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Entire solutions of two certain Fermat-type ordinary differential equations

  • Binbin Hu and Liu Yang EMAIL logo
Published/Copyright: October 11, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we investigate the precise expression forms of entire solutions for two certain Fermat-type ordinary differential equations:

( a 0 f + a 1 f ′ ) 2 + ( a 0 f + a 2 f ′ ) 2 = p

and

( a 0 f + a 1 f ′ ) 2 + ( a 0 f + a 2 f ′ ) 2 = e g

in C by making use of the Nevanlinna theory for meromorphic functions, where a 0 , a 1 , and a 2 are the complex numbers with a 0 ≠ 0 and ∣ a 1 ∣ + ∣ a 2 ∣ ≠ 0 , while p and g are the polynomials in C . Moreover, some examples are given to illustrate the existence of entire solutions for the aforementioned two certain equations.

Keywords: entire solution; ordinary differential equation; polynomial
MSC 2010: 30D20; 30D35; 34M05; 39B32

1 Introduction

Throughout this article, by meromorphic functions, we will always mean meromorphic functions in the complex plane. We adopt the standard notations in the Nevanlinna theory of meromorphic functions as explained in Hayman [1], Laine [2], and Yang [3]. It will be convenient to let E denote any set of positive real numbers of finite linear measure, not necessarily the same at each occurrence. For a nonconstant meromorphic function f , we denote by T ( r , f ) the Nevanlinna characteristic of f and by S ( r , f ) any quantity satisfying S ( r , f ) = o ( T ( r , f ) ) , as r → ∞ and r ∉ E .

In 1927, Montel [4] first studied the Fermat functional equation f n + g n = 1 and proved that the entire solutions f and g in the complex plane (and thus in C n ) must be both constants when n ≥ 3 . After a few decades, Gross [5,6] showed that all meromorphic solutions f and g of the Fermat functional equation f n + g n = 1 are characterized as follows: (i) for n = 2 , the entire solutions f = sin h and g = cos h , where h is an entire function in C . The meromorphic solutions f = 1 − β 2 1 + β 2 and g = 2 β 1 − β 2 , where β is a meromorphic function in C ; (ii) for n > 2 , there are no nonconstant entire solutions; (iii) for n = 3 , the meromorphic solutions f = 1 2 + ℘ ′ ( h ) 2 3 ⁄ ℘ ( h ) and g = 1 2 − ℘ ′ ( h ) 2 3 ⁄ ℘ ( h ) , where ℘ is the Weierstrass elliptic function satisfying ( ℘ ′ ) 2 = 4 ℘ 3 − 1 ; and (iv) for n > 3 , there are no nonconstant meromorphic solutions.

Henceforth, studying the precise forms of meromorphic solutions on the Fermat-type functional equations attracts a lot of attention. In 1970, Yang [7] investigated that the Fermat-type function equation f n + g m = 1 has no nonconstant entire solutions when 1 n + 1 m < 1 . With the development of the Nevanlinna theory, many mathematicians try to characterize meromorphic solutions of the Fermat-type differential equations. For example, Yang and Li [8] showed the transcendental meromorphic solutions of a certain type of nonlinear differential equations and obtained the following results.

Theorem A

(See [8, Theorem 1]) Let L ( f ) = ∑ k = 0 n b k f ( k ) , where b 0 , b 1 , … , b n − 1 be the polynomials, and b n be a nonzero constant. If a ( z ) ≢ 0 , then a transcendental meromorphic solution of the equation f 2 ( z ) + ( L ( f ) ) 2 = a ( z ) must have the form f ( z ) = 1 2 ( P ( z ) e R ( z ) + Q ( z ) e − R ( z ) ) , where P , Q , and R are the polynomials, and P Q = a . If all b k are the constants, then deg P + deg Q ≤ n − 1 . Moreover, R ( z ) = λ is a nonzero constant that satisfies the following equations:

∑ k = 0 n b k λ k = 1 i , ∑ k = 0 n b k k j λ k − j = 0 , j = 1 , … , p ,

∑ k = 0 n b k ( − λ ) k = − 1 i , ∑ k = 0 n b k k j ( − λ ) k − j = 0 , j = 1 , … , q .

Theorem B

(See [8, Theorem 3]) Let a 1 , a 2 , and a 3 be the nonzero meromorphic functions. Then, a necessary condition for the differential equation

a 1 f 2 + a 2 ( f ′ ) 2 = a 3

to have a transcendental meromorphic solution satisfying T ( r , a k ) = S ( r , f ) , k = 1 , 2 , 3 , and a 1 ⁄ a 3 ≡ constant .

Furthermore, Yang and Li [8] proposed a conjecture: let a 1 , a 2 , and a 3 be the nonzero polynomials. Then, the equation a 1 f 2 + a 2 ( f ′ ) 2 = a 3 has no transcendental meromorphic solution when a 1 ⁄ a 3 is a nonzero constant and a 2 ⁄ a 3 is not the square of any rational function. Later, Tang and Liao [9] denied the conjecture and characterized the precise expression form of transcendental meromorphic solutions of the Fermat-type differential equation f 2 + P 2 ( f ( k ) ) 2 = Q , where P and Q are the polynomials.

In 2019, Han and Lü [10] studied the meromorphic solutions of the differential equation f n ( z ) + ( f ′ ) n = e a z + b by discussing for positive integers n , where a , b ∈ C are the constants. They obtained the following result.

Theorem C

(See [10, Theorem 2.1]) The meromorphic solutions f of the following differential equation:

f n ( z ) + ( f ′ ( z ) ) n = e a z + b

must be entire functions, and the following assertions hold:

  1. For n = 1 , the general solutions f ( z ) = e a z + b a + 1 + c e − z for a ≠ − 1 and f ( z ) = z e − z + b + c e z ;

  2. For n = 2 , either a = 0 and the general solutions f ( z ) = e b 2 sin ( z + c ) , or f ( z ) = d e a z + b 2 ;

  3. For n ≥ 3 , the general solutions f ( z ) = d e a z + b n .

Here, a , b , c , d ∈ C with d n ( 1 + ( a n ) n ) = 1 for n ≥ 1 .

Motivated by the quadratic differential equations in [11,12], Wang et al. [13] investigated the existence and precise expression form of entire solutions for the Fermat-type differential-difference equation:

( L 1 ) 2 + 2 ω L 1 L 2 + ( L 2 ) 2 = e a z + b

by making use of a transformation: ω 1 = − ω + ω 2 − 1 and ω 2 = − ω − ω 2 − 1 , where L 1 = a 0 f ( z ) + a 1 f ′ ( z ) + a 2 f ( z + c ) , L 2 = b 0 f ( z ) + b 1 f ′ ( z ) + b 2 f ( z + c ) , and ω is a complex number with ω 2 ≠ 0 , 1 . They proved the entire solutions f with finite order of the aforementioned equation, and it only has the form f ( z ) = C 1 e ( A + a 2 ) z + C 2 e ( − A + a 2 ) z + C 3 e C 0 z , where A ≠ 0 , C i ( i = 0 , 1 , 2 , 3 ) are the constants.

On the other hand, the Nevanlinna theory of meromorphic functions has been widely used to deal with meromorphic solutions of the Fermat-type partial differential equations. The research of the Fermat-type partial differential equations goes back to the Pythagorean functional equations and induces the well-known eikonal equations u z 1 2 + u z 2 2 = 1 in C 2 , which was considered by Li in [14] and also discussed in [15] and [16]. Later on, Li discussed two types of more general partial differential equations in C 2 , such as partial differential equations (PDEs) u z 1 2 + u z 2 2 = e g in [17] and PDEs u z 1 2 + u z 2 2 = p in [18], where g and p are the polynomials in C 2 , and obtained two interesting and important results as follows:

Theorem D

(See [17, Theorem 2.1]) Let g be a polynomial in C 2 . Then, u is an entire solution of the partial differential equation:

u z 1 2 + u z 2 2 = e g

in C 2 if and only if:

  1. u = f ( c 1 z 1 + c 2 z 2 ) or

  2. u = ϕ 1 ( z 1 + i z 2 ) + ϕ 2 ( z 1 − i z 2 ) ,

where f is an entire function in C satisfying

f ′ ( c 1 z 1 + c 2 z 2 ) = ± e 1 2 g ( z ) ,

c 1 and c 2 are two constants satisfying c 1 2 + c 2 2 = 1 , and ϕ 1 and ϕ 2 are the entire functions in C satisfying

ϕ 1 ′ ( z 1 + i z 2 ) ϕ 2 ′ ( z 1 − i z 2 ) = 1 4 e g .

Furthermore, the forms of f , ϕ 1 , and ϕ 2 are also obtained in [17, Theorem 2.1].

Theorem E

(See [18, Theorem 2.1]) Let p ≢ 0 be a polynomial in C 2 . Then, u is an entire solution to the partial differential equation:

u z 1 2 + u z 2 2 = p

in C 2 if and only if:

  1. u = F ( z 1 , z 2 − i z 1 ) + f ( z 2 − i z 1 ) ; or

  2. u = ϕ 1 ( z 1 + i z 2 ) + ϕ 2 ( z 1 − i z 2 ) ,

where F ( z 1 , z 2 ) = ∫ 0 z 1 q ( t , i t + z 2 ) d t , q is a factor of p , f is a polynomial in C satisfying that f ′ ( z 2 − i z 1 ) = 1 2 i q ( z 1 , z 2 ) − p ( z 1 , z 2 ) q ( z 1 , z 2 ) − 2 i F z 2 ( z 1 , z 2 − i z 1 ) , and ϕ 1 and ϕ 2 are the polynomials in C satisfying that:

ϕ 1 ′ ( z 1 + i z 2 ) ϕ 2 ′ ( z 1 − i z 2 ) = 1 4 p ( z ) .

Moreover, the forms of f , ϕ 1 , and ϕ 2 are also obtained in [18, Theorem 2.1].

Inspired by the aforementioned results, the purpose of this study is to characterize the entire solutions of the Fermat-type ordinary differential equations:

(1.1) ( a 0 f + a 1 f ′ ) 2 + ( a 0 f + a 2 f ′ ) 2 = p ,

and

(1.2) ( a 0 f + a 1 f ′ ) 2 + ( a 0 f + a 2 f ′ ) 2 = e g ,

in C , where a 0 , a 1 , and a 2 are the complex numbers with a 0 ≠ 0 and ∣ a 1 ∣ + ∣ a 2 ∣ ≠ 0 , while p and g are the polynomials in C .

This article is organized as follows. We will describe the entire solutions for equation (1.1) in Section 2. The investigation of the entire solutions for equation (1.2) will be divided into two cases in Section 3, that is, the case where one of a 1 and a 2 is zero and the case a 1 ≠ 0 , a 2 ≠ 0 , and a 1 ≠ a 2 . The main theorems and corollary will be proved in Section 4. In addition, we shall present some examples to illustrate the existence of entire solutions for equations (1.1) and (1.2).

2 Ordinary differential equations ( a 0 f + a 1 f ′ ) 2 + ( a 0 f + a 2 f ′ ) 2 = p

In this section, we will describe the entire solutions of the Fermat-type ordinary differential equation (1.1).

Theorem 2.1

Let a 0 , a 1 , and a 2 be the complex numbers with a 0 ≠ 0 and ∣ a 1 ∣ + ∣ a 2 ∣ ≠ 0 , and p be a nonconstant polynomial in C . Then, f is an entire solution of the Fermat-type ordinary differential equation:

(2.1) ( a 0 f + a 1 f ′ ) 2 + ( a 0 f + a 2 f ′ ) 2 = p

if and only if one of the following holds:

  1. If a 1 = a 2 , then the polynomial p takes the form p = 2 φ 2 , and then, f ( z ) = e − a 0 a 1 z ∫ φ a 1 e a 0 a 1 z d z + c , where φ is a polynomial in C , and c is an arbitrary complex number;

  2. If a 1 ≠ a 2 , then f ( z ) = ( a 2 + i a 1 ) k 2 a 0 ( a 2 − a 1 ) α + a 2 − i a 1 2 k a 0 ( a 2 − a 1 ) β , where k is a nonzero constant, α and β are the factors of p such that α β = p and that:

    (2.2) k 2 { ( a 2 + i a 1 ) α ′ + ( 1 + i ) a 0 α } = − { ( a 2 − i a 1 ) β ′ + ( 1 − i ) a 0 β } .

Note that not every polynomial p such that equation (2.1) has nonconstant entire solutions. If p is an irreducible polynomial in C , then we have the following corollary.

Corollary 2.2

Let a 0 , a 1 , and a 2 be the complex numbers with a 0 ≠ 0 and ∣ a 1 ∣ + ∣ a 2 ∣ ≠ 0 , and p be an irreducible polynomial in C . Then, equation (2.1) does not have any nonconstant entire solutions.

We present four concrete examples to illustrate each of the two forms in Theorem 2.1.

Example 1

Consider the equation

( f + 2 f ′ ) 2 + ( f + 2 f ′ ) 2 = p ,

i.e., ( f + 2 f ′ ) 2 = p 2 , where p = 2 z 2 . It is easy to verify that f ( z ) = c e − 1 2 z + z − 2 is an entire solution of this equation, which is the form (i) of that in Theorem 2.1 with φ 2 = p 2 = z 2 , a 0 = 1 , a 1 = a 2 = 2 , and c is an arbitrary complex number.

Example 2

Let p = − i ( z + 1 ) 2 + 3 2 ( z + 1 ) . Then, f ( z ) = 1 − i 4 z + 2 + i 8 is an entire solution of the equation:

( 2 f + 3 f ′ ) 2 + ( 2 f ) 2 = p ,

which is the form (ii) of that in Theorem 2.1 with α = z + 1 , β = − i ( z + 1 ) + 3 2 , k = 1 , a 0 = 2 , a 1 = 3 , and a 2 = 0 . Moreover, it is easy to verify that α and β satisfy (2.2).

Example 3

Consider the equation

f 2 + ( f + 2 f ′ ) 2 = p ,

where p = − i z 2 − 2 z . One can check that f ( z ) = 1 − i 2 z − 1 is an entire solution of this equation, which is the form (ii) of that in Theorem 2.1 with α = z , β = − i z − 2 , k = a 0 = 1 , a 1 = 0 , and a 2 = 2 . Furthermore, an easy computation shows that α and β satisfy (2.2).

Example 4

Let p = − i z 2 + ( i + 1 ) z − 1 . We see at once that f ( z ) = 1 − i 2 z − 1 is an entire solution of this equation:

( f + ( 1 + i ) f ′ ) 2 + ( f + 2 i f ′ ) 2 = p ,

which is the form (ii) of that in Theorem 2.1 with α = z − 1 , β = − i z + 1 , k = a 0 = 1 , a 1 = 1 + i , and a 2 = 2 i . Moreover, one can check that α and β satisfy (2.2).

3 Ordinary differential equations ( a 0 f + a 1 f ′ ) 2 + ( a 0 f + a 2 f ′ ) 2 = e g

In this section, we will deal with entire solutions of the Fermat-type ordinary differential equation:

(3.1) ( a 0 f + a 1 f ′ ) 2 + ( a 0 f + a 2 f ′ ) 2 = e g ,

where a 0 , a 1 , and a 2 are the complex numbers with a 0 ≠ 0 and ∣ a 1 ∣ + ∣ a 2 ∣ ≠ 0 , while g is a polynomial in C .

If a 1 = a 2 , it is easy to solve the solution f ( z ) = e − a 0 a 1 z ( ∫ ± 2 2 e g 2 + a 0 a 1 z d z + c ) of equation (3.1) according to the theory of the first-order linear differential equations, where c is an arbitrary complex number. Therefore, in the following, we only consider the case a 1 ≠ a 2 . First, we consider the case where one of a 1 and a 2 is zero. Without loss of generality, one can assume that a 1 = 0 , then equation (3.1) becomes

(3.2) ( a 0 f ) 2 + ( a 0 f + a 2 f ′ ) 2 = e g .

For the aforementioned equation, we have the following theorem.

Theorem 3.1

Let a 0 and a 2 be the nonzero complex numbers, and g be a polynomial in C . The Fermat-type ordinary differential equation (3.2) has entire solutions if and only if deg g ≤ 1 . Moreover, one can assume that g ( z ) = a z + b , where a and b are the complex numbers, then the entire solutions of equation (3.2) are characterized as follows:

  1. If a = − 2 a 0 a 2 , then f ( z ) = 1 a 0 cos − a 0 a 2 z + d e a z + b 2 or f ( z ) = ± 1 a 0 e a z + b 2 , where d is an arbitrary complex number;

  2. If a ≠ − 2 a 0 a 2 , then f ( z ) = 2 { ( 2 a 0 ) 2 + ( 2 a 0 + a a 2 ) 2 } 1 2 e a z + b 2 .

The following example shows that the forms of the solutions in Theorem 3.1 are precise.

Example 5

Let g ( z ) = − z + 1 . Then, it is easy to verify that f ( z ) = ± e − z + 1 2 and f ( z ) = cos ( − 1 2 z + 1 ) e − z + 1 2 are the entire solutions of the equation:

f 2 + ( f + 2 f ′ ) 2 = e g ,

which is the form (i) of that in Theorem 3.1 with a 0 = 1 , a 2 = 2 , a = − 1 , and b = 1 .

Example 6

One can check that f ( z ) = ± 5 5 e − z + 1 2 is an entire solution of the equation:

f 2 + ( f − f ′ ) 2 = e − 2 z + 1 ,

which is the form (ii) of that in Theorem 3.1 with a 0 = 1 , a 2 = − 1 , a = − 2 , and b = 1 .

Next, for the case a 1 ≠ 0 and a 2 ≠ 0 , then we obtain the following theorem.

Theorem 3.2

Let a 0 , a 1 , and a 2 be the nonzero complex numbers with a 1 ≠ a 2 , and g be a polynomial in C . The Fermat-type ordinary differential equation (3.1) has entire solutions if and only if deg g ≤ 1 . Furthermore, it is assumed that g ( z ) = a z + b , where a and b are the complex numbers, and we summarize all the possible entire solutions of equation (3.1) as follows:

  1. If a = − 2 a 0 ( a 1 + a 2 ) a 1 2 + a 2 2 , then f ( z ) = { a 1 2 + a 2 2 } 1 2 a 0 ( a 2 − a 1 ) e a z + b 2 or

    f ( z ) = 1 a 0 ( a 2 − a 1 ) a 2 cos a 0 ( a 1 − a 2 ) a 1 2 + a 2 2 z + d − a 1 sin a 0 ( a 1 − a 2 ) a 1 2 + a 2 2 z + d e a z + b 2 ,

    where b and d are the arbitrary complex numbers;

  2. If a ≠ − 2 a 0 ( a 1 + a 2 ) a 1 2 + a 2 2 , then f ( z ) = ± a 1 a 0 ( a 2 − a 1 ) e a z + b 2 or f ( z ) = 2 { ( 2 a 0 + a a 1 ) 2 + ( 2 a 0 + a a 2 ) 2 } 1 2 e a z + b 2 .

From Theorems 3.1 and 3.2, we deduced the following corollary directly.

Corollary 3.3

If deg g ≥ 2 , then neither equation (3.1) or (3.2) has any nonconstant entire solutions.

By way of illustration, we give two explicit examples for each of the forms in Theorem 3.2.

Example 7

Let g ( z ) = 6 5 z + 1 . Then, one can check that f ( z ) = ± 5 e 3 5 z + 1 2 and

f ( z ) = e 3 5 z + 1 2 − 2 cos z 5 + 1 + sin z 5 + 1

are the entire solutions of the equation:

( f − f ′ ) 2 + ( f − 2 f ′ ) 2 = e g ,

which is the form (i) of that in Theorem 3.2 with a 0 = 1 , a 1 = − 1 , a 2 = − 2 , a = 6 5 , and b = 1 .

Example 8

Consider the equation

1 2 f + f ′ 2 + 1 2 f + 3 f ′ 2 = e g ,

where g ( z ) = 1 − 3 2 z + 1 . It is easy to verify that f ( z ) = ± ( 3 + 1 ) e 1 − 3 4 z + 1 2 is an entire solution of this equation, which is the form (ii) of that in Theorem 3.2 with a 0 = 1 2 , a 1 = 1 , a 2 = 3 , a = 1 − 3 2 , and b = 1 .

4 Proofs

Proof of Theorem 2.1

The sufficiency is clear. In fact, let f be a function defined as the form (i) in Theorem 2.1. It follows easily that f satisfies equation (2.1). On the other hand, let f be a function defined as the forms (ii) in Theorem 2.1, and one can check that f satisfies equation (2.1) by using (2.2).

To prove the necessity, let f be an entire solution of equation (2.1). If a 1 = a 2 , we now rewrite equation (2.1) as ( a 0 f + a 1 f ′ ) 2 = p 2 . Let φ ≔ a 0 f + a 1 f ′ , which is an entire function in C . We then assert that φ is a polynomial. Otherwise, if φ is a transcendental entire function, we conclude that p is also a transcendental entire function by p = 2 ( a 0 f + a 1 f ′ ) 2 = 2 φ 2 . But this contradicts that p is a nonconstant polynomial. Hence, φ is a polynomial. Furthermore, it is easy to solve the solution f ( z ) = e − a 0 a 1 z ∫ φ a 1 e a 0 a 1 z d z + c according to the theory of the first-order linear differential equations, where c is an arbitrary complex number.

Next, we consider the case a 1 ≠ a 2 . Equation (2.1) can be written as:

{ a 0 f + a 1 f ′ + i ( a 0 f + a 2 f ′ ) } { a 0 f + a 1 f ′ − i ( a 0 f + a 2 f ′ ) } = p .

From the aforementioned equation, we see that there exists a factor α of p (in the ring of polynomials) and an entire function h in C such that:

(4.1) a 0 f + a 1 f ′ + i ( a 0 f + a 2 f ′ ) = α e i h

and

(4.2) a 0 f + a 1 f ′ − i ( a 0 f + a 2 f ′ ) = β e − i h ,

where β ≔ p α is also a polynomial in C . Solving a 0 f + a 1 f ′ and a 0 f + a 2 f ′ from the aforementioned two identities yields that:

(4.3) a 0 f + a 1 f ′ = α e i h + β e − i h 2

and

(4.4) a 0 f + a 2 f ′ = α e i h − β e − i h 2 i .

In view of (4.3) and (4.4), it follows that:

(4.5) f = a 2 + i a 1 2 a 0 ( a 2 − a 1 ) α e i h + a 2 − i a 1 2 a 0 ( a 2 − a 1 ) β e − i h

and

(4.6) f ′ = − 1 − i 2 ( a 2 − a 1 ) α e i h + − 1 + i 2 ( a 2 − a 1 ) β e − i h .

Differentiating (4.5) yields that

(4.7) f ′ = a 2 + i a 1 2 a 0 ( a 2 − a 1 ) ( α ′ e i h + i α h ′ e i h ) + a 2 − i a 1 2 a 0 ( a 2 − a 1 ) ( β ′ e − i h − i β h ′ e − i h ) .

By combining (4.6) and (4.7), we have that:

(4.8) { ( a 2 + i a 1 ) ( α ′ + i α h ′ ) + ( 1 + i ) a 0 α } e i h = − { ( a 2 − i a 1 ) ( β ′ − i β h ′ ) + ( 1 − i ) a 0 β } e − i h .

We next consider two different cases in the following.

Case 1. The entire function h is not a constant.

We claim that

(4.9) ( a 2 + i a 1 ) ( α ′ + i α h ′ ) + ( i + 1 ) a 0 α = 0 .

Otherwise, by (4.8), we obtain that

(4.10) e 2 i h = − ( a 2 − i a 1 ) ( β ′ − i β h ′ ) + ( − i + 1 ) a 0 β ( a 2 + i a 1 ) ( α ′ + i α h ′ ) + ( i + 1 ) a 0 α .

Applying the Nevanlinna theory, we can obtain a contradiction. In fact, if T ( r , F ) denotes the Nevanlinna characteristic function of a meromorphic function F in C , then it follows from (4.10) that:

(4.11) T ( r , e 2 i h ) = O { T ( r , h ) + log r } ,

outside possibly a set of finite Lebesgue measure, using the results (see, e.g., [19]) that T ( r , F ′ ) = O { T ( r , F ) } for any meromorphic function F outside a set of finite Lebesgue measure and that T ( r , g ) = O { log r } for any polynomial g . If h is a nonconstant polynomial, then T ( r , h ) = O { log r } . Thus, it follows from (4.11) that T ( r , e 2 i h ) = O { log r } , which is a contradiction with the fact e 2 i h is a transcendental entire function. If h is a transcendental entire function, then T ( r , e 2 i h ) = O { T ( r , h ) } since lim r → ∞ = T ( r , h ) log r = + ∞ . This is a contradiction with the theorem [20]: If F is a transcendental meromorphic function in C and G is a transcendental entire function in C n , then lim r → ∞ = T ( r , F ( G ) ) T ( r , G ) = + ∞ . Therefore, equation (4.9) is proved.

We then deduced that

(4.12) ( a 2 − i a 1 ) ( β ′ − i β h ′ ) + ( − i + 1 ) a 0 β = 0

by (4.8) and (4.9).

If a 1 = ± i a 2 , we have ( i + 1 ) a 0 α = 0 by (4.9) and ( − i + 1 ) a 0 β = 0 by (4.12). It follows that α = 0 and β = 0 , which is impossible with the assumption that p ≢ 0 . Thus, a 2 ± i a 1 ≠ 0 , i.e., a 1 2 + a 2 2 ≠ 0 .

By combining with (4.9) and (4.12), we obtain

(4.13) ( a 1 2 + a 2 2 ) ( α ′ β + α β ′ ) + 2 a 0 ( a 1 + a 2 ) α β = 0 .

Note that p = α β , then p ′ = α ′ β + α β ′ . Furthermore, (4.13) can be written as:

(4.14) ( a 1 2 + a 2 2 ) p ′ + 2 a 0 ( a 1 + a 2 ) p = 0 ,

which shows that p ( z ) = c e − 2 a 0 ( a 1 + a 2 ) a 1 2 + a 2 2 z according to the theory of the first-order linear differential equations, where c is a nonzero complex number. If a 1 = − a 2 , then p = c is a nonzero complex number, which is a contradiction with the assumption that p is a nonconstant polynomial. If a 1 ≠ − a 2 , which is also a contradiction since the left-hand side of p ( z ) = c e − 2 a 0 ( a 1 + a 2 ) a 1 2 + a 2 2 z is a polynomial but the right-hand side is a transcendental entire function. Therefore, case 1 cannot occur.

Case 2. The entire function h is a constant.

If h is a constant, one can assume that e i h = k , where k is a nonzero complex number. From (4.5), we have

(4.15) f = k ( a 2 + i a 1 ) 2 a 0 ( a 2 − a 1 ) α + a 2 − i a 1 2 k a 0 ( a 2 − a 1 ) β .

In view of (4.8), it follows easily that α and β satisfy that:

(4.16) k 2 { ( a 2 + i a 1 ) α ′ + ( 1 + i ) a 0 α } = − { ( a 2 − i a 1 ) β ′ + ( 1 − i ) a 0 β } .

This completes the proof of Theorem 2.1.□

Proof of Corollary 2.2

Let f be an entire solution of equation (2.1). Note that p is an irreducible polynomial in C , and it may be assumed that p ( z ) = a z + b , where a and b are the complex numbers with a ≠ 0 . Given the proof of Theorem 2.1, then we only need to consider Case 2 in Theorem 2.1.

Applying (4.16) once again, that is,

(4.17) k 2 { ( a 2 + i a 1 ) α ′ + ( 1 + i ) a 0 α } = − { ( a 2 − i a 1 ) β ′ + ( 1 − i ) a 0 β } .

We set α = c , then β = p α = a z + b c , where c is a nonzero complex number. Substituting α and β into (4.17) gives

(4.18) k 2 c 2 a 0 ( 1 + i ) + a ( a 2 − i a 1 ) − b a 0 ( i − 1 ) = a a 0 ( i − 1 ) z ,

which is impossible since the left-hand side of equation (4.18) is a constant but the right-hand side is a nonconstant polynomial.

Therefore, we finish the proof of Corollary 2.2.□

Proof of Theorem 3.1

Sufficiency is readily apparent. In fact, let f be a function defined as the forms (i) and (ii) in Theorem 3.1. It is easy to check that f satisfies equation (3.2). To prove the necessity, let f be an entire solution of equation (3.2). We rewrite equation (3.2) as:

a 0 f + i ( a 0 f + a 2 f ′ ) e g 2 a 0 f − i ( a 0 f + a 2 f ′ ) e g 2 = 1 .

Therefore, there exists an entire function h in C such that:

a 0 f + i ( a 0 f + a 2 f ′ ) e g 2 = e i h

and then that

a 0 f − i ( a 0 f + a 2 f ′ ) e g 2 = e − i h ,

which it follows that:

(4.19) a 0 f = e g 2 e i h + e − i h 2 = e g 2 cos h

and

(4.20) a 0 f + a 2 f ′ = e g 2 e i h − e − i h 2 i = e g 2 sin h

by using the results (cf. [21,22]). Furthermore, we have

(4.21) f = 1 a 0 e g 2 cos h ,

which shows that:

(4.22) f ′ = 1 a 0 e g 2 1 2 g ′ cos h − h ′ sin h

by differentiating both sides of equation (4.21). Substituting f and f ′ into (4.20) shows that

(4.23) 1 + a 2 2 a 0 g ′ cos h = 1 + a 2 a 0 h ′ sin h .

We discuss the following two cases.

Case 1. Suppose first that the entire function h is not a constant.

Similar to the arguments in the proof of Theorem 2.1, we obtain that

(4.24) 1 + a 2 2 a 0 g ′ = 0 a n d 1 + a 0 a 2 h ′ = 0

by virtue of (4.23). From equation (4.24), we see that g ′ = − 2 a 0 a 2 and h ′ = − a 0 a 2 . Obviously, deg g = 1 and deg h = 1 . One can assume that g ( z ) = a z + b , h ( z ) = c z + d , then a = − 2 a 0 a 2 , c = − a 0 a 2 , where b and d are the arbitrary complex numbers. Substituting g and h into (4.21) shows that

f ( z ) = 1 a 0 cos − a 0 a 2 z + d e a z + b 2 .

Case 2. Now suppose that the entire function h is a constant.

If h is a constant, then h ′ = 0 , and (4.23) becomes

(4.25) 1 + a 2 2 a 0 g ′ cos h = sin h .

Obviously, cos h ≠ 0 . Otherwise, sin h = 0 by (4.25), which is impossible since cos h and sin h are not zero at the same time. Thus, we obtain g ′ = 2 a 0 a 2 ( tan h − 1 ) by (4.25), then deg g ≤ 1 . It is assumed that g ( z ) = a z + b with a = 2 a 0 a 2 ( tan h − 1 ) and b is an arbitrary complex number.

If a = − 2 a 0 a 2 , i.e., 2 a 0 a 2 ( tan h − 1 ) = − 2 a 0 a 2 , then tan h = 0 , and then cos h = ± 1 . Substituting g and cos h into (4.21), then we have

f ( z ) = ± 1 a 0 e a z + b 2 .

If a ≠ − 2 a 0 a 2 , then tan h can be expressed in terms of a , a 0 , and a 2 , that is, tan h = a a 2 + 2 a 0 2 a 0 , and then

cos h = 2 a 0 { ( 2 a 0 ) 2 + ( 2 a 0 + a a 2 ) 2 } 1 2 .

Substituting g and cos h into (4.21), one then has

f ( z ) = 2 { ( 2 a 0 ) 2 + ( 2 a 0 + a a 2 ) 2 } 1 2 e a z + b 2 .

Thus, the proof of Theorem 3.1 is completed.□

Proof of Theorem 3.2

The sufficiency is easy to check, which is left as an exercise for the interested reader. To prove the necessity, let f be an entire solution of equation (3.1). We rewrite equation (3.1) as:

a 0 f + a 1 f ′ + i ( a 0 f + a 2 f ′ ) e g 2 a 0 f + a 1 f ′ − i ( a 0 f + a 2 f ′ ) e g 2 = 1 .

Similar to the arguments in the proof of Theorem 3.1, we see that there exists an entire function h in C such that

(4.26) a 0 f + a 1 f ′ = e g 2 cos h

and

(4.27) a 0 f + a 2 f ′ = e g 2 sin h .

Solving f and f ′ from (4.26) and (4.27) yields that

(4.28) f = ( a 2 cos h − a 1 sinh ) e g 2 a 0 ( a 2 − a 1 )

and

(4.29) f ′ = ( sin h − cos h ) e g 2 ( a 2 − a 1 ) .

By differentiating (4.28), we have

(4.30) f ′ = 1 a 0 ( a 2 − a 1 ) 1 2 ( a 2 cos h − a 1 sin h ) g ′ + ( − a 2 sin h − a 1 cos h ) h ′ e g 2 .

By combining with (4.29) and (4.30), it follows that:

(4.31) 1 2 a 2 g ′ − a 1 h ′ + a 0 cos h = 1 2 a 1 g ′ + a 2 h ′ + a 0 sin h .

Next, we consider two different cases of the entire function h in the following.

Case 1. The entire function h is not a constant.

Similar to the arguments in the proof of Theorem 2.1, we can deduce that

(4.32) 1 2 a 2 g ′ − a 1 h ′ + a 0 = 0

and

(4.33) 1 2 a 1 g ′ + a 2 h ′ + a 0 = 0

by virtue of (4.31).

In view of (4.32) and (4.33), it follows that

(4.34) 1 2 g ′ ( a 1 2 + a 2 2 ) + a 0 ( a 1 + a 2 ) = 0

and

(4.35) h ′ ( a 1 2 + a 2 2 ) − a 0 ( a 1 − a 2 ) = 0 .

Obviously, a 1 2 + a 2 2 ≠ 0 . Otherwise, a 1 2 + a 2 2 = 0 , i.e., a 1 = ± i a 2 . In view of (4.34) and (4.35), then we deduce that a 1 = a 2 . This is a contradiction with assumption that a 1 ≠ a 2 . Therefore, a 1 2 + a 2 2 ≠ 0 . Furthermore, one can deduce

(4.36) g ′ = − 2 a 0 ( a 1 + a 2 ) a 1 2 + a 2 2 , and h ′ = a 0 ( a 1 − a 2 ) a 1 2 + a 2 2

by (4.34) and (4.35). Obviously, deg g ≤ 1 and deg h = 1 . It may be assumed that g ( z ) = a z + b , h ( z ) = c z + d , then a = − 2 a 0 ( a 1 + a 2 ) a 1 2 + a 2 2 , c = a 0 ( a 1 − a 2 ) a 1 2 + a 2 2 and b and d are the arbitrary complex numbers. Substituting g and h into (4.28) shows that:

(4.37) f ( z ) = 1 a 0 ( a 2 − a 1 ) a 2 cos a 0 ( a 1 − a 2 ) a 1 2 + a 2 2 z + d − a 1 sin a 0 ( a 1 − a 2 ) a 1 2 + a 2 2 z + d e a z + b 2 .

Case 2. The entire function h is a constant.

From (4.31), we have

(4.38) ( a 2 cos h − a 1 sin h ) g ′ = 2 a 0 ( sin h − cos h ) .

We can claim that a 2 cos h − a 1 sin h ≠ 0 by (4.38). Otherwise, if a 2 cos h − a 1 sin h = 0 , then sin h − cos h = 0 by (4.38). Furthermore, a 2 cos h − a 1 sin h = ( a 2 − a 1 ) sin h = 0 , then a 1 = a 2 , which is a contradiction with the assumption that a 1 ≠ a 2 . Thus, a 2 cos h − a 1 sin h ≠ 0 ; it follows that g ′ = 2 a 0 ( sin h − cos h ) a 2 cos h − a 1 sin h . Note that h is a constant, then deg g ≤ 1 . We can assume that g ( z ) = a z + b , then a = 2 a 0 ( sin h − cos h ) a 2 cos h − a 1 sin h and b is an arbitrary complex number.

Next, we consider two different cases.

Case 2.1: a = − 2 a 0 ( a 1 + a 2 ) a 1 2 + a 2 2 .

If a = − 2 a 0 ( a 1 + a 2 ) a 1 2 + a 2 2 , then 2 a 0 ( sin h − cos h ) a 2 cos h − a 1 sin h = − 2 a 0 ( a 1 + a 2 ) a 1 2 + a 2 2 . An easy computation shows that:

(4.39) 1 − a 1 a 2 cos h = 1 − a 2 a 1 sin h .

Obviously, we have cos h ≠ 0 by (4.39). Then, we obtain tan h = − a 1 a 2 . Furthermore, cos h = a 2 { a 1 2 + a 2 2 } 1 2 and sin h = − a 1 { a 1 2 + a 2 2 } 1 2 . Substituting g , cos h , and sin h into (4.28) shows that:

f ( z ) = { a 1 2 + a 2 2 } 1 2 a 0 ( a 2 − a 1 ) e a z + b 2 .

Case 2.2: a ≠ − 2 a 0 ( a 1 + a 2 ) a 1 2 + a 2 2 .

In this case, by a = 2 a 0 ( sin h − cos h ) a 2 cos h − a 1 sin h , then we obtain

(4.40) ( 2 a 0 + a a 2 ) cos h = ( 2 a 0 + a a 1 ) sin h .

If cos h = 0 , then a = − 2 a 0 a 1 , sin h = ± 1 by (4.40). Moreover, it is easy to check that − 2 a 0 a 1 ≠ − 2 a 0 ( a 1 + a 2 ) a 1 2 + a 2 2 . Substituting g , cos h , and sin h into (4.28), one then has

f ( z ) = ± a 1 a 0 ( a 2 − a 1 ) e a z + b 2 .

If cos h ≠ 0 , then ( 2 a 0 + a a 1 ) tan h = 2 a 0 + a a 2 by (4.40). We claim that 2 a 0 + a a 1 ≠ 0 . Otherwise, if 2 a 0 + a a 1 = 0 , then 2 a 0 + a a 2 = 0 , which is a contradiction with the assumption a 1 ≠ a 2 . Therefore, we have that tan h = 2 a 0 + a a 2 2 a 0 + a a 1 , and then,

cos h = 2 a 0 + a a 1 { ( 2 a 0 + a a 1 ) 2 + ( 2 a 0 + a a 2 ) 2 } 1 2 and sin h = 2 a 0 + a a 2 { ( 2 a 0 + a a 1 ) 2 + ( 2 a 0 + a a 2 ) 2 } 1 2 .

Substituting g , cos h , and sin h into (4.28) shows that:

f ( z ) = 2 { ( 2 a 0 + a a 1 ) 2 + ( 2 a 0 + a a 2 ) 2 } 1 2 e a z + b 2 .

Therefore, this completes the proof of Theorem 3.2.□

Acknowledgment

The authors thank the referee(s) for reading the manuscript very carefully and making several valuable and kind comments that improved the presentation.

  1. Funding information: This research was supported by NNSF of China (No. 11701006) and by Natural Science Research Project for Colleges and Universities of Anhui Province (Nos 2022AH050329 and 2022AH050290).

  2. Author contributions: Conceptualization, L. Yang; writing-original draft preparation, B. b. Hu and L. Yang; writing-review and editing, L. Yang and B. b. Hu; funding acquisition, L. Yang.

  3. Conflict of interest: The authors declare no competing interests.

  4. Data availability statement: No data were used to support this study.

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Received: 2022-10-25
Revised: 2023-08-27
Accepted: 2023-08-27
Published Online: 2023-10-11

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2023-0120
Keywords for this article
entire solution; ordinary differential equation; polynomial
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  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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