example 10 , we see that the asser-
tion inB2corresponding to*s* 1 is sharp. In this case, for *β >* 1, linearly independent
solutions*f*1*, f*2, and*f*3 satisfy*σ**M*f1 *σ**M*f2 *β*and*σ**M*f3 0. Now*a**j* ∈ H^{∞}_{p}_{j}_{13−j},
where*p*_{0} 2/3β,*p*_{1} *β*2, and*p*_{2} 0. Moreover, by2, Example 9 , we see that the
assertions in B3 are sharp for*s* 0 and *s* 2. In this case for *β >* 1, linearly indepen-
dent solutions*f*1,*f*2, and*f*3 satisfy*σ**M*f1 *σ**M*f2 *β*and*σ**M*f3 2β. Moreover,*a**j* ∈
H^{∞}_{p}

*j*13−j, where*p*_{0} 4/3β,*p*_{1} *β, andp*_{2} 2β.Example 3.2shows the sharpness of the
assertions inB3for*s*0,1,2. 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

*f*^{}*a*1zf^{}*a*0zf 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

*f*_{1}z 1−*z** ^{αβ}*exp
1

1−*z*
_{α}

1

1*z*
_{β}

*,*

*f*2z 1−*z** ^{αβ}*exp
1

1*z*

* _{β}* 3.6

are linearly independent analytic solutions of3.5, where

*a*_{0}z *β*^{2}

1*z*^{22β} *β*

*α*3β
*γz*
1−*z1z*^{2β} − *α*

*αβ*
1−*z*^{2α}

− *αβ*

1−*z*^{1α}1*z*^{1β}*β*
*αβ*
1−*z*^{2} *,*

*a*_{1}z 2β

1*z*^{1β}− *α*

1−*z*^{1α} *α*2β−1
1−*z*

3.7

belong toHD, and*γ* α*β*−2/α3β∈0,1/2.

It is clear that*a** _{j}* ∈A

^{1/2−j}

*α*

*j*, where

*α*

_{0}

*β*−1 and

*α*

_{1}

*α*−1. We calculate that

*s*

_{1}1,

*s*2 0,B

*T*1

*α*−1, andB

*T*2 2β−

*α*−1. Hence2β−

*α*−1, α−1 is the only possible interval for

*T-orders of solutions of*3.5. Since

*σ*

*f2*

_{T}*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^{∞}_{p}_{j}_{12−j}, where*p*0 *β*and*p*1 *α. We*
calculate that*s*_{1} 1,*s*_{2} 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 Theorems2.1and2.5in the case that they do not reduce to known results.

*Example 3.2. Letβ >*1, and denote*gz 5/1*−*z** ^{β}*. Then the functions

*f** _{j}*z 1−

*z*

*exp*

^{β}*gz*^{j}*,* *j* 1,2,3, 3.8

are linearly independent solutions of*f*^{}*a*2zf^{}*a*1zf^{}*a*0zf 0, where

*a*2z *P*_{2}

*gz*

1−*zQ*

*gz,* *a*1z *P*_{1}

*gz*

1−*z*^{2}*Q*

*gz,* *a*0z *β*^{3}*P*_{0}

*gz*

1−*z*^{3}*Q*

*gz* 3.9

are such that

*P*_{2}ζ 54βζ^{8}−27βζ^{7}−24βζ^{6}

108β54
*ζ*^{5}−

82β63

*ζ*^{4}3βζ^{3}

22β39
*ζ*^{2}−

6β24
*ζ*6,
*P*_{1}ζ −108β^{2}*ζ*^{10}72β^{2}*ζ*^{9}

27β^{2}−54β *ζ*^{8}

27β−135β^{2} *ζ*^{7}

24β15β^{2} *ζ*^{6}

64β^{2}−108β−18 *ζ*^{5}

2182β−51β^{2} *ζ*^{4}

−

3β15β^{2} *ζ*^{3}

31β^{2}−22β−13 *ζ*^{2}

86β−14β^{2} *ζ*2β^{2}−2,
*P*_{0}ζ 108ζ^{11}−234ζ^{10}126ζ^{9}123ζ^{8}−276ζ^{7}183ζ^{6}−104ζ^{5}40ζ^{4}

−6ζ^{3}−4ζ^{2}*,*

*Qζ *−18ζ^{5}21ζ^{4}−13ζ^{2}8ζ−2.

3.10

The zeros of*Qζ*lie 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 all*z*∈D, we conclude that*a*0*, a*1*, a*2 ∈
HD. In fact, the coeﬃcients*a*_{0}*, a*_{1}*,* and *a*_{2}satisfy

*a*2z∼
1

1−*z*
_{3β1}

*,* *a*1z∼

1
1−*z*

_{5β2}

*,* *a*0z∼

1
1−*z*

_{6β3}

*,* 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^{1/3−j}*α**j* , where*α*2 3β−1,*α*1 5/2β−1, and*α*0 2β−1. Evidently
*σ** _{T}*f

*j*

*βj*−1 for

*j*1,2,3. We deduce that there is one solution

*f*

_{3}such that

*σ*

*f3*

_{T}*α*

_{2}3β−1, two solutions

*f*2and

*f*3such that

*σ**T*

*f*3

*> σ**T*

*f*2

min{2α1−*α*2*, α*1}2β−1, 3.12

and three solutions*f*_{1},*f*_{2}, and*f*_{3}such that

*σ**T*

*f*3

*> σ**T*

*f*2

*> σ**T*

*f*1

min

3α0−2α1*,*3
2*α*0− 1

2*α*2*, α*0

*β*−1. 3.13

That is, in all cases*s* 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 solutions*f*1,*f*2, and*f*3

are of strictly positive*T-order, and in this cases*^{∗}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*, f*_{2}*, f*_{3}}equality holds in the first inequality in2.9, and for the
solution base{f1*f*_{3}*, f*_{2}*f*_{3}*, f*_{3}}equality holds in the last inequality in2.9. This shows the
sharpness ofCorollary 2.3.

On the other hand,*a**j* ∈H^{∞}_{p}

*j*13−j, where*p*2 3β,*p*1 5β/2, and*p*0 2β. Evidently
*σ** _{M}*f

*j*

*βj*for

*j*1,2,3. We deduce that there is one solution

*f*

_{3}such that

*σ*

*f3*

_{M}*p*

_{2}3β >1, two solutions

*f*2and

*f*3such that

*σ**M*

*f*3

*> σ**M*

*f*2

min

2p1−*p*2*, p*1

2β >1, 3.14

and three solutions*f*_{1},*f*_{2}, and*f*_{3}such that

*σ**M*

*f*3

*> σ**M*

*f*2

*> σ**M*

*f*1

min

3p0−2p1*,*3
2*p*0− 1

2*p*2*, p*0

*β.* 3.15

That is, in all cases*s* 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 solutions*f*1,*f*2, and*f*3

are of*M-order strictly greater than 1, and in this cases** ^{}*0.

Now*γ** _{M}*0

*β*

*0*

_{M}*β,γ*

*1*

_{M}*β*

*1 2β, and*

_{M}*γ*

*2*

_{M}*β*

*2 3β. It follows that for the solution base{f1*

_{M}*, f*2

*, f*3}equality holds in2.27, and for the solution base{f1

*f*3

*, f*2

*f*3

*, f*3}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 fromC.

**Lemma C**see4, 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}zy^{k−1}· · ·*a*_{0,0}zy0, *k*≥*m,* 4.1

*wherea*0,0z, . . . , a0,k−1z*are meromorphic functions in*D. For 1≤*p*≤*m*−*1, set*

*f*_{p,j}

*f*_{p−1,j1}*f*_{p−1,1}

_{}

*,* *j*1, . . . , m−*p.* 4.2

*Thenf*_{p,1}*, f*_{p,2}*, . . . , f*_{p,m−p}*are linearly independent meromorphic solutions of*

*y*^{k−p}*a** _{p,k−p−1}*zy

^{k−p−1}· · ·

*a*

*p,0*zy0, 4.3

*where*

*a**p,j*z ^{k−p1}

*nj1*

*n*
*j*1

*a** _{p−1,n}*z

*f*

_{p−1,1}^{n−j−1}z

*f** _{p−1,1}*z 4.4

*forj* 0, . . . , k−*p*−*1. Herea** _{n,k−n}*z≡

*1 for alln*0, . . . , p.

**Lemma D**see15, Lemma Eb . Let*kandjbe integers satisfyingk > j*≥*0, and letε >0. Iff*
*is meromorphic in*D*such thatσ** _{T}*f

*<*∞, and

*f*

^{j}

*/*≡

*0, then*

D

*f*^{k}z
*f*^{j}z

1/k−j

1− |*z|*^{σ}^{T}^{f}^{ε}*dmz<*∞. 4.5

* 4.1. Cases*1

Let *k* ≥ 3, *q* ≥ 2, *s* 1, and *β** _{T}*1

*>*0, since otherwise there is nothing to prove.

In particular, if *α*1 ≤ 0, then 2.4 is trivial, since by taking *j* *k* in 2.6, we obtain
*β**T*1≤*α*1≤0. Let{f0,1*, f*0,2*, . . . , f*0,k}be a solution base of1.1, and assume on the contrary
to the assertion that there exist*s*12 linearly independent solutions*f*_{0,1}and*f*_{0,2}such that
max{σ*T*f0,1, σ*T*f0,2}:*σ < β**T*1. Then the meromorphic function*g*: f0,1*/f*0,2^{}satisfies
*σ**T*g≤*σ. Moreover, Lemma C implies thatg*satisfies

*g*^{k−1}*a*_{1,k−2}zg^{k−2}· · ·*a*1,0zg0, 4.6

where

*a*_{1,j}z *a*_{0,j1}z ^{k}

*nj2*

*n*
*j*1

*a*_{0,n}z*f*_{0,1}^{n−j−1}z

*f*0,1z 4.7

for*j*0,1, . . . , k−2, and*a*_{0,k}z≡1. Therefore

|a0,1z| ≤ |a1,0z|^{k}

*n2*

*n*
1

|a0,nz|

*f*_{0,1}^{n−1}z
*f*0,1z

*,* 4.8

where

|a1,0z| ≤

*g*^{k−1}z
*gz*

|a1,k−2z|

*g*^{k−2}z
*gz*

· · ·|a1,1z|

*g*^{}z
*gz*

*,* 4.9

since*g*satisfies4.6. Putting the last two inequalities together, we obtain

|a0,1z|^{k−1}

*j1*

*a*1,jz

*g*^{j}z
*gz*

^{k}

*n2*

|a0,nz|

*f*_{0,1}^{n−1}z
*f*0,1z

*.* 4.10

Let*ε >*0. Raising both sides to the power 1/k−1and integrating over the disc*D0, r*of
radius*r*∈0,1with respect to1− |z|^{2}^{α}^{1}^{−ε}*dmz, we obtain*

*D0,r*|a0,1z|^{1/k−1}

1− |z|^{2} ^{α}^{1}^{−ε}*dmz*

^{k−1}

*j1*

D

*a*1,jz^{1/k−1}

*g*^{j}z
*gz*

1/k−1

1− |*z|*^{2} ^{α}^{1}^{−ε}*dmz*

^{k}

*n2*

D|a0,nz|^{1/k−1}

*f*_{0,1}^{n−1}z
*f*0,1z

1/k−1

1− |z|^{2} ^{α}^{1}^{−ε}*dmz.*

4.11

To deal with the second sum in4.11, consider

*I** _{n}*:

D|a0,nz|^{1/k−1}

*f*_{0,1}^{n−1}z
*f*0,1z

1/k−1

1− |*z|*^{2} ^{α}^{1}^{−ε}*dmz,* *n*2, . . . , k. 4.12

By Lemma D,

*I*_{k}

D

*f*_{0,1}^{k−1}z
*f*0,1z

1/k−1

1− |*z|*^{2} ^{α}^{1}^{−ε}*dmz<*∞ 4.13

for *ε >* 0 being small enough since *σ**T*f0,1 ≤ *σ < β**T*1 ≤ *α*1. Moreover, by H ¨older’s
inequality, with indicesk−1/k−*n*andk−1/n−1, we have

*I** _{n}*≤

D|a0,nz|^{1/k−n}

1− |*z|*^{2} ^{α}^{n}^{ε}*dmz*

_{k−n/k−1}

·

⎛

⎜⎝

D

*f*_{0,1}^{n−1}z
*f*_{0,1}z

1/n−1

1− |*z|*^{2} ^{ω}^{1}^{n}*dmz*

⎞

⎟⎠

n−1/k−1 4.14

for all*n*2, . . . , k−1, where

*ω** _{s}*n: k−

*sα*

*s*−

*α*

_{n}*n*−*s* *α** _{n}*−2k−

*n*−

*s*

*n*−*s* *ε.* 4.15

The first member in the product is finite since*a*0,n ∈ A^{1/k−n}*α**n* for all*n* 2, . . . , k−1 by the
assumption, and so is the second one for*ε >*0 small enough since

*σ**T*

*f*0,1

≤*σ < β**T*1≤ k−1α1−*α**n*

*n*−1 *α**n**,* *n*2, . . . , k−1, 4.16
by the antithesis. Thus_{k}

*n2**I** _{n}*is finite for

*ε >*0 being small enough.