On a theorem of A. and C. Rényi and a conjecture of C. C. Yang concerning periodicity of entire functions

Abstract

A theorem of A. and C. Rényi on periodic entire functions states that an entire function f(z) must be periodic if P(f(z)) is periodic, where P(z) is a nonconstant polynomial. By extending this theorem, we can answer some open questions related to the conjecture of C. C. Yang concerning periodicity of entire functions. Moreover, we give more general forms for this conjecture and we prove, in particular, that f(z) is periodic if either P(f (z))f(k)(z) or P(f(z))/f(k)(z) is periodic, provided that f(z) has a finite Picard exceptional value. We also investigate the periodicity of f(z) when f(z)n + a1f′(z) + ⋯ + akf(k)(z) is periodic. In all our results, the possibilities for the period of f(z) are determined precisely.

Full text

arXiv:2103.12182v2 [math.CV] 2 Apr 2021
ON A THEOREM OF A. AND C. R ´
ENYI AND A CONJECTURE OF C. C.
YANG CONCERNING PERIODICITY OF ENTIRE FUNCTIONS
Z. LATREUCH AND M. A. ZEMIRNI∗
ABS TR ACT. A theorem of A. and C. R´enyi on periodic entire functions states that an en-
tire function f(z)must be periodic if P(f(z)) is periodic, where P(z)is a non-constant
polynomial. By extending this theorem, we can answer some open questions related to the
conjecture of C. C. Yang concerning periodicity of entire functions. Moreover, we give
more general forms for this conjecture and we prove, in particular, that f(z)is periodic if
either P(f(z))f(k)(z)or P(f(z))/f(k)(z)is periodic, provided that f(z)has a finite Pi-
card exceptional value. We also investigate the periodicity of f(z)when f(z)n+a1f′(z) +
···+akf(k)(z)is periodic. In all our results, the possibilities for the period of f(z)are
determined precisely.
1. INTRO DUCTI ON
Periodicity of entire functions is associated with numerous difficult problems despite its
simple concept. It has been studied from different aspects, such as uniqueness theory, com-
posite functions, differential and functional equations; see [8] and references therein. In
this paper, special attention is paid to the problem of the periodicity of entire functions f(z)
when particular differential polynomials generated by f(z)are given to be periodic. By us-
ing concepts from Nevanlinna theory (see, e.g., [4,15]), we can extend the following result
due to Alfr´ed and Catherine R´enyi.
Theorem A ([12, Theorem 2]).Let Q(z)be an non-constant polynomial and f(z)be an
entire function. If Q(f(z)) is a periodic function, then f(z)must be periodic.
With the help of the extensions of Theorem Atogether with other results from Nevanlinna
theory, we study the problem mentioned above and answer some related open questions. In
particular, we are interested in the following conjecture and its variations.
Conjecture 1 (Generalized Yang’s conjecture).Let f(z)be a transcendental entire function
and n, k be positive integers. If f(z)nf(k)(z)is a periodic function, then f(z)is also a
periodic function.
This conjecture is originally stated in [5] for n= 1 and has been known as Conjecture of
C. C. Yang. The present form of Conjecture 1is stated in [7]. The case k= 1 of Conjecture 1
was settled in [13, Theorem 1] for n= 1, and in [7, Theorem 1.2] for n≥2.
In the following, we will present three results related to Conjecture 1, which are the main
subject of this paper. Before that, we recall that the order and the hyper-order of an entire
function f(z)are defined, respectively, by
ρ(f) = lim sup
r→∞
log T(r, f)
log r, ρ2(f) = lim sup
r→∞
log log T(r, f)
log r,
2020 Mathematics Subject Classification. 30D20, 30D35.
Key words and phrases. Differential polynomials, entire functions, Nevanlinna theory, order of growth,
periodic functions, Yang’s conjecture.
∗Corresponding author.
1

2 Z. LATREUCH AND M. A. ZEMIRNI∗
where T(r, f)is the Nevanlinna characteristic function of f(z).
Regarding the case when k≥2in Conjecture 1, we mention the following result.
Theorem B ([7, Theorem 1.2]).Let f(z)be a transcendental entire function and n, k be
positive integers. Suppose that f(z)nf(k)(z)is a periodic function with period c,f(z)has
a finite Picard exceptional value, and that ρ(f)<∞, then f(z)is a periodic function of
period cor (n+ 1)c.
If n= 1 and f(z)has a non-zero Picard exceptional value, then [6, Theorem 1.1] reveals
that the conclusion of Theorem Boccurs without condition on the order. The case when
n= 1 and f(z)has 0as a Picard exceptional value is proved in [9, p. 455]. Recently,
Theorem Bwas improved in [11, Theorem 1.1] to hold for entire functions with ρ2(f)<1
having a finite Borel exceptional value.
The idea of allowing nto be negative integer in Conjecture 1is considered in [10], where
the following particular result is obtained.
Theorem C ([10, Theorem 1.5]).Let f(z)be a transcendental entire and kbe a positive
integer. Suppose that f(k)(z)/f (z)is a periodic function with period c, and f(z)has a
Picard exceptional value d6= 0. Then f(z)is a periodic function with period c.
The conclusion of Theorem Cis not always true when d= 0, as shown in [10] by taking
f(z) = eeiz+z.
Another variation of Conjecture 1is addressed in [11], where the differential polynomial
f(z)nf(k)(z)is replaced with f(z)n+a1f′(z) + ···+akf(k)(z). In this regard, we present
the following result, which combines [11, Theorem 1.2, Remark 1.2(ii)].
Theorem D. Let f(z)be a transcendental entire function, and let k≥1be an integer and
a1,...,akare constants. If f(z)n+a1f′(z) + ··· +akf(k)(z)is a periodic function with
period c, and if one of the following conditions holds
(1) n= 2 or n≥4,
(2) n= 3 and ρ2(f)<1,
then f(z)is periodic of period cor nc.
The rest of the paper is devoted to improving Theorems A–D, and it is organized as fol-
lows. We prove two extensions of Theorem Ain Section 2, which will be used to prove the
results of Section 3. In Section 3, we improve and generalize Theorems Band C. Mean-
while, Section 4contains results improving the Case (2) in Theorem D.
2. ON A THEO RE M O F A. AND C . R ´
ENYI
Let f(z)be an entire function. The notation S(r, f )stands for any quantity satisfy-
ing S(r, f) = o(T(r, f)) as r→ ∞ possibly outside an exceptional set of finite lin-
ear/logarithmic measure. A meromorphic function g(z)is said to be small function of f(z)
if and only if T(r, g) = S(r, f).
The question addressed here is whether the conclusion of Theorem Acan still hold if the
polynomial in f(z),Q(f), is replaced with a rational function in f(z),P1(z, f )/P2(z, f ),
with coefficients being small functions of f(z). In this section, we treat this question for
particular rational functions, and we discuss the sufficient conditions that we offered. Before
stating our results, we recall some notations. First, we introduce a quantity δPassigned to a
polynomial
P(z) = cν1zν1+···+cνℓzνℓ, ℓ ≥2, ν1<···< νℓ,

ON THE PERIODICITY OF ENTIRE FUNCTIONS 3
where cνs6= 0 for all s= 1,...,ℓ, and defined by δP:= gcd (ν2−ν1,...,νℓ−νℓ−1). For
example, if P(z) = zn+zm+n(m > 0), then δP=m. Notice that when ν1= 0, then
δP= gcd(ν2,...,νℓ). The functions N(r, f)and N(r, 1/f)are, respectively, the integrated
counting functions for the poles and zeros of f(z)in the disc |z| ≤ rwith counting the
multiplicities. Analogously, the functions N(r, f)and N(r, 1/f)are, respectively, the in-
tegrated counting functions for the poles and zeros of f(z)in the disc |z| ≤ rignoring the
multiplicities.
We proceed now to state and prove our results.
Theorem 2.1. Let g(z)be a transcendental entire function such that N(r, 1/g) = S(r, g),
A(z)be non-vanishing meromorphic function satisfying T(r, A) = S(r, g), and let Q(z)be
a polynomial with at least two non-zero terms. If A(z)Q(g(z)) is periodic of period c, then
g(z)is periodic of period cor δQc.
From Theorem A, we see that the conditions on Q(z)and on zeros of g(z)can be dropped
in Theorem 2.1 if A(z)is constant. In case of Q(z)is a monomial and A(z)is non-constant,
the conclusion of Theorem 2.1 does not hold in general. In fact, we can always find functions
A(z)and g(z)for which A(z)g(z)nis periodic, but g(z)is not periodic, where nis a positive
integer. This can be seen, for example, by taking A(z) = e−nz and g(z) = eeiz+z.
The occurrence of the possibility when g(z)is of period δQccan be seen by taking g(z) =
ez/3and Q(z) = z3+z6+z9, where δQ= 3. In this case, Q(g(z)) = ez+e2z+e3zis of
period c= 2πi while g(z)is of period 3c= 6πi.
The following result is a modification of Theorem 2.1, which treats the case when irre-
ducible rational function in f(z)is periodic.
Theorem 2.2. Let g(z)be a transcendental entire function such that N(r, 1/g) = S(r, g),
A(z)be non-vanishing meromorphic function satisfying T(r, A) = S(r, g), and let Q(z)be
a polynomial with at least two non-zero terms. Suppose that A(z)Q(g(z)) /g(z)is periodic
of period c. Then g(z)is periodic of period 2cif Q(z) = c0+c1z+c2z2, where c0c26= 0.
Otherwise, g(z)is periodic of period cor δQc.
The case when Q(z)has the form Q(z) = c0+c1z+c2z2in Theorem 2.2 and g(z)
is of period 2cmay occur. For example, the function g(z) = eezis of period 2πi while
Q(g(z))/g(z)is of period πi, where Q(z) = 1 + cz +z2and c∈C.
In the proofs Theorems 2.1 and 2.2, we will use gcand Acto stand for g(z+c)and
A(z+c), respectively. Also, we will frequently use the following lemma.
Lemma 2.3 ([15, Theorem 1.62]).Let f1(z),...,fn(z), where n≥3, be meromorphic
functions which are non-constant except probably fn(z). Suppose that
n
P
j=1
fj(z) = 1 and
fn(z)6≡ 0. If
n
X
i=1
Nr, 1
fi+ (n−1)
n
X
i=1
i6=j
N(r, fi)<(λ+o(1))T(r, fj), j = 1,2,...,n−1,
as r→ ∞ and r∈I, where λ < 1and I⊂[0,∞)is a set of infinite linear measure, then
fn(z)≡1.

4 Z. LATREUCH AND M. A. ZEMIRNI∗
Proof of Theorem 2.1.Let Q(z) = Pℓ
s=1 cνszνs, where ℓ≥2,ν1< . . . < νℓand cνs6= 0
for all s= 1,...,ℓ. Since A(z)Q(g(z)) is periodic of period c, it follows that
Ac
ℓ
X
s=1
cνsgνs
c=A
ℓ
X
s=1
cνsgνs.(2.1)
Since T(r, A) = S(r, g), it follows from (2.1) that T(r, gc)∼T(r, g)as r→ ∞ probably
outside an exceptional set of finite linear measure. Let m∈ {1,...,ℓ}. Dividing (2.1) by
cνmA(z)g(z)νmresults in
Ac
A
ℓ
X
s=1
cνs
cνm
gνs
c
gνm−
ℓ
X
s=1
s6=m
cνs
cνm
gνs−νm= 1.(2.2)
Clearly Ac
A
gνs
c
gνmis non-constant for all s6=m. Then by Lemma 2.3, we obtain
gc
gνm
=A
Ac
.
Since this is true for every m= 1,...,ℓ, it follows that
gc
gν2−ν1
=···=gc
gνℓ−νℓ−1
= 1.
This means g(z)is periodic of period cor δQc.
Proof of Theorem 2.2.Let Q(z)be as in Proof of Theorem 2.1. If ν16= 0, then Q(z)z−1is
a polynomial, and hence the results follows from Theorem 2.1. So, we assume that ν1= 0.
Similarly as in the proof of Theorem 2.1, one can see that T(r, gc)∼T(r, g)as r→ ∞
probably outside an exceptional set of finite linear measure. Next we distinguish two cases:
(1) Case ν2= 1: By periodicity of A(z)Q(f(z)) /f (z), we have
Acc0
gc
+Acc1+Ac
ℓ
X
s=3
cνsgνs−1
c−Ac0
g−Ac1−A
ℓ
X
s=3
cνsgνs−1= 0.(2.3)
Dividing (2.3) by c1A(z)and using Lemma 2.3 yield A(z+c)≡A(z). Using this, and
multiplying (2.3) by g(z)/c0, we find
g
gc
+
ℓ
X
s=3
cνs
c0
ggνs−1
c−
ℓ
X
s=3
cνs
c0
gνs= 1.(2.4)
If ℓ= 2, then the sums in (2.4) are vanished, and therefore g(z)is periodic of period c. We
suppose then that ℓ≥3. Notice that ggνs−1
cis non-constant unless ν3= 2. Also, notice that
g/gcand ggccannot be constants simultaneously. Hence, if ggcis non-constant or ν3>2,
then by applying Lemma 2.3 on (2.4), we obtain that g(z)is periodic of period c. Next, we
suppose that ν3= 2 and g/gcis non-constant. Then again from Lemma 2.3 and (2.4) we
obtain that
ggc=c0/c2,(2.5)
which implies that g(z)is periodic of period 2c. We show next that this case can occur only
if ℓ= 3. Assume that ℓ≥4. We obtain from (2.5) that g(z) = eα(z)and g(z+c) =
(c0/c2)e−α(z), where α(z)is an entire function. Then from (2.4) we obtain
ℓ
X
s=4
cνs
c0
(c0/c2)νs−1e−(νs−2)α(z)−
ℓ
X
s=4
cνs
c0
eνsα(z)= 0,
which is not possible by Borel’s lemma. Thus ℓ= 3, and therefore Q(z) = c0+c1z+c2z2.

ON THE PERIODICITY OF ENTIRE FUNCTIONS 5
(2) Case ν2≥2: In this case, we have
Acc0
gc
+Ac
ℓ
X
s=2
cνsgνs−1
c−Ac0
g−A
ℓ
X
s=2
cνsgνs−1= 0.(2.6)
Dividing (2.6) by Ac0
gresults in
Acg
Agc
+Ac
A
ℓ
X
s=2
cνs
c0
ggνs−1
c−
ℓ
X
s=2
cνs
c0
gνs= 1.(2.7)
Here we notice that Ac
Aggνs−1
cis non-constant unless ν2= 2. Also, Acg
Agcand Ac
Aggccan-
not be constants simultaneously. If Ac
Aggcis non-constant or ν2>2, then from (2.7) and
Lemma 2.3 we obtain that Ac/A =gc/g. Then replacing Ac/A with gc/g in (2.6) results in
ℓ
X
s=2
cνsgνs
c−
ℓ
X
s=2
cνsgνs= 0.
By Theorem 2.1, we deduce that g(z)is periodic of cor δQc.
Now, if Acg
Agcis non-constant and ν2= 2, then from (2.7) and Lemma 2.3 we obtain
ggc=c0
c2
A
Ac
.(2.8)
On the other hand, dividing (2.6) by Acc0
gcresults in
−
ℓ
X
s=2
cνs
c0
gνs
c+Agc
Acg+A
Ac
ℓ
X
s=2
cνs
c0
gcgνs−1= 1.(2.9)
Hence from (2.9) and Lemma 2.3 we obtain
gcg=c0
c2
Ac
A.(2.10)
From (2.8) and (2.10), it follows that gcg=±c0/c2. Similarly as previous case (1), we
deduce that g(z)is periodic of period 2cand Q(z)has, in this case, the form Q(z) =
c0+c2z2.
3. RESULTS ON YANG’S CONJECTUR E
We start this section by proving the following result, which improves Theorem B.
Theorem 3.1. Let f(z)be a transcendental entire function and n, k be positive integers.
Suppose that f(z)nf(k)(z)is a periodic function with period c, and one of the following
holds.
(i) f(z)has the value 0as Picard exceptional value, and ρ2(f)<∞.
(ii) f(z)has a non-zero Picard exceptional value.
Then f(z)is a periodic function of period cor (n+ 1)c.
Proof. By the assumption, f(z)has the form f(z) = eh(z)+d, where h(z)is an entire
function, and d∈C.
From Theorem B, we need only to consider the case when h(z)is a transcendental entire
function.
(i) Case d= 0: Since f(z)nf(k)(z)is a periodic function of period c, it follows
f(z)nf(k)(z) = f(z+c)nf(k)(z+c).

6 Z. LATREUCH AND M. A. ZEMIRNI∗
Substituting f(z) = eh(z), where ρ(h)<∞, into this equation gives
e(n+1)∆h(z)=Bk(z)
Bk(z+c),(3.1)
where ∆h(z) = h(z+c)−h(z)and
Bk(z) = (h′(z))k+Qk−1(z),(3.2)
and Qk−1(z)is a differential polynomial in h′with constant coefficients and of degree k−1.
Assume that ∆h(z)is a transcendental entire function. Then (3.1) yields
∞=ρe(n+1)∆h=ρBk(z)
Bk(z+c)≤ρ(h)<∞,
which is a contradiction. Thus ∆h(z)is a polynomial. Now, assume that ∆h(z)is non-
constant polynomial of degree p≥1. Then (3.1) can be seen as a linear difference equation
Bk(z+c) = e−(n+1)∆h(z)Bk(z).(3.3)
From this and [1, Theorem 9.2], we obtain that ρ(Bk)≥ρe(n+1)∆h+ 1 = p+ 1. Hence
ρ(h′)≥ρ(Bk)≥p+ 1.(3.4)
Since ∆h(z)is polynomial, it follows from (3.2) and (3.3) that
(h′(z))k=−Qk−1(z) + Lk−1(z)
1−e−(n+1)∆h(z),(3.5)
where Lk−1(z)is is a differential polynomial in h′with polynomial coefficients of degree
k−1. Therefore, (3.5) results in
kT (r, h′)≤(k−1)T(r, h′) + O(rp) + O(log r),
and hence ρ(h′)≤p, which contradicts (3.4). Thus ∆h(z)is a constant. In this case, one
can easily see from (3.2) that Bk(z+c) = Bk(z). Thus e(n+1)(h(z+c)−h(z)) = 1, that is, f(z)
is a periodic function of period cor (n+ 1)c.
(ii) Case d6= 0: In this case, we have
f(z)nf(k)(z) = Bk(z)
n
X
j=0 n
jdn−je(j+1)h(z),(3.6)
where d6= 0 and Bk(z)is defined as in (3.2). From Theorem 2.1, we obtain that f(z)is
periodic of period c. This completes the proof. 
In the following result, we extend Theorem Cto hold for f(k)(z)/f(z)n.
Theorem 3.2. Let f(z)be a transcendental entire function, and let k≥1and n≥2be
integers. Suppose that f(k)(z)/f (z)nis a periodic function with period c, and one of the
following holds.
(i) f(z)has the value 0as Picard exceptional value, and ρ2(f)<∞.
(ii) f(z)has a non-zero Picard exceptional value.
Then f(z)is a periodic function of period c,2cor (n−1)c.
Proof. Since f(k)(z)/f(z)nis periodic with period c, it follows that
f(z+c)n
f(k)(z+c)=f(z)n
f(k)(z).(3.7)
By substituting the form f(z) = eh(z)+din (3.7), we obtain
1
Bk(z+c)eh(z+c)+dne−h(z+c)=1
Bk(z)eh(z)+dne−h(z),(3.8)

ON THE PERIODICITY OF ENTIRE FUNCTIONS 7
where Bk(z)is defined in (3.2). Next, we distinguish the cases related to d.
(i) Case d= 0: In this case, (3.8) becomes
e(n−1)∆h(z)=Bk(z+c)
Bk(z).
Then, from proof of Theorem Bwhen h(z)is polynomial, and from proof Theorem 3.1(i)
when h(z)is transcendental, we obtain that f(z)is a periodic function with period cor
(n−1)c.
(ii) Case d6= 0: In this case, the result follows directly from Theorem 2.2. This completes
the proof. 
If we replace the monomial f(z)nby a polynomial with at least two non-zero terms in
Theorem 3.1 and Theorem 3.2, then the restriction on the growth in Case (i) in both theorems
is no more needed.
Theorem 3.3. Let f(z)be a transcendental entire function, P(z)be a polynomial with
at least two non-zero terms, and k≥1be an integer. Suppose that P(f(z))f(k)(z)is a
periodic function with period c, and f(z)has a finite Picard exceptional value. Then f(z)
is a periodic function of period cor δPc.
Proof. Let P(z) = Pn
j=1 bνjzνj.
(i) Case d= 0: In this case we have
P(f(z))f(k)(z) = Bk(z)
n
X
j=1
bνje(νj+1)h(z),(3.9)
where Bk(z)is defined in (3.2). From Theorem 2.1, we deduce that f(z)is periodic with
period cor δPc.
(ii) Case d6= 0: In this case, we have
P(f(z))f(k)(z) = Bk(z)
n
X
j=1
bνj νj
X
s=0
ase(s+1)h(z)!,
where as=νj
sdνj−s. By changing the order of the sums in the previous equality, we get
P(f(z))f(k)(z) = Bk(z)
νn
X
s=0
cse(s+1)h(z), cs=as
νn
X
j=s
bj.
From Theorem 2.1, we conclude that f(z)is a periodic function with period c.
Theorem 3.3 generalizes and improves [14, Corollary 1.5 and Theorem 1.6] (see also [8,
Corollary 5.1 and Theorem 5.14]).
Theorem 3.4. Let f(z)be a transcendental entire function, P(z)be a polynomial with at
least two non-zero terms, and k≥1be an integer. Suppose that f(k)(z)/P (f(z)) is a
periodic function with period c, and f(z)has a finite Picard exceptional value. Then f(z)
is a periodic function of period c,2cor δPc.
Proof. We follow the proof of Theorem 3.3 by using Theorem 2.2.
Theorem 3.4 generalizes and improves [14, Theorem 1.7] (see also [8, Theorem 5.16]).

8 Z. LATREUCH AND M. A. ZEMIRNI∗
4. ON A VARI ATION O F YANG’S CONJECTUR E
Recall that the p-iterated order of an entire function f(z)is defined by
ρp(f) = lim sup
r→∞
logpT(r, f)
log r,
where log1r= log rand logp= log logp−1r. The finiteness degree i(f)of f(z)is defined
to be i(f) := 0 if f(z)is a polynomial, i(f) := min {j∈N:ρj(f)<∞} if there exists
some j∈Nfor which ρj(f)<∞, or otherwise i(f) := ∞.
We find that the condition on the growth in Case (2) of Theorem Dcan be relaxed to
i(f)<∞.
Theorem 4.1. Let f(z)be a transcendental entire function, and let k≥1be an integer and
a1,...,akare constants. If f(z)3+a1f′(z) + ···+akf(k)(z)is a periodic function with
period cand i(f)<∞, then f(z)is periodic with period c,2cor 3c.
We find also that the condition i(f)<∞can be dropped at the expense of allowing some
restrictions on the zeros of f. In fact, we replace the condition i(f)<∞with Θ(0, f )>0,
where
Θ(0, f ) := 1 −lim sup
r→∞
Nr, 1
f
T(r, f).
Theorem 4.2. Let f(z)be a transcendental entire function, and let k≥1be an integer and
a1,...,akare constants. If f(z)3+a1f′(z) + ···+akf(k)(z)is a periodic function with
period cand Θ(0, f )>0, then f(z)is periodic with period cor 3c.
To prove Theorems 4.1 and 4.2, we start first with general preparations.
Since f(z)3+a1f′(z) + ···+akf(k)(z)is periodic with period c, it follows
f(z+c)3−f(z)3=−
k
X
j=1
aj[f(j)(z+c)−f(j)(z)].(4.1)
If Pk
j=1 aj[f(j)(z+c)−f(j)(z)] ≡0, then f(z+c)3−f(z)3≡0. Therefore, f(z)is periodic
with period cor 3c. Thus, in what follows we suppose that Pk
j=1 aj[f(j)(z+c)−f(j)(z)] 6≡ 0.
Moreover, we suppose that f(z+c)−f(z)6≡ 0. Then (4.1) can be written as
f(z+c)2+f(z+c)f(z) + f(z)2=−
k
X
j=1
aj
f(j)(z+c)−f(j)(z)
f(z+c)−f(z).(4.2)
Let H(z)denotes the right hand side of (4.2). Then H(z)is an entire function satisfying
T(r, H) = O(log T(r, ∆f) + log r), r /∈E, (4.3)
where ∆f(z) = f(z+c)−f(z)and E⊂[0,∞)is a subset of finite linear measure. By
using the notation w(z) := f(z+c)/f(z), (4.2) is rewritten as
w2+w+ 1 = H
f2.(4.4)
By definition of ∆f, we have
f=∆f
w−1.(4.5)
Substituting (4.5) into (4.4), we get
w2+w+ 1
(w−1)2=H
(∆f)2.(4.6)

ON THE PERIODICITY OF ENTIRE FUNCTIONS 9
Suppose now that wis constant. Then from (4.3), (4.4) and (4.5) we obtain
2T(r, f) = T(r, H) = O(log T(r, ∆f) + log r) = O(log T(r, f ) + log r), r 6∈ E,
which is a contradiction. Thus we may suppose that wis non-constant in all what follows.
Proof of Theorem 4.1.Since i(f)<∞, it follows that there exists some integer p≥1for
which ρp(f)<∞and ρp−1(f) = ∞.
From (4.3), (4.4) and (4.6), we get
T(r, f) = T(r, ∆f) + O(log T(r, ∆f) + log r), r /∈E,
which results in i(∆f) = p, and therefore i(H)≤p−1. We rewrite (4.2) as follows
(f(z+c)−qf (z)) f(z+c)−q2f(z)=H(z),
where q=−1+i√3
2. By Weierstrass factorization, we obtain
f(z+c)−qf (z) = Π(z)eα(z),(4.7)
and then
f(z+c)−q2f(z) = H(z)
Π(z)e−α(z),(4.8)
where Π(z)is the canonical product, and α(z)is an entire function. From the properties of
canonical products, it follows that ρp−1(Π) = λp−1(Π) ≤ρp−1(H)<∞, see [3, Satz 12.3].
If i(eα)≤p−1, then from (4.7) and (4.8) we have
(q2−q)f(z) = Π(z)eα(z)−H(z)
Π(z)e−α(z),(4.9)
and this yields i(f)≤p−1, which is a contradiction. Thus i(eα) = p. In this case,
by applying [2, Corollary 4.5], we obtain that N(r, 1/Π) ≤N(r, 1/H)≤T(r, H) =
o(T(r, eα)) as r→ ∞ and r∈F, where logdens(F) = 1. From (4.7) and (4.8), we obtain
1
q
H(z)
Π(z)2e−2α(z)+1
q2
Π(z+c)
Π(z)eα(z+c)−α(z)
−1
q2
H(z+c)
Π(z)Π(z+c)e−α(z+c)−α(z)= 1.
(4.10)
By using [15, Theorem 1.56], we distinguish two cases:
(i) We have:
−1
q2
H(z+c)
Π(z)Π(z+c)e−α(z+c)−α(z)= 1
and
1
q
H(z)
Π(z)2e−2α(z)+1
q2
Π(z+c)
Π(z)eα(z+c)−α(z)= 0,
which results in
H(z) = −1
qeα(z+c)+α(z)Π(z)Π(z+c)
and
H(z+c) = −q2eα(z+c)+α(z)Π(z)Π(z+c).
Since q3= 1, it follows that H(z+c) = H(z)and Π(z+ 2c)eα(z+2c)= Π(z)eα(z).
Therefore, (4.9) reveals that f(z)is a periodic function of period 2c.

10 Z. LATREUCH AND M. A. ZEMIRNI∗
(ii) We have
1
q2
Π(z+c)
Π(z)eα(z+c)−α(z)≡1
and
1
q
H(z)
Π(z)2e−2α(z)−1
q2
H(z+c)
Π(z)Π(z+c)e−α(z+c)−α(z)≡0,
that is
Π(z+c)eα(z+c)=q2Π(z)eα(z)
and
H(z+c)
Π(z+c)e−α(z+c)=qH(z)
Π(z)e−α(z).
Since q3= 1, we can easily deduce from this and from (4.9) that f(z)is periodic
with period 3c.
Proof of Theorem 4.2.Here, we prove that under the assumption that f(z+c)−f(z)6≡ 0,
we get a contradiction. From (4.3) and (4.6), we have S(r, w) = S(r, ∆f). Therefore, (4.4)
leads to
T(r, f) = T(r, w) + S(r, w).(4.11)
Using the second main theorem, together with (4.4), we obtain
T(r, w)≤N(r, w) + Nr, 1
w−q+Nr, 1
w−q2+S(r, w)
=N(r, w) + Nr, 1
w2+w+ 1+S(r, w)
≤Nr, 1
f+Nr, 1
H+S(r, w) = Nr, 1
f+S(r, w).
Hence, from (4.11), we obtain
T(r, f)≤Nr, 1
f+S(r, w).(4.12)
Since Θ(0, f )>0, it follows for any ε∈(0,1) sufficiently small, there exists R > 0such
that N(r, 1/f)<(1 −ε)T(r, f). This together with (4.12) results in T(r, f) = S(r, w),
which contradicts (4.11). 
REFEREN CES
[1] Y. M. Chiang and S. Feng, On the Nevanlinna characteristic of f(z+η)and difference equations in the
complex plane. Ramanujan J. 16 (2008), no. 1, 105–129.
[2] J. Heittokangas, Z. Latreuch, J. Wang and M. A. Zemirni, On the limits of real-valued functions in sets
involving ψ-density, and applications. arXiv (2020), 15 pp.
https://arxiv.org/abs/2006.02066.
[3] G. Jank and L. Volkmann, Einf¨
uhrung in die Theorie der ganzen und meromorphen Funktionen mit
Anwendungen auf Differentialgleichungen. Birkh¨auser Verlag, Basel, 1985.
[4] I. Laine, Nevanlinna Theory and Complex Differential Equations. De Gruyter Stud. Math., vol. 15, Walter
de Gruyter & Co., Berlin, 1993.
[5] P. Li, W. R. L¨u and C. C. Yang, Entire solutions of certain types of nonlinear differential equations.
Houston J. Math. 45 (2019), No. 2, 431–437.
[6] K. Liu and P. Y. Yu, A note on the periodicity of entire functions. Bull. Aust. Math. Soc. 100 (2019),
no. 2, 290–296.
[7] K. Liu, Y. Wei and P. Y. Yu, Generalized Yang’s Conjecture on the periodicity of entire functions. Bull.
Korean Math. Soc. 57 (2020), no. 5, 1259–1267.

ON THE PERIODICITY OF ENTIRE FUNCTIONS 11
[8] X. L. Liu, The periodicity of transcendental meromorphic functions. Publications Univ. Eastern Finland,
Diss. Forestry and Natural Sci. 419 (2021).
http://urn.fi/URN:ISBN:978-952-61-3727-8
[9] X. L. Liu and R. Korhonen, On the periodicity of transcendental entire functions. Bull. Aust. Math. Soc.
101 (2020), no. 3, 453–465.
[10] X. L. Liu, R. Korhonen and K. Liu, Variations on a conjecture of C. C. Yang concerning periodicity.
Comput. Methods Funct. Theory (2021).
https://doi.org/10.1007/s40315-020-00359-0
[11] W. L ¨u and X. Zhang, On the periodicity of transcendental entire functions. Results Math. 75 (2020),
no. 4, Paper No. 176. 13 pp.
[12] A. R´enyi and C. R´enyi, Some remarks on periodic entire functions. J. Anal. Math. 14(1) (1965), 303–310.
[13] Q. Wang and P. C. Hu, Zeros and periodicity of entire functions. Acta Math. Sci. Ser. A (Chin. Ed.) 38
(2018), 209–214.
[14] Y. M. Wei, K. Liu and X. L. Liu, The periodicity on a transcendental entire function with its differential
polynomials. To appear in Analysis Math.
[15] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions. Science Press and Kluwer
Acad. Publ., Beijing, 2003.
(Z. Latreuch) UNIVE RS IT Y OF MO STAG AN EM , DE PART ME NT O F MATH EM ATIC S, L ABORATORY OF
PUR E AN D APP LI ED MATHE MATI CS , B. P. 227 MO STAG AN EM , AL GE RI A.
Email address:z.latreuch@gmail.com
(M. A. Zemirni) UNI VE RS IT Y OF EA STERN FINL AN D, DEPARTM EN T OF PH YS ICS AND MATH EM ATICS,
P.O. BOX 111, 80101 JOE NS UU , FI NL AN D.
Email address:amine.zemirni@uef.fi