# On a characteristic property of periodic entire functions

01 Jan 1980-Kodai Mathematical Journal (Department of Mathematics, Tokyo Institute of Technology)-Vol. 3, Iss: 2, pp 253-286

About: This article is published in Kodai Mathematical Journal.The article was published on 1980-01-01 and is currently open access. It has received 1 citation(s) till now. The article focuses on the topic(s): Characteristic property & Entire function.

Topics: Characteristic property (65%), Entire function (58%)

...read more

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

##### Citations

##### References

More filters

••

289 citations

••

198 citations

••

127 citations

••

32 citations