Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 52, pp. 1–16.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.52

CROSSED DIFFERENTIAL SYSTEMS OF EQUATIONS AND

CLUNIE LEMMA

YINGCHUN GAO, KAI LIU, XIAOGUANG QI

Abstract. We study properties of transcendental meromorphic solutions of

crossed complex differential systems of equations. For instance, we study the
crossed Riccati differential system

f(z)2 = 1− g′(z),

g(z)2 = 1− f ′(z),

and the crossed Weierstrass differential system

f(z)3 = 1− g′(z)2,

g(z)3 = 1− f ′(z)2.

In addition, we establish a crossed version of Clunie lemma.

1. Introduction

Let f(z) and g(z) be non-constant meromorphic functions in the complex plane.
Define a differential monomial

Mj(z, f) = f(z)λ0j (f ′(z))λ1j · · · (f (n)(z))λnj ,

where γMj := λ0j+λ1j+ · · ·+λnj is the degree of Mj(z, f). We define a differential
polynomial

L(z, f) =

k∑
j=1

αj(z)Mj(z, f),

where αj(z) are small functions with respect to f(z) in the sense of Nevanlinna
theory. The degree γL of L(z, f) is defined by γL = max1≤j≤k{γMj

}. In particular,

L(z, f) = α0(z)f(z) + α1(z)f
′(z) + · · ·+ αk(z)f

(k)(z) is called a linear differential
polynomial of f(z).

Definition 1.1. Let L(z, f) be a non-constant differential polynomial. If

L(z, f) = L(z, g)

implies that f = g, where f(z) and g(z) are two meromorphic functions, then
L(z, f) is called a unique differential polynomial of meromorphic functions (UDPM).

2020 Mathematics Subject Classification. 30D35, 34M05.
Key words and phrases. Meromorphic solutions; Clunie lemma; Riccati equations;

crossed differential systems of equations.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted March 20, 2024. Published September 10, 2024.

1



2 Y. C. GAO, K. LIU, X. G. QI EJDE-2024/52

Remark 1.2. (1) The definition of UDPM is a generalization of UPM. A non-
constant polynomial P (z) is called a unique polynomial of meromorphic functions
(UPM for abbreviation), whenever P (f) = P (g) implies that f = g. Li and Yang
[18, Theorem 7] proved that any non-linear polynomial with degree two or three is
not a UPM. Li and Yang [18, Theorem 8] obtained that

P (z) = z4 + a3z
3 + a2z

2 + a1z + a0

is not a UPM except that
a3
3

8 − a2a3

2 + a1 ̸= 0. If P (z) = zm + azn + b and m ≥ 5,
1 ≤ n ≤ m − 2 and (m,n) = 1, then P (z) is a UPM, see [37]. The singularity
theory, the concept of genus for algebraic curves and Nevanlinna theory always be
applied to consider the UPM, see [10] and references therein.

(2) Yang and Hua [32] obtained that f = tg if fnf ′ = gng′ and n ≥ 3, where
tn+1 = 1. Fang and Fang [4] obtained that f = g if fn(f − 1)2f ′ = gn(g − 1)2g′

and n ≥ 8. Thus, fn(f − 1)2f ′(n ≥ 8) is a UDPM, however, fnf ′ is not a UDPM.

As far as we know describing UDPM completely is more difficult than UPM for
the reason of the complexity of differential polynomials. Nevanlinna theory plays
an important part in considering the uniqueness of meromorphic functions. The
standard notations of Nevanlinna theory such as the counting function N(r, f),
the proximity function m(r, f) and the characteristic function T (r, f) can be found
in [13, 36] for details or [16, 21] for a short introduction. Heittokangas, Korho-
nen and Laine [15, Theorem 4.2] described the uniqueness of complex differential
polynomials of certain types, their results can be stated as follows.

Theorem 1.3. Let n, k be positive integers with n ≥ 4, let f(z) be a meromorphic
function with N(r, f) = S(r, f) and let L(z, f) be a non-zero linear differential
polynomial of f(z) with small coefficients with respect to f(z), and let h(z) be a
meromorphic function. For the differential equation

fn + L(z, f) = h, (1.1)

then one of the following two situations holds:

(a) equation (1.1) has f(z) as its unique transcendental meromorphic solution
such that N(r, f) = S(r, f);

(b) equation (1.1) has exactly n transcendental meromorphic solutions fj (j =
1, 2, · · · , n) such that N(r, fj) = S(r, fj). In this case, L(z, fj) ≡ 0 and
h = fn

j .

Remark 1.4. (1) Theorem 1.3 is not true for n = 3 by observing that f1(z) = sin z,

f2(z) = −
√
3
2 cos z− 1

2 sin z, f3(z) =
√
3
2 cos z− 1

2 sin z are the entire solutions of the

non-linear differential equation 4f3 + 3f ′′ = − sin 3z, see [15], [34, Theorem 4].
(2) Some results on transcendental meromorphic solutions to non-linear differen-

tial equations of certain types can be found in [22, 23] and references therein, which
is an active research topic recently. Yang and Laine [33, Theorem 2.6] obtained the
uniqueness of complex delay-differential equations of certain type.

(3) Theorem 1.3 also shows that fn + L(z, f) (n ≥ 4) is UDPM in the sense of
N(r, f) = S(r, f) and L(z, f) is a non-zero linear differential polynomial of f(z)
with small coefficients. Remark that the linearity of L(z, f) in Theorem 1.3, it
seems that there are no further results when L(z, f) is a non-linear differential
polynomial as far as we know.



EJDE-2024/52 CROSSED DIFFERENTIAL SYSTEMS AND CLUNIE LEMMA 3

Making a small modification of UDPM will allow us to consider the main systems
in the following. For example, the existence of meromorphic solutions f(z) and g(z)
of the functional equation fn + L(z, f) = gn + L(z, g) is considered in Theorem
1.3, the above equation is equivalent to fn − L(z, g) = gn − L(z, f). According to
Theorem 1.3, we obtain the following corollary.

Corollary 1.5. Let L(z, h) be a non-zero linear differential polynomial. If n ≥ 4,
then the differential system

f(z)n − L(z, g) = h(z),

g(z)n − L(z, f) = h(z),

has at most a pair admissible transcendental entire solution (f, g) and L(z, f) ̸≡ 0
and L(z, g) ̸≡ 0. In fact, we have f ≡ g provided that the pair admissible transcen-
dental entire solution exists.

Definition 1.6. The following system of equations will be called the crossed system
of equations

P (z, f) = Q(z, g) + α(z),

P (z, g) = Q(z, f) + α(z),
(1.2)

in other words, two equations in the system will be consistent if we interchange the
functions f and g for the equations, where P (z, h) and Q(z, h) are differential or
difference or delay-differential polynomials of h(z) with the small coefficients with
respect to h and α(z) is a meromorphic function.

An application of Nevanlinna theory to the systems of complex differential, dif-
ference and delay-differential equations is an interesting research. The research on
the complex differential systems of equations can be found in Gao [5, 6], Tu and
Xiao [27], Liu, Laine and Yang [21, Chapter 11]. Recently, Wang and Liu [28], Guo
and Liu [8], Li and Liu [20] have also considered the systems of complex differential
and difference equations. In fact, the systems of equations considered in Li and Liu
[20, Theorems 1.1, 1.3, 1.5] are the crossed systems of equations, such as Fermat
type crossed differential system

f(z)2 + g′(z)2 = 1,

f ′(z)2 + g(z)2 = 1.
(1.3)

Additionally, some systems of partial difference or differential equations in several
variables can be found in [29, 30, 31]. Barsegian, Laine and Yang [1] raised seven
problems, related to the complex differential systems of equations, one of them is
stated below.

Problem. What conditions can assure that all meromorphic solutions (f, g) of
complex differential system

P1

(
z, f, g, f (n), g(m)

)
= 0,

P2

(
z, f, g, f (n), g(m)

)
= 0,

(1.4)

are of finite order, namely f(z) and g(z) are finite order meromorphic functions,
where n,m ≥ 2 and P1, P2 are polynomials in all variables.



4 Y. C. GAO, K. LIU, X. G. QI EJDE-2024/52

Remark 1.7. (1) If m = n = 1, then the entire solutions of (1.4) may be of infinite
order. For example

f ′ − fg = 0,

f ′ − fg′ = 0,
(1.5)

can be solved by f(z) = ee
z

and g(z) = ez.
(2) We remark that (1.5) is not the crossed differential systems of equations from

the point of view of this paper. Therefore, we may consider the system

f ′ − f2g = 0,

g′ − g2f = 0.

Then, we conclude that f = ag and ag′ = a2g3, which implies that f(z) is a
constant, and so g(z) is. We may also consider the system

f ′ − fg′ = 0,

g′ − gf ′ = 0.

Then, we obtain f = 1/g, by the basic computation, we obtain that g(z) must be
a constant.

This article is organized as follows. In Section 2, we discuss the meromorphic
solutions of the crossed Riccati differential systems of equations. In Section 3,
we consider some other crossed differential systems of equations related to some
classical complex differential equations. In Section 4, we give a crossed version of
Clunie lemma.

2. Crossed Riccati differential systems of equations

In this section, we first consider the crossed Riccati differential system of equa-
tions. Gao [5] considered a Riccati differential system of equations, but it is not
the crossed Riccati differential system of equations. The crossed complex Riccati
differential system of equations with small coefficients can be expressed as

f ′(z) = a0(z) + a1(z)g(z) + a2(z)g(z)
2,

g′(z) = a0(z) + a1(z)f(z) + a2(z)f(z)
2.

(2.1)

If a2(z) is a constant, then (2.1) can be changed to the special case

u′(z) = A(z) + v(z)2,

v′(z) = A(z) + u(z)2,
(2.2)

by replacing f(z) = u(z)
a2

− a1(z)
2a2

, g(z) = v(z)
a2

− a1(z)
2a2

, where A(z) = a0(z)a2 −
a1(z)

2

4 +
a′
1(z)
2 . We will deal with the case of a0(z) ≡ 1, a1(z) ≡ 0 and a2(z) ≡ −1 in

the following theorem. A meromorphic function f(z) in the complex plane is called
properly meromorphic if f(z) has at least one pole.

Theorem 2.1. Let f(z) and g(z) be meromorphic solutions of the complex differ-
ential system of equations

f(z)2 = 1− g′(z),

g(z)2 = 1− f ′(z).
(2.3)

Then, we obtain



EJDE-2024/52 CROSSED DIFFERENTIAL SYSTEMS AND CLUNIE LEMMA 5

(1) f(z) and g(z) must be properly meromorphic functions and all poles are
simple poles;

(2) m(r, f) = S(r, f), m(r, g) = S(r, g) and T (r, f) = T (r, g) + S(r, f).

Proof. (1) Firstly, f(z) and g(z) are not polynomials by the basic observations.
Assume that f(z) is a transcendental entire function, then so g(z) is. By Valiron-
Mohon’ko theorem [16, Theorem 2.2.5], we obtain

2T (r, f) = T (r, g′) +O(1) ≤ T (r, g) + S(r, g) =
1

2
T (r, g2) + S(r, g)

≤ 1

2
T (r, f ′) + S(r, g) ≤ 1

2
T (r, f) + S(r, f) + S(r, g).

Similarly, we have 2T (r, g) ≤ 1
2T (r, g) + S(r, f) + S(r, g). Hence,

2(T (r, f) + T (r, g)) ≤ 1

2
(T (r, f) + T (r, g)) + S(r, f) + S(r, g),

which is impossible. Thus, f(z) and g(z) must be properly meromorphic functions.
Let z0 be the pole of f(z) with multiplicity p. Therefore, z0 must be a pole of g′(z)
with multiplicity 2p, then p = 1 follows by the estimation of poles of the second
equation of (2.3).

(2) To estimate the proximate functions of f(z) and g(z), taking the first deriv-
ative of the second equation of (2.3), we conclude that

2gg′ = −f ′′.

Combining the first equation of (2.3) with the above equation, we have

2g(1− f2) = −f ′′.

Hence, we have 4g2(1 − f2)2 = (f ′′)2 and 4(1 − f ′)(1 − f2)2 = (f ′′)2. Using the
lemma of the logarithmic derivative, we obtain

m(r, 1− f2) ≤ m
(
r,

(f ′′)2

(1− f ′)(1− f2)

)
+O(1)

≤ m
(
r,

f ′′

1− f ′

)
+m

(
r,

f ′′

1− f2

)
+O(1)

≤ m
(
r,

f ′′

1− f ′

)
+m

(
r,

f ′′

1− f

)
+m

(
r,

f ′′

1 + f

)
+O(1)

= S(r, f),

then 2m(r, f) = m(r, f2) ≤ m(r, 1− f2) + O(1) = S(r, f). Exchange f with g, we
obtain

m(r, g) = S(r, g).

Since f and g are properly meromorphic functions, then

2T (r, f) ≤ T (r, g′) +O(1) ≤ 2T (r, g) + S(r, g) = T (r, g2) + S(r, g)

≤ T (r, f ′) + S(r, g) ≤ 2T (r, f) + S(r, f) + S(r, g).

Hence, T (r, f) = T (r, g) + S(r, f). □

So far, not all transcendental meromorphic solutions of (2.3) have been presented
completely. We will give the partial consideration below. From (2.3), we conclude
that

f(z)2(1− f ′(z)) = g(z)2(1− g′(z)),



6 Y. C. GAO, K. LIU, X. G. QI EJDE-2024/52

and

f(z)2 − g(z)2 = f ′(z)− g′(z),

hence

f(z)2f ′(z)− g(z)2g′(z) = f ′(z)− g′(z). (2.4)

Integrating (2.4), it follows that

1

3
f(z)3 − 1

3
g(z)3 = f(z)− g(z) +B. (2.5)

Case 1. If B = 0, then

1

3
(f(z)3 − g(z)3) = f(z)− g(z).

If f(z) ̸= g(z), then the above equation implies that

f(z)2 + f(z)g(z) + g(z)2 = 3.

We rewrite the above equation as(f(z)√
3

)2

+
f(z)√

3

g(z)√
3

+
(g(z)√

3

)2

= 1.

Thus, we have F (z)2 + F (z)G(z) +G(z)2 = 1 by setting F (z) = f(z)√
3

and G(z) =
g(z)√

3
. From Saleeby’s result [26, Theorem 2.1.] with f(z) and g(z) are properly

meromorphic functions, we have

F (z) =
1 +

√
3i

2
√
3i

h+
−1 +

√
3i

2
√
3i

1

h
,

G(z) = − 1√
3i
h+

1√
3ih

.

(2.6)

Therefore,

f(z) =
1 +

√
3i

2i
h+

−1 +
√
3i

2i

1

h
,

g(z) = ih− i

h
,

(2.7)

where h is a non-zero meromorphic function. Substituting (2.7) into (2.3), we
obtain (1 +√

3i

2i
h+

−1 +
√
3i

2i

1

h

)2

= 1−
(
ih− i

h

)′
,(

ih− i

h

)2

= 1−
(1 +√

3i

2i
h+

−1 +
√
3i

2i

1

h

)′
.

(2.8)

Hence,

1−
√
3i

2
h2 +

1 +
√
3i

2

1

h2
= −1− ih′ − i

h′

h2
,

−h2 − 1

h2
= −1− 1 +

√
3i

2i
h′ +

−1 +
√
3i

2i

h′

h2
.

(2.9)

Cancelling the term h′

h2 , we have(1−√
3i

2

−1 +
√
3i

2i
− i

)
h2+

(1 +√
3i

2

−1 +
√
3i

2i
− i

) 1

h2
=

(1−√
3i

2i
− i

)
−
√
3ih′,



EJDE-2024/52 CROSSED DIFFERENTIAL SYSTEMS AND CLUNIE LEMMA 7

which implies that
√
3− 3i

2
h2 =

(1−√
3i

2i
− i

)
−

√
3ih′ =

3−
√
3i

2i
−
√
3ih′.

Thus, h satisfies

h′ =

√
3 + i

2
h2 −

√
3− i

2
which is a Riccati differential equation with constant coefficients and can be solved
completely.

Case 2. If B ̸= 0, [19, Corollary 1] affirms that there exist meromorphic solutions
of (2.5), but it is difficult to obtain all meromorphic solutions at present. Here,
we give the following details to describe the meromorphic solutions. Consider the
equation

f3

3
− g3

3
− f + g = B.

Let f = 3
√
3
(
G−F

)
and g = 3

√
3
(
G+F

)
. Then, the above equation changes into(

3
√
3(G− F )

)3
3

−
(

3
√
3(G+ F )

)3
3

− 3
√
3(G− F ) +

3
√
3(G+ F )−B = 0,

that is

2F 3 + 6G2F − 2
3
√
3F +B = 0.

Let F = 1/F1 and G = G1/F1. We obtain

1

F 3
1

(
BF 3

1 − 2
3
√
3F 2

1 + 6G2
1 + 2

)
= 0.

Thus, we have

G2
1 =

−B

6
F 3
1 +

1

3
3
√
3F 2

1 − 1

3
=

−B

6
(F1 − b1)(F1 − b2)(F1 − b3).

If two of b1, b2, b3 are equal, say b1 = b2, the above equation is( G1

F1 − b1

)2

=
B

6
(F1 − b3),

which admits obviously the non-constant meromorphic solutions F1 and G1, for
example F1 = b3 + H(z)2. If b1, b2, b3 are different, then we can assume that
G1 = F ′

1, thus there exists Weierstrass elliptic function F1 satisfying

F ′2
1 =

−B

6
(F1 − b1)(F1 − b2)(F1 − b3).

Remark 2.2. (1) Obviously, f = g = tanh z satisfies (2.3). If f = g, then (2.3)
reduces to Riccati differential equation f ′(z) = 1 − f(z)2 and all meromorphic

solutions of the above equation can be stated as f(z) = e2z+c+1
e2z+c−1 , where c is any

constant.
(2) [16, Proposition 9.1.11 ] shows that all meromorphic solutions of (4.1) with

rational coefficients are of finite order on growth and pseudo-prime with polynomials
coefficients in [24]. It is an interesting question for the growth and pseudo-primeness
of transcendental meromorphic solutions of (2.3), where the pseudo-primeness of
F (z) means that every factorization of F = f ◦ g implies that f is rational or g is
a polynomial.



8 Y. C. GAO, K. LIU, X. G. QI EJDE-2024/52

3. Crossed differential systems of equations in certain types

Recall the Weierstrass ℘-function which satisfies the Weierstrass differential
equation ℘′(z)2 = 4℘(z)3 − 1. The following theorem is to present the properties
on the crossed Weierstrass differential system of equations. Unfortunately, we will
only describe the proximate functions of the transcendental meromorphic solutions
and can not describe the precise expressions of the transcendental meromorphic
solutions.

Theorem 3.1. Let f(z) and g(z) be transcendental meromorphic solutions of the
crossed Weierstrass differential system

f(z)3 = 1− g′(z)2,

g(z)3 = 1− f ′(z)2.
(3.1)

Then,

(1) f(z) and g(z) are properly transcendental meromorphic functions and all
poles are double poles;

(2) m(r, f) = S(r, f) and m(r, g) = S(r, g).

Proof. (1) By the basic observations from (3.1), then f(z) and g(z) are not poly-
nomials. Let f(z) be a transcendental entire function. So g(z) is entire. By
Valiron-Mohon’ko theorem [16, Theorem 2.2.5] again, we obtain

3T (r, f) = 2T (r, g′) +O(1) ≤ 2T (r, g) + S(r, g) =
2

3
T (r, g3) + S(r, g)

≤ 4

3
T (r, f ′) + S(r, g) ≤ 4

3
T (r, f) + S(r, f) + S(r, g).

Similarly, we have

3T (r, g) ≤ 4

3
T (r, g) + S(r, f) + S(r, g).

Combining the above two inequalities implies that

3(T (r, f) + T (r, g)) ≤ 4

3
(T (r, f) + T (r, g)) + S(r, f) + S(r, g),

which is impossible. Hence, f(z) and g(z) must be properly transcendental mero-
morphic functions. Let z0 be a pole of f(z) and g(z) with multiplicity p and q,
respectively. Then p = q = 2 follows by (3.1).

(2) Taking the first derivative of the second equation of (3.1), we have

3g2g′ = −2f ′f ′′.

Taking the first derivative of the first equation of (3.1), we obtain

3f2f ′ = −2g′g′′.

Then
3f3f ′ = 3(1− g′2)f ′ = −2fg′g′′,

we obtain
9(1− g′2)2f ′2 = 4f2g′2g′′2.

Hence, 1
f2 = 4g′2g′′2

9(1−g′2)2(1−g3) . Then, we conclude that

m(r, g) = m
(
r,
−2f ′f ′′

3gg′

)
= m

(
r,
4g′g′′f ′′

9gg′f2

)



EJDE-2024/52 CROSSED DIFFERENTIAL SYSTEMS AND CLUNIE LEMMA 9

≤ m
(
r,
f ′′

f2

)
+ S(r, g)

≤ m
(
r,

1

f

)
+ S(r, g) + S(r, f).

It follows that

m(r, g2) ≤ m
(
r,

1

f2

)
+ S(r, f) + S(r, g)

≤ m
(
r,

4g′2g′′2

9(1− g′2)2(1− g3)

)
+ S(r, f) + S(r, g)

≤ m
(
r,

g′g′′

1− g′2

)
+m

(
r,

g′′

1− g′

)
+m

(
r,

g′′

1 + g′

)
+m

(
r,

g′

1− g3

)
+ S(r, f) + S(r, g)

≤ S(r, f) + S(r, g).

Furthermore,

3T (r, f) = 2T (r, g′) +O(1)

≤ 4T (r, g) + S(r, g) =
4

3
T (r, g3) + S(r, g)

≤ 4

3
T (r, f ′2) + S(r, g)

≤ 16

3
T (r, f) + S(r, f) + S(r, g).

So, we havem(r, g) = S(r, g). In addition,m(r, f) = S(r, f) can be proved similarly.
□

Theorem 3.2. Let f(z) and g(z) be transcendental meromorphic solutions of the
complex differential system

f(z)4 = 1− g′(z)2,

g(z)4 = 1− f ′(z)2.
(3.2)

Then m(r, f) = S(r, f) and m(r, g) = S(r, g).

Proof. Assume that f(z) and g(z) are transcendental meromorphic solutions of
(3.2). We have that T (r, f) = T (r, g) + S(r, g). Taking the first derivative of the
first equation of (3.2), we have 4f3f ′ = −2g′g′′. Thus 4f6f ′2 = g′2g′′2 and

4f6(1− g4) = g′2g′′2.

Therefore,

m(r, f) ≤ m
(
r,

g′2g′′2

1− g4

)
≤ m

(
r,

g′2

1− g2

)
+m

(
r,

g′′2

1 + g2

)
≤ m

(
r,

g′

1− g

)
+m

(
r,

g′

1 + g

)
+m

(
r,

g′′

1 + ig

)
+m

(
r,

g′′

1− ig

)
≤ S(r, g) = S(r, f).

And m(r, g) = S(r, g) can be proved in the same way. □



10 Y. C. GAO, K. LIU, X. G. QI EJDE-2024/52

Remark 3.3. If f = g in the system (3.2), then it reduces to the equation f(z)4+

f ′(z)2 = 1. By the result given by Gross [9], we assume that f(z)2 = 1−β(z)2

1+β(z)2 and

f ′(z) = 2β(z)
1+β(z)2 , where β(z) is any meromorphic function. Furthermore, taking the

first derivative of f(z)2 and the basic computations, β(z) must satisfy β′(z)2 =
2β(z)(1 + β(z)2) which is also a Weierstrass differential equation.

Theorem 3.4. The complex differential system

f(z)n = 1− g′(z)2,

g(z)n = 1− f ′(z)2,
(3.3)

has no transcendental meromorphic solutions f(z) and g(z) if n ̸= 2, 3, 4.

Proof. If n ≥ 5, by Picard theorem and the completely ramified theorem (A non-
constant meromorphic functions f(z) can have at most four completely ramified
values), we can get a contradiction from the first equation of (3.3). If n = 1, we
will consider the existence of meromorphic solutions of

f(z) = 1− g′(z)2,

g(z) = 1− f ′(z)2.
(3.4)

Obviously, if (3.4) admits transcendental meromorphic solutions f(z) and g(z),
then f(z) and g(z) are transcendental entire functions. Taking the first derivative
of the first equation of (3.4), we obtain f ′(z) = −2g′(z)g′′(z), then g(z) = 1 −
4g′(z)2g′′(z)2. Furthermore, taking the derivative of the above equation, we have

g′(z) = −8g′(z)g′′(z)3 − 8g′(z)2g′′(z)g′′′(z)

and

−8g′′(z)3 − 8g′(z)g′′(z)g′′′(z) = 1.

For the reason that g(z) is a transcendental entire function and the above equation,
then g′′(z) has no zeros and g′′(z) = eP (z), where P (z) is any entire function. Thus,

g′′(z)2 + g′(z)g′′′(z) = −1

8
e−P (z). (3.5)

By a result given by Mues [25] or see Gundersen and Yang [7, Theorem 2], we obtain
that g′(z) = α1e

λ1z. Thus, g′′(z) = α1λ1e
λ1z and g′′′(z) = α1λ

2
1e

λ1z. Substituting
the above into (3.5), we have 2α3

1λ
3
1e

3λ1z = − 1
8 , which is impossible. □

Remark 3.5. (1) All transcendental meromorphic solutions of (3.3) can be ex-
pressed when n = 2, see Li and Liu [20, Theorem 1.1]. We also have discussed the
properties of transcendental meromorphic solutions of (3.3) when n = 3, 4. The
properties are deserved to considering when the first derivative is replaced with the
higher derivatives in (3.3).

(2) Gundersen and Yang [7] obtained many results on the generalizations of
(3.5).

Next, we present two results concerning the existence of transcendental meromor-
phic solutions on two crossed complex differential systems of equations in certain
types.



EJDE-2024/52 CROSSED DIFFERENTIAL SYSTEMS AND CLUNIE LEMMA 11

Theorem 3.6. The complex differential system

f(z)f ′(z) = 1− g′(z),

g(z)g′(z) = 1− f ′(z),
(3.6)

has no non-constant transcendental meromorphic solutions.

Proof. If (3.6) admits transcendental meromorphic solutions f(z) and g(z), then
f(z) and g(z) must be transcendental entire functions by checking the poles mul-
tiplicities of f(z) and g(z). By the addition and subtraction of two equations in
(3.6), we have ff ′+ gg′ = 2− f ′− g′ and ff ′− gg′ = f ′− g′. Integrating the above
two equations, we have

1

2
f2 − 1

2
g2 = f − g +A1,

and
1

2
f2 +

1

2
g2 = 2z − f − g +A2,

thus we have T (r, f) = T (r, g)+S(r, f) by Valiron-Mohon’ko theorem [16, Theorem
2.2.5]. However, we obtain f2 = 2z−2g+A1+A2 by adding the above two equations,
which implies that 2T (r, f) = T (r, g) + S(r, f), which is a contradiction. □

Theorem 3.7. The complex differential system

f(z)ng′(z) = 1,

g(z)nf ′(z) = 1,
(3.7)

has no meromorphic solutions.

Proof. We assume that z0 is a pole of f(z) with multiplicity p, then z0 is a pole of
f ′(z) with multiplicity p+ 1. From (3.7), we obtain that z0 is a zero of g′(z) with
multiplicity np and z0 is a zero of g(z) with multiplicity p+1

n . Thus, np+ 1 = p+1
n .

Therefore, we have that either n = 1 or f(z) and g(z) are all entire functions with
no zeros for avoiding a contradiction. We discuss two cases below.

Case 1. If n = 1, then
f(z)g′(z) = 1,

g(z)f ′(z) = 1.
(3.8)

Subtracting the second from the equation equations of (3.8), we have
(
g
f

)′
= 0 and

g = cf follows; however, cff ′ = 1 has no any meromorphic solutions, where c is a
constant.

Case 2. If f(z) and g(z) are all entire functions with no zeros, then we assume that
f(z) = eP (z) and g(z) = eQ(z), where P (z) and Q(z) are entire functions. From
(3.7), we have

Q′(z)enP (z)+Q(z) = 1,

P ′(z)enQ(z)+P (z) = 1.
(3.9)

Case 2.1. If f(z) and g(z) are entire functions of finite order, that is P (z) and
Q(z) are polynomials. From (3.9), we have P (z) = Az+a, Q(z) = Bz+b (AB ̸= 0),
substituting P (z) and Q(z) into (3.9), we have A = −B, n = 1. However, ea+b = 1

A

and ea+b = 1
B , thus A = B follows, which is impossible.

Case 2.2. If f(z) and g(z) are entire functions of infinite order, that is P (z)
and Q(z) are transcendental entire functions. From the equations of (3.7), we



12 Y. C. GAO, K. LIU, X. G. QI EJDE-2024/52

have S(r, f) = S(r, g). From the equations of (3.9), by letting P ′(z) = es(z) and
Q′(z) = et(z), we have

enP (z)+Q(z) = e−t(z),

enQ(z)+P (z) = e−s(z).
(3.10)

From [36, Theorem 1.47] and the first main theorem of Nevanlinna theory, we obtain

T (r,Q′(z)) = T (r, e−t(z)) +O(1) = S(r, eQ(z)) = S(r, g(z)) = S(r, f(z))

and

T (r, P ′(z)) = T (r, e−s(z)) +O(1) = S(r, eP (z)) = S(r, f(z)) = S(r, g(z)).

The system (3.10) implies that e(n
2−1)P (z) = e−nt(z)+s(z) and n = 1 follows for

avoiding a contradiction, this is the Case 1. □

4. Crossed version of Clunie lemma

The Clunie lemma is an efficient tool to study the properties of meromorphic
solutions of complex differential equations and has many variants with numerous
applications in the bibliography. The original lemma is due to Clunie [2, Lemma
1] and the following version can be found in [16, Lemma 2.4.2].

Lemma 4.1. Let f(z) be a transcendental meromorphic solution of

fnP (z, f) = Q(z, f),

where P (z, f), Q(z, f) are differential polynomials with small coefficients, say {aλ :
λ ∈ I}, such that m(r, aλ) = S(r, f) for all λ ∈ I. If deg(Q(z, f)) ≤ n, then

m(r, P (z, f)) = S(r, f),

where S(r, f) = o(T (r, f)) for all r outside of an exceptional set with finite linear
measure.

Obviously, the Clunie lemma can be applied to estimate the counting function of
poles of P (z, f), namely N(r, P (z, f)), to determine the existence of transcendental
meromorphic or entire solutions of complex differential equations. For instance,

f ′(z) = a0(z) + a1(z)f(z) + f(z)2 (4.1)

has no transcendental entire solutions, where a0(z), a1(z) are small functions with
respect to f(z), otherwise m(r, f) = S(r, f) follows immediately by the Clunie
lemma. Equation (4.1) is called Riccati differential equation and admits actually
transcendental meromorphic solutions, more details on the existence of meromor-
phic solutions of Riccati differential equation can be found in [16, Chapter 9].
Doeringer [3], He-Xiao [14], Korhonen [11], Yang and Ye [35, Theorem 1] also de-
rived the different estimate on the proximity functions. The difference version of
Clunie lemma is given by Halburd and Korhonen [12], Laine and Yang [17]. The
delay-differential version of Clunie lemmas can be found in [21, Chapter 1]. These
different versions of Clunie lemma have many applications on complex differential
equations, complex difference equations and complex delay-differential equations.
Theorems 2.1, 3.1, 3.2 inspire us to consider a corresponding version of Clunie
lemma for the crossed complex differential systems of equations.



EJDE-2024/52 CROSSED DIFFERENTIAL SYSTEMS AND CLUNIE LEMMA 13

Theorem 4.2. Let f(z) and g(z) be transcendental meromorphic solutions of the
system

f(z)nL1(z, f) = L2(z, g),

g(z)nL1(z, g) = L2(z, f),
(4.2)

where L1(z, f) and L2(z, f) are differential polynomials in f(z) with small coeffi-
cients, L1(z, g) and L2(z, g) are differential polynomials in g(z) with small coeffi-
cients. If γL2 ≤ n and |L1(z, f)| ≥ 1 when |f | ≥ 1, then m(r, L1(z, f)) = S1(r);
if γL2

≤ n and |L1(z, g)| ≥ 1 when |g| ≥ 1, then m(r, L1(z, g)) = S1(r), where
S1(r) = S(r, f) + S(r, g).

Proof. Assume that E1 and E2 satisfy

E1 = {φ ∈ [0, 2π]||f(reiφ)| < 1},
E2 = [0, 2π] \ E1.

According to the definition of m(r, L1(z, f)), we have

m(r, L1(z, f)) =
1

2π

∫
E1

log+ |L1(z, f)|dφ+
1

2π

∫
E2

log+ |L1(z, f)|dφ := I1 + I2.

To estimate the first part I1, we rewrite L1(z, f) as

|L1(z, f)| =
∣∣∑
i∈I

Mi(z, f)
∣∣

=
∣∣∑
i∈I

aif
l0i(f ′)l1i · · · (f (ν))lνi

∣∣
≤

∑
i∈I

aλ
∣∣f ′

f

∣∣l1i · · · ∣∣f (ν)

f

∣∣lνi
,

by the lemma of the logarithmic derivative, we conclude that

I1 =
1

2π

∫
E1

log+ |L1(z, f)|dφ = S(r, f).

To consider the second part I2, we see that

|L1(z, f)| =
∣∣L2(z, g)

fn

∣∣
=

∣∣L2(z, f)

fn

∣∣ ∣∣L2(z, g)

L2(z, f)

∣∣ ∣∣gn
gn

∣∣
=

∣∣L2(z, f)

fn

∣∣ ∣∣ 1

L1(z, g)

∣∣ ∣∣L2(z, g)

gn
∣∣.

(4.3)

Furthermore, if φ ∈ E2 and |g(reiφ)| < 1, by the first equation of (4.2), we obtain

|L1(z, f)| =
∣∣L2(z, g)

fn

∣∣.
Hence,

1

2π

∫
E2

log+ |L1(z, f)|dφ =
1

2π

∫
E2

log+
∣∣L2(z, g)

fn

∣∣dφ = S(r, g),

then m(r, L1(z, f)) = S(r, f) + S(r, g) follows.



14 Y. C. GAO, K. LIU, X. G. QI EJDE-2024/52

We proceed to assume that φ ∈ E2 and |g(reiφ)| ≥ 1. Since γL2
≤ n, we can

conclude that ∣∣L2(z, f)

fn

∣∣ = ∣∣∑j∈J Nj(z, f)

fn

∣∣
=

∣∣∑j∈J bjf
s0j (f ′)s1j · · · (f (µ))sµj

fn

∣∣
≤

∑
j∈J

bj
∣∣f ′

f

∣∣s1j · · · ∣∣f (µ)

f

∣∣sµj
.

By the lemma of the logarithmic derivative again, we obtain

1

2π

∫
E2

log+
∣∣L2(z, f)

fn

∣∣dφ = S(r, f).

Similarly, for φ ∈ E2 and |g(reiφ)| ≥ 1, we have

1

2π

∫
E2

log+
∣∣L2(z, g)

gn
∣∣dφ = S(r, g).

By

|L1(z, f)| =
∣∣L2(z, f)

fn

∣∣ ∣∣ 1

L1(z, g)

∣∣ ∣∣L2(z, g)

gn
∣∣,

we have

I2 =
1

2π

∫
E2

log+ |L1(z, f)|dφ = S(r, f) + S(r, g)

with the condition that |L1(z, g)| ≥ 1 when |g| ≥ 1. In addition, m(r, L1(z, g)) =
S1(r) can be proved in the same way. □

Let us see the condition that |L1(z, f)| ≥ 1 when |f | ≥ 1, |L1(z, g)| ≥ 1 when
|g| ≥ 1. In fact, this condition is satisfied in Theorems 2.1, 3.1, 3.2.

5. Discussion

Some properties on of crossed differential systems of certain types have been
considered in Sections 2 and 3. We pose the following question for the further
studies.

Question 5.1. Can we include all transcendental meromorphic solutions of the
crossed differential systems in Theorems 2.1, 3.1, and 3.2? Furthermore, how can
we describe the uniqueness, growth or pseudo-primeness of transcendental mero-
morphic solutions of the crossed differential system (1.2)?

The complex differential polynomials f(z)2 − f ′(z) and f(z)n − f ′(z)2(n = 3, 4)
are considered in Theorems 2.1, 3.1, and 3.2, however f(z)2−f ′(z) is not a UDPM.
It remains an open question whether f(z)n−f ′(z)2 (n ≥ 3) are UDPM or not. The
following question may be also worth considering.

Question 5.2. Can we obtain conditions to describe completely the UDPM?

Acknowledgments. This work was partial supported by the NSFC (No. 12061042)
and the NSF of Shandong Province (No. ZR2022MA071). The authors are grateful
to Professor Elias G. Saleeby for useful discussions in Section 2 and the reviewer
for useful comments and suggestions for the paper.



EJDE-2024/52 CROSSED DIFFERENTIAL SYSTEMS AND CLUNIE LEMMA 15

References

[1] G. A. Barsegian, I. Laine, C. C. Yang; Stability phenomenon and problems for complex
differential equations with related to shared values, Matematychni. Studii., 13 (2000), 224–

228.

[2] J. Clunie; On integral and meromorphic functions, J. London Math. Soc., 37 (1962), 17–27.
[3] W. Doeringer; Exceptional values of differential polynomials, Pacific J. Math., 98 (1982),

55–62.

[4] C. Y. Fang, M. L. Fang; Uniqueness of meromorphic functions and differential polynomials,
Comput. Math. Appl., 44 (2002), 607–617.

[5] L. Y. Gao; On m admissible components of solutions, Complex Variables Theory Appl., 35
(1998), 297–306.

[6] L. Y. Gao; On the growth of components of meromorphic solutions of systems of complex

differential equations, Acta Math. Appl. Sin. Engl. Ser., 21 (2005), (3), 499–504.
[7] G. G. Gundersen, C. C. Yang; Entire solutions of binomial differential equations, Comput.

Methods Funct. Theory, 21 (2021), 1–13.

[8] Y. H. Guo, K. Liu; Meromorphic solutions of Fermat type differential and difference equations
of certain types, Ann. Polon. Math., 131 (2023), 1–19.

[9] F. Gross; On the equation fn + gn = 1, Bull. Amer. Math. Soc., 72 (1966), 86–88.

[10] H. H. Khoai, C. C. Yang; On the functional equation P (F ) = Q(G), Kluwer Academic
Publishers, 2004. Value Distribution Theory and Related Topics, Book series: Advanced in

Complex Analysis and Its Applications, 201–207.

[11] R. Korhonen; Sharp forms of Nevanlinna error terms in differential equations, Symposium
on Complex Differential and Functional Equations, 117–133, Univ. Joensuu Dept. Math. Rep.

Ser., 6, Univ. Joensuu, Joensuu, 2004.
[12] R. G. Halburd, R. J. Korhonen; Difference analogue of the lemma on the logarithmic deriv-

ative with applications to difference equations, J. Math. Anal. Appl., 314 (2006), 477–487.

[13] W. K. Hayman; Meromorphic Functions, Clarendon Press, Oxford, 1964.
[14] Y. Z. He, X. Z. Xiao; Algeborid functions and ordinary differential equations, Science Press

Beijing, 1988.

[15] J. Heittokangas, R. Korhonen, I. Laine; On meromorphic solutions of certain nonlinear
differential equations, Bull. Austral. Math. Soc., 66 (2002), 331–343.

[16] I. Laine; Nevanlinna Theory and Complex Differential Equations. Studies in Mathematics

15, Walter de Gruyter, Berlin, 1993.
[17] I. Laine, C. C. Yang; Clunie theorems for difference and q-difference polynomials. J. Lond.

Math. Soc., 76 (2007), (2), 556–566.

[18] P. Li, C. C. Yang; Some further results on the unique range set of meromorphic functions,
Kodai Math. J., 18 (1995), 437–450.

[19] P. Li, C. C. Yang; Some further results on the functional equation P (F ) = Q(G), Kluwer

Academic Publishers, 2004. Value Distribution Theory and Related Topics, Book series: Ad-
vanced in Complex Analysis and Its Applications, 220–231.

[20] Y. X. Li, K. Liu; Meromorphic solutions of nonlinear systems of Fermat type, Bull. Malays.
Math. Sci. Soc., 46 (2023), 196.

[21] K. Liu, I. Laine, L. Z. Yang; Complex Delay-Differential Equations, De Gruyter Stud. Math.,
78 De Gruyter, Berlin, 2021.

[22] H. F. Liu, Z. Q. Mao; The exact meromorphic solutions of some nonlinear differential equa-

tions, Acta Math. Sci. Ser. B (Engl. Ed.), 44 (2024), 103–114.

[23] H. F. Liu, Z. Q. Mao; On the forms of meromorphic solutions of some type of non-linear
differential equations, Comput. Methods Funct. Theory. DOI 10.1007/s40315-023-00509-0.

[24] E. Mues; Über faktorisierbare Lösungen von Riccatischen Differentialgleichungen, Math. Z.,
121 (1971), 145–156.

[25] E. Mues; Über die Nullstellen homogener Differentialpolynome, Manuscripta Math. 23

(1978), 325–341.
[26] E. G. Saleeby; On complex analytic solutions of certain trinomial functional and partial

differential equations, Aequat. Math. 85 (2013), 553–562.
[27] Z. H. Tu, X. Z. Xiao; On the meromorphic solutions of system of higher order algebraic

differential equations, Complex Variables Theory Appl., 15 (1990), 197–209.



16 Y. C. GAO, K. LIU, X. G. QI EJDE-2024/52

[28] F. N. Wang, K. Liu; Crossing Malmquist systems with certain types, J. Contemp. Math.

Anal., 58 (2023), 459–469.

[29] H. Y. Xu, S. Y. Liu, Q. P. Li; Entire solutions for several systems of nonlinear difference
and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl., 483 (2020),

123641.

[30] H. Y. Xu, L. Xu; Transcendental entire solutions for several quadratic binomial and trinomial
PDEs with constant coefficients. Anal. Math. Phys., 12 (2022), 64.

[31] L. Xu, T. B. Cao; Solutions of complex Fermat-type partial difference and differential-

difference equations, Mediterr. J. Math., 15 (2018), 1-14. Correction: Mediterr. J. Math.,
17 (2020), 8.

[32] C. C. Yang, X. H. Hua; Uniqueness and value sharing of meromorphic functions, Ann. Acad.

Sci. Fenn. Math., 22 (1997), 395–406.
[33] C. C. Yang, I. Laine; On analogies between non-linear difference and differential equations,

Proc. Japan Acad. Ser. A Math. Sci., 86 (2010), 10–14.
[34] C. C. Yang, P. Li; On the transcendental solutions of a certain type of nonlinear differential

equations, Arch. Math., 82 (2004), 442–448.

[35] C. C. Yang, Z. Ye; Estimates of the proximate function of differential polynomials, Proc.
Japan Acad. Ser. A Math. Sci., 83 (2007), 50–55.

[36] C. C. Yang, H. X. Yi; Uniqueness Theory of Meromorphic Functions, Kluwer Academic

Publishers, 2003.
[37] H. X. Yi; Meromorphic functions that share one or two values, Complex Variables Theory

Appl., 28 (1995), (1), 1–11.

Yingchun Gao

Department of Mathematics, Nanchang University, Nanchang, Jiangxi, 330031, China
Email address: gaoyingchun97@163.com

Kai Liu (corresponding author)
Department of Mathematics, Nanchang University, Nanchang, Jiangxi, 330031, China

Email address: liukai418@126.com, liukai@ncu.edu.cn

Xiaoguang Qi

School of Mathematics, University of Jinan, Jinan, Shandong, 250022, China

Email address: xiaoguang.202@163.com, xiaogqi@mail.sdu.edu.cn


	1. Introduction
	Problem

	2. Crossed Riccati differential systems of equations
	3. Crossed differential systems of equations in certain types
	4. Crossed version of Clunie lemma
	5. Discussion
	Acknowledgments

	References