Further results of the estimate of growth of entire solutions of some classes of algebraic differential equations

Abstract

In this article, by means of the normal family theory we estimate the growth order of entire solutions of some algebraic differential equations and improve the related results of Bergweiler, Barsegian, and others. We also estimate the growth order of entire solutions of a type system of a special algebraic differential equations. We give some examples to show that our results are sharp in special cases.

Mathematica Subject Classification (2000): Primary 34A20; Secondary 30D35.

1. Introduction and main results

Let f(z) be a meromorphic function in the complex plane. We use the standard notation of the Nevanlinna theory of meromorphic functions and denotes the order of f(z) by λ(f) (see [1–3]).

Let ℂ be the whole complex domain. Let D be a domain in ℂ and $ℱ$ be a family of meromorphic functions defined in D. $ℱ$ is said to be normal in D, in the sense of Montel, if each sequence $\left\{f_n\right\}\subset ℱ$ has a subsequence $\left\{f_{n_j}\right\}$ which converse spherically locally uniformly in D, to a meromorphic function or ∞ (see [1]).

In general, it is not easy to have an estimate on the growth of an entire or meromorphic solution of a nonlinear algebraic differential equation of the form

$$P\left(z,w,w^{\prime },\dots ,w^{\left(k\right)}\right)=0,$$

(1.1)

where P is a polynomial in each of its variables.

A general result was obtained by Gol'dberg [4]. He obtained

Theorem 1.1. All meromorphic solutions of algebraic differential equation ( 1 .1) have finite order of growth, when k = 1.

For a half century Bank and Kaufman [5] and Barsegian [6] gave some extensions or different proofs, but the results have not changed. Barsegian [7] and Bergweiler [8] have extended Gol'dberg's result to certain algebraic differential equations of higher order. In 2009, Yuan et al. [9], improved their results and gave a general estimate of order of w(z), which depends on the degrees of coefficients of differential polynomial for w(z). In order to state these results, we must introduce some notations: m ∈ ℕ = {1, 2, 3,...}, r _j ∈ ℕ₀ = ℕ ∪ {0} for j = 1, 2,..., m, and put r = (r₁, r₂,..., r _m ). Define M _r [w](z) by

$$M_r\left[w\right]\left(z\right):={\left[w^{\prime }\left(z\right)\right]}^{r_1}{\left[w^{″}\left(z\right)\right]}^{r_2}\cdots {\left[w^{\left(m\right)}\left(z\right)\right]}^{r_m},$$

with the convention that M_{0}[w] = 1. We call p(r) := r₁ + 2r₂ + ⋯ + mr _m the weight of M _r [w]. A differential polynomial P[w] is an expression of the form

$$P\left[w\right]\left(z\right):=\sum _{r\in I}{a}_r\left(z,w\left(z\right)\right)M_r\left[w\right]$$

(1.2)

where the a _r are rational in two variables and I is a finite index set. The weight deg P[w] of P[w] is given by deg P[w] := max_r∈lp(r). deg_z,∞a _r denotes the degree at infinity in variable z concerning a _r (z, w). deg_z,∞a := max_r∈lmax{deg_z,∞a _r , 0}.

Theorem 1.2. [9]Let w(z) be a meromorphic function in the complex plane, n ∈ ℕ, P[w] be a polynomial with the form (1.2) n > deg P[w]. If w(z) satisfies the differential equation [w'(z)]ⁿ= P[w], then the growth order λ := λ(w) of w(z) satisfies

$$\lambda \le 2+\frac{2{\text{deg}}_{z,\infty }a}{n-\text{deg}P\left[w\right]}.$$

Recently Qi et al. [10] further improved Theorem 1.2 as below.

Theorem 1.3. Let w(z) be a meromorphic function in the complex plane and all zeros of w(z) have multiplicity at least k (k ∈ ℕ), P[w] be a polynomial with the form (1.2) and nkq > deg P[w] (n ∈ ℕ). If w(z) satisfies the differential equation [Q(w^(k)(z))]ⁿ= P[w], then the growth order λ := λ(w) of w(z) satisfies

$$\lambda \le 2+\frac{2{\text{deg}}_{z,\infty }a}{nqk-\text{deg}P\left[w\right]},$$

where Q(z) is a polynomial with degree q.

In this article, we first give a small upper bound for entire solutions.

Theorem 1.4. Let w(z) be an entire function in the complex plane and all zeros of w(z) have multiplicity at least k (k ∈ ℕ), P[w] be a polynomial with the form (1.2) and nkq > deg P[w] (n ∈ ℕ). If w(z) satisfies the differential equation [Q(w^(k)(z))]ⁿ= P[w], then the growth order λ := λ(w) of w(z) satisfies

$$\lambda \le 1+\frac{{\text{deg}}_{z,\infty }a}{nqk-\text{deg}P\left[w\right]},$$

where Q(z) is a polynomial with degree q.

Example 1 For n = 2, entire function $w\left(z\right)=e^{z^2}$ satisfies the following algebraic differential equation

$${\left(w^{″}\right)}^2=4w^2+16z^2{w}^2+8z^3{w}^{\prime }w,$$

we know deg_z,∞a = 3, deg P[w] = 2, So $\lambda =2\le 1+\frac{3}{2\times 2-1}=2$. This example illustrates that Theorem 1.4 is an extending result of Theorem 1.3 and our result is sharp in the special cases.

By Theorem 1.4, we immediately have the following corollaries.

Corollary 1.5. Let w(z) be an entire function in the complex plane and all zeros of w(z) have multiplicity at least k (k ∈ ℕ), P[w] be a differential polynomial with constant coefficients in variable w or deg_z,∞a _t ≤ 0(t ∈ I) in the (1.2) and nkq > deg P[w] (n ∈ ℕ). If w(z) satisfies the differential equation [Q(w^(k)(z))]ⁿ= P[w], then the growth order λ := λ(w) of w(z) satisfies λ ≤ 1, where Q(z) is a polynomial with degree q.

Corollary 1.6. Let w(z) be an entire function in the complex plane and all zeros of w(z) have multiplicity at least k (k ∈ ℕ), P[w] be a polynomial with the form (1.2) and nk > deg P[w] (n ∈ ℕ). If w(z) satisfies the differential equation [H(w(z))]ⁿ= P[w], then the growth order λ := λ(w) of w(z) satisfies

$$\lambda \le 1+\frac{{\text{deg}}_{z,\infty }a}{nk-\text{deg}P\left[w\right]},$$

where H(w(z)) = w^(k)(z) + b_k-1w^(k-1)(z) + b_k-2w^(k-2)(z) + ⋯ + b₁w(z) + b₀and b_k-1,..., b₀are constants.

In 2009, Gu et al. [11] investigated the growth order of solutions of a type systems of algebraic differential equations of the form

$$\left\{\begin{array}{ccc}\hfill {\left({w^{\prime }}_2\right)}^{m_1}\hfill & \hfill =\hfill & \hfill a\left(z\right)w_1^{\left(n\right)},\hfill \\ \hfill {\left(w_1^{\left(n\right)}\right)}^{m_2}\hfill & \hfill =\hfill & \hfill P\left[w_2\right]\hfill \end{array}\right.$$

(1.3)

where m₁, m₂ are the non-negative integer, a(z) is a polynomial, P[w₂] is defined by (1.2).

They obtained the following result.

Theorem 1.7. Let w = (w₁, w₂) be the meromorphic solution vector of a type systems of algebraic differential equations of the form (1.3), if m₁m₂ > deg P(w₂), then the growth orders λ(w _i ) of w _i (z) for i = 1,2 satisfy

$$\lambda \left(w_1\right)=\lambda \left(w_2\right)\le 2+\frac{2\left(\nu +{\text{deg}}_{z,\infty }a\right)}{m_1{m}_2-\text{deg}P\left(w_2\right)}$$

where$\nu =\text{deg}{\left(a\left(z\right)\right)}^{m_2}$.

Qi et al. [10] also consider the similar result to Theorem 1.7 for the systems of the algebraic differential equations

$$\left\{\begin{array}{ccc}\hfill {\left(Q\left(w_2^{\left(k\right)}\left(z\right)\right)\right)}^{m_1}\hfill & \hfill =\hfill & \hfill a\left(z\right)w_1^{\left(n\right)}\hfill \\ \hfill {\left(w_1^{\left(n\right)}\right)}^{m_2}\hfill & \hfill =\hfill & \hfill P\left(w_2\right),\hfill \end{array}\right.$$

(1.4)

where Q(z) is a polynomial with degree q.

They obtained the following result.

Theorem 1.8. Let w = (w₁, w₂) be a meromorphic solution of a type systems of algebraic differential equations of the form (1.4), if m₁m₂qk > deg P(w₂), and all zeros of w₂(z) have multiplicity at least k (k ∈ ℕ), then the growth orders λ(w _i ) of w _i (z) for i = 1,2 satisfy

$$\lambda \left(w_1\right)=\lambda \left(w_2\right)\le 2+\frac{2\left(\nu +{\text{deg}}_{z,\infty }a\right)}{m_1{m}_{2}qk-\text{deg}P\left(w_2\right)},$$

where$\nu =\text{deg}{\left(a\left(z\right)\right)}^{m_2}$.

Similarly we have a small upper bounded estimate for entire solutions below.

Theorem 1.9. Let w = (w₁, w₂) be an entire solution of a type systems of algebraic differential equations of the form (1.4), if m₁m₂qk > deg P(w₂), and all zeros of w₂(z) have multiplicity at least k (k ∈ ℕ), then the growth orders λ(w _i ) of w _i (z) for i = 1, 2 satisfy

$$\lambda \left(w_1\right)=\lambda \left(w_2\right)\le 1+\frac{\nu +{\text{deg}}_{z,\infty }a}{m_1{m}_{2}qk-\text{deg}P\left(w_2\right)},$$

where $\nu =\text{deg}{\left(a\left(z\right)\right)}^{m_2}$.

By Theorem 1.9, we immediately obtain a corollary below.

Corollary 1.10. Let w = (w₁, w₂) be an entire solution of a type systems of algebraic differential equations of the form

$$\left\{\begin{array}{c}\hfill {\left(H\left(w_2\right)\right)}^{m_1}=a\left(z\right)w_1^{\left(n\right)}\hfill \\ \hfill {\left(w_1^{\left(n\right)}\right)}^{m_2}=p\left(w_2\right),\hfill \end{array}\right.$$

(1.5)

where H(w(z)) = w^(k)(z)+b_k-1w^(k-1)(z)+b_k-2w^(k-2)(z)+⋯+b₀and b_k-1, ..., b₀are constants. If m₁m₂qk > deg P(w₂), and all zeros of w₂(z) have multiplicity at least k (k ∈ ℕ), then the growth orders λ(w _i ) of w _i (z) for i = 1, 2 satisfy

$$\lambda \left(w_1\right)=\lambda \left(w_2\right)\le 1+\frac{\nu +{\text{deg}}_{z,\infty }a}{m_1{m}_{2}qk-\text{deg}P\left(W_2\right)},$$

where$\nu =\text{deg}{\left(a\left(z\right)\right)}^{m_2}$.

Example 2 Set w₁(z) = e^z+ c, w₂(z) = e^zsatisfy a type systems of algebraic differential equations of the form

$$\left\{\begin{array}{c}\hfill \left(w_2^{\left(k\right)}\right)=w_1^{\left(n\right)}\hfill \\ \hfill {\left(w_1^{\left(n\right)}\right)}^5={\left(w_2\right)}^3{\left({w^{\prime }}_2\right)}^2\hfill \end{array},\right.$$

(1.6)

where c is a constant, m₁ = 1, m₂ = 5, ν = 0, deg_z,∞a = 0, and deg P(w₂) = 2. The (1.6) satisfies the m₁m₂ = 5 > 2 = deg P(w₂). So λ(w₁) = λ(w₂) = 1 ≤ 1. So the conclusion of Theorem 1.9, Corollary 1.10 may occur and our results are sharp in the special cases.

2. Preliminary lemmas

In order to prove our result, we need the following lemmas. The first one extends a famous result by Zalcman [12] concerning normal families. Zalcman's lemma is a very important tool in the study of normal families. It has also undergone various extensions and improvements. The following is one up-to-date local version, which is due to Pang and Zaclman [13].

Lemma 2.1[13, 14] Let $ℱ$ be a family of meromorphic (analytic) functions in the unit disc Δ with the property that for each $f\in ℱ$, all zeros of multiplicity at least k. Suppose that there exists a number A ≥ 1 such that |f^(k)(z)| ≤ A whenever $f\in ℱ$ and f = 0. If $ℱ$ is not normal in Δ, then for 0 ≤ α ≤ k, there exist

• 1. a number r ∈ (0,1);

• 2. a sequence of complex numbers z _n , |z _n | < r;

• 3. a sequence of functions $f_n\in ℱ$;

• 4. a sequence of positive numbers ρ _n → 0⁺;

such that $g_n\left(\xi \right)={\rho }_n^{-\alpha }{f}_n\left(z_n+{\rho }_n\xi \right)$ converges locally uniformly (with respect to the spherical metric) to a non-constant meromorphic (entire) function g(ξ) on ℂ, and moreover, the zeros of g(ξ) are of multiplicity at least k, g^#(ξ) ≤ g^#(0) = kA + 1. In particular, g has order at most 2. In particular, we may choose w _n and ρ _n , such that

$${\rho }_n\le \frac{2}{{\left[f_n^{\#}\left(w_n\right)\right]}^{\frac{1}{1+\left|\alpha \right|}}},f_n^{\#}\left(w_n\right)\ge f_n^{\#}\left(0\right).$$

Here, as usual, $g^{\#}\left(\xi \right)=\frac{\left|g^{\prime }\left(\xi \right)\right|}{1+{\left|g\left(\xi \right)\right|}^2}$ is the spherical derivative. For 0 ≤ α < k, the hypothesis on f^(k)(z) can be dropped, and kA + 1 can be replaced by an arbitrary positive constant.

Lemma 2.2[15] Let f(z) be holomorphic in whole complex plane with growth order λ := λ(f) > 1, then for each 0 < μ < λ - 1, there exists a sequence a _n → ∞, such that

$$\underset{n\to \infty }{\text{lim}}\frac{f^{\#}\left(a_n\right)}{{\left|a_n\right|}^{\mu }}=+\infty .$$

(2.1)

3. Proof of the results

Proof of Theorem 1.4 Suppose that the conclusion of theorem is not true, then there exists an entire solution w(z) satisfies the equation [Q(w(z))]ⁿ= P[w]. such that

$$\lambda >1+\frac{{\text{deg}}_{z,\infty }a}{nqk-\text{deg}P\left[w\right]}.$$

(3.1)

By Lemma 2.2 we know that for each 0 < ρ < λ - 1, there exists a sequence of points a _m → ∞(m → ∞), such that (2.1) is right. This implies that the family {w _m (z) := w(a _m + z)}_m∈ℕis not normal at z = 0. By Lemma 2.1, there exist sequences {b _m } and {ρ _m } such that

$$\left|a_m-b_m\right|<1,{\rho }_m\to 0,$$

(3.2)

and g _m (ζ) := w _m (b _m - a _m + ρ _m ζ) = w(b _m + ρ _m ζ) converges locally uniformly to a nonconstant entire function g(ζ), which order is at most 2, all zeros of g(ζ ) have multiplicity at least k. In particular, we may choose b _m and ρ _m , such that

$${\rho }_m\le \frac{2}{w^{\#}\left(b_m\right)},w^{\#}\left(b_m\right)\ge w^{\#}\left(a_m\right).$$

(3.3)

According to (2.1) and (3.1)-(3.3), we can get the following conclusion:

For any fixed constant 0 ≤ ρ < λ - 1, we have

$$\underset{m\to \infty }{\text{lim}}{b}_m^{\rho }{\rho }_m=0.$$

(3.4)

In the differential equation [Q(w^(k)(z))]ⁿ= P[w(z)], we now replace z by b _m + ρ _m ζ. Assuming that P[w] has the form (1.2). Then we obtain

$${\left(Q\left(w^{\left(k\right)}\left(b_m+{\rho }_m\zeta \right)\right)\right)}^n=\sum _{r\in I}{a}_r\left(b_m+{\rho }_m\zeta ,g_m\left(\zeta \right)\right){\rho }_m^{-p\left(r\right)}{M}_r\left[g_m\right]\left(\zeta \right),$$

where

$$\begin{array}{c}\hfill Q\left(w^{\left(k\right)}\left(b_m+{\rho }_m\zeta \right)\right)={\rho }_m^{-qk}\left[{\left(g_m^{\left(k\right)}\right)}^q\left(\zeta \right)+\right.{\rho }_m^k{a}_{q-1}{\left(g_m^{\left(k\right)}\right)}^{q-1}\left(\zeta \right)+\\ \hfill \left.+\cdots +{\rho }_m^{\left(q-1\right)k}{a}_1{g}_m^{\left(k\right)}\left(\zeta \right)+{\rho }_m^{qk}{a}_0\right].\end{array}$$

Hence we deduce that

$$\begin{array}{c}{\rho }_m^{-nqk}{\left[{\left(g_m^{\left(k\right)}\right)}^q\left(\zeta \right)+{\rho }_m^k{a}_{q-1}{\left(g_m^{\left(k\right)}\right)}^{q-1}\left(\zeta \right)+\cdots +{\rho }_m^{qk}{a}_0\right]}^n\\ =\sum _{r\in I}{a}_r\left(b_m+{\rho }_m\zeta ,g_m\left(\zeta \right)\right){\rho }_m^{-p\left(r\right)}{M}_r\left[g_m\right]\left(\zeta \right).\end{array}$$

Therefore

$$\begin{array}{c}{\left[{\left(g_m^{\left(k\right)}\right)}^q\left(\zeta \right)+{\rho }_m^k{a}_{q-1}{\left(g_m^{\left(k\right)}\right)}^{q-1}\left(\zeta \right)+\cdots +{\rho }_m^{qk}{a}_0\right]}^n\\ ={\sum }_{r\in I}\frac{a_r\left(b_m+p_m\zeta ,g_m\left(\zeta \right)\right)}{b_m^{{\text{deg}}_{z,\infty }{a}_r}}{\left[b_m^{\frac{{\text{deg}}_{z,\infty }{a}_r}{nqk-p\left(r\right)}}{\rho }_m\right]}^{nqk-p\left(r\right)}{M}_r\left[g_m\right]\left(\zeta \right).\end{array}$$

(3.5)

Because $0\le \rho =\frac{{\text{deg}}_{z,\infty }{a}_r}{nqk-p\left(r\right)}\le \frac{{\text{deg}}_{z,\infty }a}{nqk-\text{deg}P\left[w\right]}<\lambda -1,p\left(r\right)<nqk$, for every fixed ζ ∈ ℂ, if ζ is not the zero of g(ζ), by (3.4) then we can get g^(k)(ζ) = 0 from (3.5). By the all zeros of g(ζ) have multiplicity at least k, this is a contradiction.

The proof of Theorem 1.4 is complete.

Proof of Theorem 1.9 By the first equation of the systems of algebraic differential equations (1.4), we know

$$w_1^{\left(n\right)}=\frac{{\left(Q\left(w_2^{\left(k\right)}\left(z\right)\right)\right)}^{m_1}}{a\left(z\right)}.$$

Therefore we have

$$\lambda \left(w_1\right)=\lambda \left(w_2\right).$$

If w₂ is a rational function, then w₁ must be a rational function, so that the conclusion of Theorem 2 is right. If w₂ is a transcendental meromorphic function, by the systems of algebraic differential equations (1.3), then we have

$${\left(Q\left(w_2^{\left(k\right)}\right)\right)}^{m_1{m}_2}={\left(a\left(z\right)\right)}^{m_2}P\left(w_2\right).$$

(3.6)

Suppose that the conclusion of Theorem 2 is not true, then there exists an entire vector w(z) = (w₁(z),w₂(z)) which satisfies the system of equations (1.4) such that

$$\lambda :=\lambda \left(w_2\right)>1+\frac{\nu +{\text{deg}}_{z,\infty }a}{m_1{m}_{2}qk-\text{deg}P\left(w_2\right)},$$

(3.7)

By Lemma 2.2 we know that for each 0 < ρ < λ - 1, there exists a sequence of points a _m → ∞ (m → ∞), such that (2.1) is right. This implies that the family {w _m (z) := w(a _m + z)}_m∈ℕis not normal at z = 0. By Lemma 2.1, there exist sequences {b _m } and {ρ _m } such that

$$\left|a_m-b_m\right|<1,{\rho }_m\to 0,$$

(3.8)

and g _m (ζ) := w_2,m(b _m - a _m + ρ _m ζ) = w₂(b _m + ρ _m ζ) converges locally uniformly to a nonconstant entire function g(ζ), which order is at most 2, all zeros of g(ζ) have multiplicity at least k. In particular, we may choose b _m and ρ _m , such that

$${\rho }_m\le \frac{2}{w_2^{\#}\left(b_m\right)},w_2^{\#}\left(b_m\right)\ge w_2^{\#}\left(a_m\right).$$

(3.9)

According to (3.6) and (3.7)-(3.9), we can get the following conclusion:

For any fixed constant 0 ≤ ρ < λ - 1, we have

$$\underset{m\to \infty }{\text{lim}}{b}_m^{\rho }{\rho }_m=0.$$

(3.10)

In the differential equation (3.6) we now replace z by b _m + ρ _m ζ, then we obtain

$$\begin{array}{c}{\left(Q\left(w_2^{\left(k\right)}\left(b_m+{\rho }_m\zeta \right)\right)\right)}^{m_1{m}_2}\\ =\sum _{r\in I}a{\left(b_m+{\rho }_m\zeta \right)}^{m_2}{a}_r\left(b_m+{\rho }_m\zeta ,g_m\left(\zeta \right)\right){\rho }_m^{-p\left(r\right)}{M}_r\left[g_m\right]\left(\zeta \right).\end{array}$$

where

$$\begin{array}{c}Q\left(w_2^{\left(k\right)}\left(b_m+{\rho }_m\zeta \right)\right)={\rho }_m^{-qk}\left[{\left(g_m^{\left(k\right)}\right)}^q\left(\zeta \right)+\right.{\rho }_m^k{a}_{q-1}{\left(g_m^{\left(k\right)}\right)}^{q-1}\left(\zeta \right)+\\ \left.+\cdots +{\rho }_m^{qk}{a}_1{g}_m^{\left(k\right)}\left(\zeta \right)\right].\end{array}$$

Namely

$$\begin{array}{c}{\left[{\left(g_m^{\left(k\right)}\right)}^q\left(\zeta \right)+{\rho }_m^k{a}_{q-1}{\left(g_m^{\left(k\right)}\right)}^{q-1}\left(\zeta \right)+\cdots +{\rho }_m^{qk}{a}_1{g}_m^{\left(k\right)}\left(\zeta \right)\right]}^{m_1{m}_2}\\ =\sum _{r\in I}\frac{a{\left(b_m+{\rho }_m\zeta \right)}^{m_2}{a}_r\left(b_m+{\rho }_m\zeta ,g_m\left(\zeta \right)\right)}{b_m^{a+{\text{deg}}_{z,\infty }{a}_r}}{\left\{b_m^{\frac{a+{\text{deg}}_{z,\infty }{a}_r}{m_1{m}_{2}qk-p\left(r\right)}}{\rho }_m\right\}}^{m_1{m}_{2}qk-p\left(r\right)}{M}_r\left[g_m\right]\left(\zeta \right).\end{array}$$

(3.11)

For every fixed ζ ∈ ℂ, if ζ is not zero of g(ζ), for m → ∞ and $0\le \rho =\frac{a+{\text{deg}}_{z,\infty }{a}_r}{m_1{m}_{2}qk-p\left(r\right)}\le \frac{a+{\text{deg}}_{z,\infty }a}{m_1{m}_{2}qk-\text{deg}P\left(w_2\right)}<\lambda -1$ then we have ${\left(g^{\left(k\right)}\right)}^{m_1{m}_2}=0$, which contradicts with all zeros of g(ζ) have multiplicity at least k. So $\lambda \left(w_2\right)\le 1+\frac{a+{\text{deg}}_{z,\infty }a}{m_1{m}_{2}qk-\text{deg}P\left(w_2\right)}$.

The proof of Theorem 1.9 is complete.