Unicity of Meromorphic Solutions of the Pielou Logistic Equation

<jats:p>This paper mainly considers the unicity of meromorphic solutions of the Pielou logistic equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>y</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>z</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>R</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mi>y</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mo>/</mml:mo><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>Q</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>P</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mi>y</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>P</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mi>Q</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>R</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>z</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math> are nonzero polynomials. It shows that the finite order transcendental meromorphic solution of the Pielou logistic equation is mainly determined by its poles and 1-value points. Examples are given for the sharpness of our result.</jats:p>


Introduction
For a meromorphic function f(z), we use standard notations of the Nevanlinna theory, such as T(r, f), m(r, f), and N(r, f) (see, e.g., [1][2][3]). Let S(r, f) denote any quantity that satisfies S(r, f) � o(T(r, f)) as r ⟶ ∞ possibly outside of an exceptional set of finite logarithmic measure. And we define the order of growth of f(z) by Also we know that the unicity of solutions of a given equation is always one of its most essential properties. is paper is to discuss the unicity of meromorphic solutions of the Pielou logistic equation where P(z), Q(z), and R(z) are nonzero polynomials. Equation (2) is an important equation generalized from the famous Verhulst-Pearl equation, which is the most popular continuous model of growth of a population: x ′ (t) � x(t)[a − bx(t)], a, b > 0.
By denoting f(z) � 1/y(z), we can get from (2) that which is a linear difference equation. On the growth, zeros, and poles of meromorphic solutions of (2) and (4), Chen proved numbers of significant results in [4]. en, Cui and Chen [5,6] began to consider the unicity of meromorphic solutions concerning their zeros, 1-value points, and poles and proved.
Theorem 1 (see [5]). Let f(z) be a finite order transcendental meromorphic solution of the equation where P 1 (z) and P 2 (z) are nonzero polynomials such that Theorem 2 (see [6]). Let f(z) be a finite order transcendental meromorphic solution of the equation where P 1 (z), P 2 (z), and P 3 (z) are nonzero polynomials such that P 1 (z) + P 2 (z) ≡ 0. If a meromorphic function g(z) shares 0, 1, ∞ CM with f(z), then one of the following cases holds: ere exists a polynomial β(z) � az + b 0 and a constant a 0 satisfying e a 0 ≠ e b 0 such that where a 0 ≠ 0, b 0 are constants.
Here and in the following, f(z) and g(z) are said to share the value a CM (IM), provided that f(z) − a and g(z) − a have the same zeros counting multiplicities (ignoring multiplicities). And f(z) and g(z) are said to share the value ∞ CM (IM), provided that f(z) and g(z) have the same poles with the same multiplicities (ignoring multiplicities).
Cui and Chen's work is a natural product of generalization work (see, e.g., [1,3,[7][8][9][10][11]) on the famous Nevanlinna's 5 IM (4 CM) eorem (see, e.g., [3,12]) during the past, about 90 years, especially of the hot research studies on the complex differences and complex difference equations (see, e.g., [1,4,[8][9][10][13][14][15]) recently. ey have given examples to show that all cases of eorem A and eorem B can happen, and the numbers of shared values cannot be reduced. Li and Chen [16] turned to consider the following question: What can we say about the unicity of finite order transcendental meromorphic solutions of the equation where R 1 (z) ≡ 0, R 2 (z), R 3 (z) are rational functions? And we proved some interesting results and also provided some examples for sharpness of them. Two of those results read are as follows.

Remark 1.
Notice that f(z) and g(z) share 0 CM if and only if 1/f(z) and 1/g(z) share ∞ CM; f(z) and g(z) share ∞ CM if and only if 1/f(z) and 1/g(z) share 0 CM; and f(z) and g(z) share 1 CM if and only if 1/f(z) and 1/g(z) share 1 CM. As a result, for the unicity of finite order transcendental meromorphic solutions equation (2), we only need to consider the case that two CM shared values are 1, ∞. Indeed, we prove the following eorem 5, whose proof is different from that in [5,6,16].
Theorem 5. Let x(z) and y(z) be two finite order transcendental meromorphic solutions of equation (2). If x(z) and y(z) share 1, ∞ CM and one of the following cases holds: is not an integer, and x(z) has at most finitely many simple poles, then x(z) ≡ y(z) We give some examples for the sharpness of eorem 5 as follows.
(2) x(z) � 1/(e πiz + 1) and y(z) � 1/(e − πiz + 1) satisfy the equation Here, x(z) and y(z) share 1, ∞ CM such that they have infinitely many simple poles and ρ(x) � ρ(y) � 1 and is example shows that eorem 5 may not hold for the case R(z) ≡ P(z) if most (except finitely many) poles of x(z) are simple or ρ(x) is an integer.

Remark 2.
It is interesting to ask a question: whether the shared condition "CM" is replaced by "IM" in eorem 5. We have tried hard but failed to find some negative examples for this question. We conjecture that the conclusions in eorem 5 still hold when the shared condition "CM" is replaced by "IM."

Proof of Theorem 5
To prove eorem 5, we need the following lemma of Clunie (see, e.g., [1,2]). Lemma 1 (see [1,2]). Let f(z) be a transcendental meromorphic solution of the equation where P(z, f) and Q(z, f) are polynomials in f and its derivatives with meromorphic coefficients, say a λ | λ ∈ I , such that m(r, a λ ) � S(r, f) for all λ ∈ I. If the total degree of Q(z, f) as a polynomial in f and its derivatives is ≤ n, then m(r, P(z, f)) � S(r, f).
Proof of eorem 5. Since x(z) and y(z) are finite order transcendental solutions of equation (2) and share 1, ∞ CM, without loss of generality, assume that ρ(x) ≥ ρ(y), and we get where

h(z) is a polynomial such that deg h(z) ≤ ρ(x), and A(z) � Q(z)/R(z), B(z) � P(z)/R(z) are rational functions.
If e h ≡ 1, then our conclusion holds. If e h ≡ 1, then e h ≡ 1, and from (17), we have Here and in the following, we use the notations for any given meromorphic function f(z) for convenience. Submitting (18) into (16), we have where From (15) and (20), we obtain or equally, (23) Next, we discuss three cases.
Case 2. R(z) ≡ P(z) and x(z) has infinitely many poles of multiplicity ≥ 2. From (23), we have where en, D(z) and E(z) are rational functions and hence have at most finitely many poles. Choose a pole of x(z) with multiplicity k 1 ≥ 1, denoted by z 3 , such that D(z 3 ) ≠ ∞, E(z 3 ) ≠ ∞. en, z 3 is a pole of x 2 (z) with multiplicity 2k 1 and a pole of D(z)x(z) + E(z) with multiplicity k 1 . However, from (35), we see that it is impossible.
we see that e h − 1 has at most n zeros of multiplicity ≥ 2. en, D(z) and E(z) are meromorphic functions which have at most finitely many poles of multiplicity ≥ 2. Choose a pole of x(z) with multiplicity k 2 ≥ 2 denoted by z 4 such that z 4 is not a pole of D(z), E(z) of multiplicity ≥ 2. en, z 4 is a pole of x 2 (z) with multiplicity 2k 2 ≥ 4 and a pole of D(z)x(z) + E(z) with multiplicity at most k 2 + 1. However, from (35) and k 2 + 1 < 2k 2 , we find that it is also impossible.

Case 3. R(z) ≡ P(z),ρ(x)
is not an integer, and x(z) has at most finitely many simple poles. en, deg h(z) < ρ(x) since deg h(z) ≤ ρ(x). From Case 2, we can suppose that x(z) has at most finitely many poles of multiplicity ≥ 2 and use (35) directly. Now, x(z) has at most finitely many poles.
On the one hand, we have m(r, x) � T(r, x) − N(r, x) � T(r, x) + S(r, x).

Conclusion
Our result shows that the finite order transcendental meromorphic solution of equation (2) is mainly determined by its poles and 1-value points.

Data Availability
e data used to support the findings of this study are included within the article.