Abstract
The generalized Yang’s Conjecture states that if, given an entire function f(z) and positive integers n and k, \(f(z)^nf^{(k)}(z)\) is a periodic function, then f(z) is also a periodic function. In this paper, it is shown that the generalized Yang’s conjecture is true for meromorphic functions in the case \(k=1\). When \(k\ge 2\) the conjecture is shown to be true under certain conditions even if n is allowed to have negative integer values.
Introduction and Main Results
Yang’s Conjecture on the periodicity of transcendental entire functions, proposed in [8] and [15, Conj. 1.1], has recently prompted intensive research activity in the field of periodic entire functions, see, e.g., [10,11,12, 15], which include results both on Yang’s Conjecture and its difference version. For related problems on the periodicity of transcendental meromorphic functions, we refer the reader to [1, 4,5,6, 9, 13, 14, 16, 17].
Yang’s Conjecture. Let f(z) be a transcendental entire function and k be a positive integer. If \(f(z)f^{(k)}(z)\) is a periodic function, then f(z) is also a periodic function.
The generalized Yang’s Conjecture has been considered by Liu, Wei and Yu [10].
Generalized Yang’s Conjecture. Let f(z) be a transcendental entire function and n, k be positive integers. If \(f(z)^{n}f^{(k)}(z)\) is a periodic function, then f(z) is also a periodic function.
Liu, Wei and Yu [10, Thm. 1.2] have shown that if f is a transcendental entire function and, in addition, if \(k = 1\), or if \(f(z) = e^{h(z)}\) where h(z) is a nonconstant polynomial, or if f(z) has a nonzero Picard exceptional value and f(z) is of finite order, then the Generalized Yang’s Conjecture is true. The aforementioned conjectures and results are all concerned with transcendental entire functions. The following results are, to the best of our knowledge, the first to address the meromorphic case. In addition, we allow the exponent n to have negative values as an extension of the Generalized Yang’s Conjecture. Actually, we consider the following question.
Question 1 Let f(z) be a transcendental meromorphic function, let \(n\in {\mathbb {Z}}\), and let \(k\in {\mathbb {N}}\). If \(f(z)^{n}f^{(k)}(z)\) is a periodic function, does it follow that f(z) is also a periodic function?
Our first theorem gives a complete description of the case \(k=1\) of Question 1.
Theorem 1.1
Let f(z) be a transcendental meromorphic function and \(n\in {\mathbb {Z}}\). Suppose that \(f(z)^{n}f'(z)\) is a periodic function with period c.

(1)
If \(n\ge 1\) or \(n\le 3\), then f(z) is a periodic function with period \((n+1)c\).

(2)
If \(n=0\), then \(f(z)=\varphi (z)+Az/c\), where \(\varphi (z)\) is a periodic function with period c and A is a constant.

(3)
If \(n=1\), then \(f(z)=e^{Az/c}\varphi (z)\), where \(\varphi (z)\) is a periodic function with period c and A is a constant.

(4)
If \(n=2\), then \(f(z)=1/(\varphi (z)+Az/c)\), where \(\varphi (z)\) is a periodic function with period c and A is a constant. Furthermore, if f(z) is transcendental entire, then f(z) is a periodic function with period c.
Remark 1.2
(1) In Theorem 1.1 (1), f may have period qc for a minimal divisor q of \(n+1\), where \(\omega ^{q}=1\) and \(\omega ^{n+1}=1\) for some root of unity \(\omega \).
(2) The function \(f(z)=\varphi (z)+Az/c\), \(A\not =0\), in Theorem 1.1 (2) is called a pseudoperiodic function mod Az/c, see [2, Def. 3.4].
We proceed to consider Question 1. Here we will restrict our consideration to the entire case. The claim in case of \(n=1\) and \(k\ge 1\) in Question 1 is not true, which can be seen by looking at \(f(z)=e^{e^{iz}+z}\). Here f(z) is not a periodic function, but \(f^{(k)}/f\) is a periodic function with period \(2m\pi \), where m is a nonzero integer. Observe that the transcendental entire function \(f(z)=e^{e^{iz}+z}\) has a Picard exceptional value 0. We give the following result to show that this is, in a sense, essentially the only type of an example to be given here.
Theorem 1.3
Let f be a transcendental entire function with a Picard exceptional value 0. If \(f''/f\) is a periodic function with period c, then \(f(z)=e^{h_{1}(z)+Az/(2c)}\), where \(h_{1}(z)\) is a periodic function with period 2c and A is a constant.
Remark 1.4
(1) If f is of finite order in Theorem 1.3, then \(f(z)=e^{Az+B}\), where A is a nonzero constant.
(2) Assume that \(f^{(k)}/f=A(z)\), that is \(f^{(k)}A(z)f=0\), where A(z) is a periodic function. Hence, we also can consider the periodicity of \(f^{(k)}/f\) from the perspective of differential equations with periodic coefficients. Such results can be found in, e.g., [7, Lem. 5.19] and [3, Thm. 1].
We proceed to consider the case of f(z) admitting a nonzero Picard exceptional value d, i.e. \(f(z)=e^{h(z)}+d\), where h(z) is an entire function.
Theorem 1.5
Let f be a transcendental entire function with a Picard exceptional value \(d\not =0\). If \(f^{(k)}/f\) is a periodic function with period c, then f(z) is a periodic function with period c.
The method of proof of Theorem 1.5 below cannot be used to extend the statement to hold for \(f^{(k)}/f^{n}\) instead of \(f^{(k)}/f\). However, we obtain the following result.
Theorem 1.6
Let f(z) be a transcendental meromorphic function, let \(n\in {\mathbb {Z}}\), and let \(k\in {\mathbb {N}}\). Suppose that \(f^{(k)}(z)/f(z)^{n}\) and \(f^{(k+1)}(z)/f(z)^{n}\) are periodic functions with period c.

(1)
If \(n=0\), then \(f(z)=\varphi (z)+p(z)\), where \(\varphi (z)\) is a periodic function with period c and p(z) is a polynomial with degree at most k.

(2)
If \(n=1\), then \(f(z)=e^{Az/c}\varphi (z)\), where \(\varphi (z)\) is a periodic function with period c and A is a constant.

(3)
If \(n\ne 0,1\), then f(z) is a periodic function with period \((n1)c\).
Remark 1.7
We conjecture that there doesn’t exist a transcendental meromorphic function f such that \(f^{(k)}/f^{n}\) and \(f^{(k+1)}/f^{n}\) are periodic functions, where \(d_{1}\) is a period of \(f^{(k)}/f^{n}\) but not of \(f^{(k+1)}/f^{n}\) while \(d_{2}\) is a period of \(f^{(k+1)}/f^{n}\) but not of \(f^{(k)}/f^{n}\), and \(d_{1}/d_{2}\) is not a rational number.
Proofs of Theorems
Proof of Theorem 1.1
We first consider the case \(n\ge 1\). Assume that \(f(z)^{n}f'(z)\) is a periodic function with period c, then
Integrating the above equation, we have
where B is a constant. Let \(F(z)=f(z)^{n+1}\). Thus F(z) is a nonconstant meromorphic function with no simple zeros. From (2.1), we obtain
Suppose that \(B\ne 0\) and let \(2\le m\in {\mathbb {N}}\). Then
This implies that all roots of \(F(z)=mB\) are multiple for all \(m\ge 2\), which is impossible, since a transcendental meromorphic function has at most four completely ramified values. Hence, \(B=0\) and F(z) is a periodic function with period c. So \(f(z+c)^{n+1}=f(z)^{n+1}\), and it follows that \(f(z+c)=\omega f(z)\), where \(\omega ^{n+1}=1\). Therefore, f(z) is a periodic function with period \((n+1)c\).
The case of \(n=0\) is trivial. Namely, the assertion follows from \(f'(z+c)=f'(z)\) by integrating and then solving the resulting first order nonhomogeneous difference equation.
We proceed to consider the case \(n=1\). Now \(f'/f\) is a periodic function with period c, and so
Integrating the above equation, we have \(f(z+c)=e^{A}f(z)\), where A is a constant. Thus f(z) can be written as \(f(z)=e^{Az/c}\varphi (z)\), where \(\varphi (z)\) is a periodic function with period c.
In the remaining case \(n\le 2\), we may write \(f^{n}f'=g^{n2}g'\) where \(f=1/g\). Hence \(f^{n}f'\) and \(g^{n2}g'\) have the same periodicity. Thus, if \(n=2,\) we immediately obtain, by applying the case \(n=0\), that \(f(z)=1/(\varphi (z)+Az/c)\), where \(\varphi (z)\) is a periodic function with period c and A is a constant. If \(n\le 3\), then, by the first part of the proof, g(z) is a periodic function with period \((1n)c\), thus \((n+1)c\) is also a period of g(z) and so of f(z) as well.
Finally, we consider the case \(n=2\) and f(z) is transcendental entire. By
we obtain
We affirm that \(A=0\), and thus f(z) is a periodic function with period c. If f is entire, then 1/f has no zeros. Let \(G=1/f\). Then,
and G has no zeros. Let \(2\le m\in {\mathbb {N}}\) again. So the fact that \(G(z)mA\) has no zeros follows by the same argument we used following (2.1). But this is impossible, since a meromorphic function has at most two finite Picard exceptional values. \(\square \)
Proof of Theorem 1.3
By assuming that \(f(z)=e^{h(z)}\) and \({f''}/{f}\) is a periodic function with period c, it follows that
By a recent result given by Liu, Wei and Yu [10, Thm. 1.6], then \(h'(z)\) must be a periodic function with period 2c. Thus, \(h(z)=h_{1}(z)+Az/(2c)\), where \(h_{1}(z)\) is a periodic function with period 2c and A is a constant. \(\square \)
Proof of Theorem 1.5
Since \(f(z)=e^{h(z)}+d\) and \(f^{(k)}/f\) is a periodic function with period c, it follows by an elementary computation, that
where \(H_{k}(z)\) is a differential polynomial of h(z) and
From (2.3), we also have
The Eq. (2.3) can be written as
Let
Using the second main theorem, (2.4), (2.5) and (2.6), we obtain
In order to avoid a contradiction, we see that \(f_{1}\) must be a constant. Since f(z) is a transcendental entire function, we have
and
Hence, the fact that f(z) is a periodic function with period c follows from (2.3). \(\square \)
Proof of Theorem 1.6
We first consider the case \(n=0\), that is \(f^{(k)}(z)\) and \(f^{(k+1)}(z)\) are periodic functions with period c. Then \(f^{(k)}(z)\equiv f^{(k)}(z+c)\), thus \(f(z+c)f(z)=Q(z)\), where Q(z) is a polynomial of degree less than k. By choosing a polynomial p(z) with the degree \(\deg (Q)+1\) such that \(p(z+c)p(z)=Q(z)\) it follows that \(f(z)=\varphi (z)+p(z)\), where \(\varphi (z)\) is a periodic function with period c and p(z) is a polynomial with degree at most k.
We proceed to consider the case \(n\ne 0\). Then we know that
Since \(f^{(k)}(z)/f(z)^{n}\) and \(f^{(k+1)}(z)/f(z)^{n}\) are periodic functions with period c, we obtain \(f'/f\) is also a periodic function with period c. From Theorem 1.1 (3) we have that \(f(z)=e^{Az/c}\varphi (z)\), where \(\varphi \) is a periodic function with period c and A is a constant. Furthermore, if \(n\ne 1\), rewrite (2.7) as
By the classical formula, see, e.g., [7, Lem. 2.3.7], we have
where \(P_{k1}\left( f'/f\right) \) is a differential polynomial in \(f'/f\) and its derivatives with constant coefficients and of total degree \(\le k1\). Recall that \(f'/f\) is periodic with period c, thus \(f^{(k)}/f\) is also a periodic function with period c. From (2.8), we see that \(1/f^{n1}\) is also a periodic function with period c, which implies that f(z) has period \((n1)c\), and case (3) is thus proved. \(\square \)
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Xinling Liu was partially supported by the Edufi Fellowship (TM1811020). Risto Korhonen was partially supported by the Academy of Finland grant \((\sharp 286877)\). Xinling Liu and Kai Liu were partially supported by the NSFC (No.11661052) and the NSF of Jiangxi (No. 20202BAB201003)
Communicated by James K. Langley.
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Liu, X., Korhonen, R. & Liu, K. Variations on a Conjecture of C. C. Yang Concerning Periodicity. Comput. Methods Funct. Theory (2021). https://doi.org/10.1007/s40315020003590
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Keywords
 Meromorphic functions
 Periodicity
 Yang’s Conjecture
Mathematical Subject Classification
 30D35
 39A05