H.
URABE
AND C. C.
YANG
KODAI
MATH.
J.
3
(1980),
253-2B6
ON
A CHARACTERISTIC PROPERTY OF
PERIODIC
ENTIRE
FUNCTIONS
BY
HIRONOBU
URABE AND
CHUNG-CHUN
YANG
Introduction.
We shall pursue an investigation on a certain functional
equation treated in [8] with some overmuch restrictions. The functional equa-
tion
is related to the
following
problem: If two entire functions in a certain
class
have the same zero-sets (including multiplicities), then what can be said
about these functions ?
Denoting
by Gφ) the
class
of all entire functions each of which is periodic
with period b (ΦO) mod a non-constant entire function of order
less
than one
(cf. Def. in § 1), in this paper, we shall prove that if two entire functions
belonging to Gφj) 0—1, 2) have the same zero-sets (essentially), then they must
coincide up to a non-zero multiplicative constant (Theorem 1). In the
proof,
we
use the Borel-Nevanlinna type unicity theorem.
Note
that, together with Gφ), the
class
Jφ), consisting of the entire func-
tions each of which is periodic mod a non-constant polynomial of degree one, is
significant in factorization theory (under composition) of transcendental entire
functions (cf. for example, [1] or [7]).
Now recall some of the results of Gross ([2]). Among others, he proved
that
any non-constant, periodic, entire function H(z) has an infinite number of
fixed
points, that is, the zeros of H{z)—z. Further the
fixed
points play an
important
role especially in cases concerning periodic entire functions (in factor-
ization theory,
etc.).
So one might expect that periodic entire functions would
be uniquely determined by the sets of their
fixed
points. In this paper,
we'll
show that this is the case (Theorem 2).
Here,
the
first
author
wishes
to
express
his gratitude to Professors Y. Kusu-
noki (Kyoto Univ.) and M. Ozawa (Tokyo Inst. of Technology) for their encour-
agement and suggestions.
1. Statement of results.
At
first,
we'll
give
the definition of the
class
Gφ) explicitly.
DEFINITION.
For a non-zero constant b, we denote by Gφ) the
class
of
entire functions of the form
Received April 16, 1979
253
254 HIRONOBU URABE AND
CHUNG-CHUN
YANG
f(z)=h(z)+H(z),
where H{z) and h{z) are non-constant entire functions such that H{z) is periodic
with period b, H(z+b)=H(z), and that h{z) is of order
less
than one.
After
we've
written a short summary [9], we get a generalization. Our
generalized results are stated as
follows.
THEOREM
1. Let
f{z)^G{b
x
)
and
g{z)^G{b
2
)
for
some
non-zero
constants
b
3
0 = 1, 2).
Assume
that the
sets
of the
zeros
of f{z) and g{z) are
identical
{includ-
ing
multiplicities)
except
at
most
a
{sequence)
set
whose
exponent
of
convergence
is
less
than one. Then we
must
have
(1)
f{z)=cg{z)
for
some
non-zero
constant
c and b
1
/b
2
is a
rational
number.
Remark. Let
f{z)^G{b
λ
)
and
g{z)^G{b
2
)
be represented as
(2) f{z)=h(z)+H(z),
where H{zΛ~b
ι
)^H{z), K{z-\-b
2
)=K{z), h{z) and k{z) are non-constant entire
functions such that h and k are both of order
less
than one. Then the condi-
tion
of Theorem 1 concerning the zero-sets of / and g means that the identical
relation
(3) h(z)+H(z)=(k(z)+K(z))R(z)e
p
™
is
valid
for some meromorphic function R{z)
(Ξ£0)
of order
less
than one and for
some entire function p{z).
The conclusion (1) is equivalent to show that R{z) and p{z) are both constant.
Also
note that any meromorphic function R{z) (^0) of order
less
than one
can
be represented as
(4) R{z)=v{z)/u{z)
for some entire functions u{z) and v{z) (^0 and without common zeros), both of
which are of order
less
than one.
Periodic entire functions have the
following
characteristic property, as is
mentioned
in Introduction.
THEOREM
2. Let H
0
(z) 0 = 1, 2) be
non-constant
periodic
entire
functions
with
period
b
d
{resp.).
Assume
that the
sets
of the
fixed
points
of H
3
{z) are
identical
{including
multiplicities)
except
at
most
a
{sequence)
set
whose
exponent
of con-
vergence
is
less
than one. Then we have
necessarily
that
(5)
Hlz^niz)
and
bjb
2
is a
rational
number.
ON
A CHARACTERISTIC PROPERTY OF PERIODIC ENTIRE FUNCTIONS 255
2.
Lemmas.
For
the proof of Theorem 1, we shall need the following unicity theorem of
Borel-Nevanlinna type due to Niino-Ozawa [5].
LEMMA
A. Let G
3
(z) be a
transcendental
entire
function
and c
3
be a
non-zero
constant,
and let g(z) be an
entire
function, Ξ£0
such
that T(r, g)~o(T(r, Gj)) as
r tends lo infinity for any j
with
l^j^n.
Assume
that
there
exists
an
identical
relation
such
as
n
Σ
CjGj(z)
= g(z) ,
then we have
necessarily
Σ
δ(0, Gj)^n-1.
Here
T(r, *) and δ(a, *)
denote
the Nevanlinna
characteristic
function
and
defi-
ciency,
respectively.
Remark. Compared with the original Niino-Ozawa's lemma, the above
Lemma
A
will
seem slightly general. But the proof is essentially same.
Also
we use the following simple fact.
LEMMA
B. Let w{z) be a
meromorphic
function
of
order
less
than one and
assume
that
(6) w
for
some
constants
b (Φθ) and c. Then w{z) must be
constant.
For
completeness, we prove this. Indeed, if we consider the function
(7)
F(z)=w(z)-exp[-jz],
then
the assumption (6) implies
(8)
F(z+b)=F(z).
If w(z) is non-constant, then w(z) has at least one zero or one pole, since w(z)
is of order
less
than one by assumption. Then from (7) and (8), the exponent
of convergence of the zeros or poles of w(z) cannot be
less
than one, so that
the
order of w{z) cannot be so. This is a contradiction. Hence w(z) must be
constant.
3.
To prove Theorem 1, at the
first
step, we note here the following facts.
PROPOSITION
1. Let the identity (3) be
valid.
And
assume
that p{z) is con-
256
HIRONOBU
URABE
AND
CHUNG-CHUN
YANG
stant.
Then
R(z) is
constant
(ΦO)
and b
1
/b
2
is a
rational
number.
PROPOSITION
2. Let the
identity
(3) be
valid.
And
assume
that
b
L
/b
2
is a
rational
number.
Then
R(z) and p(z) are
constants.
Proof
of
Prop.
1. In
this case,
the
identity
(3) may be
written
as
(9) h(z)+H{z)={Kz)+K{z))R{z).
We
shall
introduce here
the
notations;
(10) h
J
(z)=h(z+jb
1
), k
J
(z)=Kz+jb
1
),
R
J
(z)=R(z+jb
1
),
for
any
natural number
j.
Since H(z+b
x
)=H{z)
by
assumption, noting
(10),
from
(9) we
have
so that
(11)
K(z+b
1
)R
1
-K(z)R-=(hi-h)-(k
1
R
1
-kR).
Setting
(12) S(z)^R(z+b
2
), r(z)=h(z+b
2
), s(z)=k{z+b
2
),
and
noting K(z+b
2
)—K(z), from
(11) we
obtain
(13)
K
(
z+bl)Sl
-K(z)S=(r
1
-r)-(s
1
S
1
-sS).
Here
we put
(14) r,(z)=r(*+7W, s,(2r)=sU+yW,
S
j
(z)=S(z+jb
1
)
for
any
natural number
j as in (10).
Note that
Ri(z+b
2
)=R(z+b
1
+b
2
)=S(z+b
1
)=S
1
(
<
z)
etc.,
by (10), (12) and (14).
Cancelling K(z+b
x
) from
the
identities
(11) and (13), we get
(15) (RS
ί
-R
1
S)K(z)=^R
1
ί(r
1
-r)-(s
1
S
1
-sS)-]
Now
the
right hand
side
of
this identity
is of
order
less
than
one,
while
if
(RS
1
—R
1
S)
is not
identically zero,
the
left
hand
side
is of
order
not
less
than
one,
since
K{z) is a
non-constant periodic entire function
(by
assumption). There-
fore
we
conclude
RS
1
-R
1
S=0
t
or
S/R-SJR^O.
Noting
(10) and (14),
this means that
S/R is
periodic with period
b
x
.
Since
S/R
is
a
meromorphic function
of
order
less
than
one, an
application
of
Lemma
B
ON
A
CHARACTERISTIC PROPERTY
OF
PERIODIC
ENTIRE
FUNCTIONS
257
gives
(16)
S/R=const.=c,
or
cR-S = 0.
Setting R(z)=v(z)/u(z)
as
in (4),
then
(16) can be
written
as
(17)
cu(z+bMz)-u(z)v(z+b
2
)=0.
Assume
that
R(z)
is
non-constant. Then
we
may
assume without
loss
of
generality that
u(z)
is
non-constant.
In
this case, since
u(z)
is of
order
less
than
one, u(z) has
zeros.
Assume
u(z
o
)=O. From
(17),
dividing
the
relation
by
u{z),
we
have
the
identity
(18) cu(z+b
2
)v(z)/u(z)=υ(z+b
2
).
Here
the
right hand side
of
(18)
is
entire,
and
since
u{z) and v(z)
have
no
common
zeros, u(z
o
)=O implies u(z
o
+b
2
)=O,
and so
again
by (18),
changing
the
variable
if
necessary, u(z
o
+2b
2
)=O. Repeating this argument,
we
obtain that
(19) u(z
o
+mb
2
)=O,
for any
natural number
m.
Then
from
(19) we
conclude that
the
exponent
of
convergence
of
the
zeros
of
u(z)
is
not
less
than
one, and
hence
the
order
of
u(z)
is
so.
This
is a
contra-
diction. Thus
we
have proved that
R(z)
is
constant.
Putting R(z)=const.=c (ΦO), from
(11) we
have (since i?
1
(z)=const.
=
c also)
(20)
dKiz+bJ-Kiz^i^-V-cik^k).
Here
the
left
hand side
of
(20)
is
periodic with period
b
2i
while
the
right hand
side
is of
order
less
than
one, so
that
we
conclude again that
(21)
#(z+^)-#Cε)=const.
Then
K(z+b
2
)=K(z)
and (21)
imply that
K\z) (the
first
derived function
of K(z),
non-constant
since
K(z)
is
non-constant
and
periodic)
is
periodic with periods
b
λ
and
b
2
.
Since non-constant entire function cannot
be
doubly periodic,
bjb
2
must
be
a
rational number, which
is
to be
proved.
Proof
of
Prop.
2.
By the
assumption,
we may pur
for some non-zero integers
m
and
n.
In
this case,
(22)
H(z+jb)=H(z), K(z+jb)=K(z)
for
any
natural number
j.
Putting
(23)
h^z)=h(z+jb)
f
kj(z)=k(z+jb),
Rj(z)=R(z+jb), p£z)=p
from
(3)
wepiave