Linear differential equations in the unit disc with analytic solutions of finite order
Authors: Korhonen Risto , Rättyä Jouni 
Published: February 2007

A function

g
g

, analytic in the unit disc

D
D

, belongs to the weighted Hardy space

H
q
∞

H_q^\infty

if

sup

0
≤
r
>
1

M
(
r
,
g
)
(
1
−

r
2

)
q

>
∞

\sup _{0\le r>1}M(r,g)(1-r^2)^q>\infty

, where

M
(
r
,
g
)

M(r,g)

is the maximum modulus of

g
(
z
)

g(z)

in the circle of radius

r
r

centered at the origin. If

g
g

belongs to

H
q
∞

H_q^\infty

for some

q
≥
0

q\geq 0

, then it is said to be an

H

\mathcal {H}

-function. Heittokangas has shown that all solutions of the linear differential equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis reverse-solidus dagger right-parenthesis EndLabel f Superscript left-parenthesis k right-parenthesis Baseline plus upper A Subscript k minus 1 Baseline left-parenthesis z right-parenthesis f Superscript left-parenthesis k minus 1 right-parenthesis Baseline plus midline-horizontal-ellipsis plus upper A 1 left-parenthesis z right-parenthesis f Superscript prime Baseline plus upper A 0 left-parenthesis z right-parenthesis f equals 0 comma EndLayout">

(\dagger)

f

(
k
)

+

A

k
−
1

(
z
)

f

(
k
−
1
)

+
⋯
+

A
1

(
z
)

f
′

+

A
0

(
z
)
f
=
0
,

\begin{equation} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots +A_1(z)f’+A_0(z)f=0,\tag {\dagger } \end{equation}

where

A
j

(
z
)

A_j(z)

is analytic in

D
D

for all

j
=
0
,
…
,
k
−
1

j=0,\ldots ,k-1

, are of finite order of growth in

D
D

if and only if all coefficients

A
j

(
z
)

A_j(z)

are

H

\mathcal {H}

-functions. It is said that

g
∈

G
p

g\in G_p

when

p
=
inf
{
q
≥
0
:
g
∈

H
q
∞

}

p=\inf \{q\geq 0 : g\in H^\infty _q\}

. In this study it is shown that if all coefficients

A
j

(
z
)

A_j(z)

of

(
†
)

(\dagger )

satisfy

A
j

∈

G

p
j

A_j\in G_{p_j}

for all

j
=
0
,
…
,
k
−
1

j=0,\ldots ,k-1

, then all nontrivial solutions

f
f

of

(
†
)

(\dagger )

satisfy
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="min Underscript j equals 1 comma ellipsis comma k Endscripts StartFraction p 0 minus p Subscript j Baseline Over j EndFraction minus 2 less-than-or-equal-to sigma Subscript upper M Baseline left-parenthesis f right-parenthesis less-than-or-equal-to max left-brace right-brace comma 0 comma minus minus of maxmax comma comma equals equals j 0 comma ellipsis comma minus minus k 1 pj minus minus kj 1 comma">

min

j
=
1
,
…
,
k

p
0

−

p
j

j

−
2
≤

σ
M

(
f
)
≤
max

{
0
,

max

j
=
0
,
…
,
k
−
1

p
j

k
−
j

−
1
}

,

\begin{equation*} \min _{j=1,\ldots ,k} \frac {p_0-p_j}{j}-2 \le \sigma _M(f) \le \max \left \{0, \max _{j=0,\ldots ,k-1} \frac {p_j}{k-j}-1\right \}, \end{equation*}

where

p
k

:=
0

p_k:=0

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript upper M Baseline left-parenthesis f right-parenthesis colon equals limit sup StartFraction log Superscript plus Baseline log Superscript plus Baseline upper M left-parenthesis r comma f right-parenthesis Over minus log left-parenthesis 1 minus r right-parenthesis EndFraction period">

σ
M

(
f
)
:=

lim sup

r
→

1
−

log
+

⁡

log
+

⁡
M
(
r
,
f
)

−
log
⁡
(
1
−
r
)

.

\begin{equation*} \sigma _M(f):=\limsup _{r\to 1^-}\frac {\log ^+\log ^+ M(r,f)}{-\log (1-r)}. \end{equation*}

In addition, if

n
∈
{
0
,
…
,
k
−
1
}

n\in \{0,\ldots ,k-1\}

is the smallest index for which

p
n

k
−
n

=

max

j
=
0
,
…
,
k
−
1

p
j

k
−
j

,

\begin{equation*} \frac {p_n}{k-n} = \max _{j=0,\ldots ,k-1} \frac {p_j}{k-j}, \end{equation*}

then there are at least

k
−
n

k-n

linearly independent solutions of

(
†
)

(\dagger )

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript upper M Baseline left-parenthesis f right-parenthesis greater-than-or-equal-to max Underscript j equals 0 comma ellipsis comma k minus 1 Endscripts StartFraction p Subscript j Baseline Over k minus j EndFraction minus 2 period">

σ
M

(
f
)
≥

max

j
=
0
,
…
,
k
−
1

p
j

k
−
j

−
2.

\begin{equation*} \sigma _M(f)\geq \max _{j=0,\ldots ,k-1} \frac {p_j}{k-j} - 2. \end{equation*}

These results are a generalization of a recent result due to Chyzhykov, Gundersen and Heittokangas.