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Further results of the estimate of growth of entire solutions of some classes of algebraic differential equations
Advances in Difference Equations volume 2012, Article number: 6 (2012)
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 \mathcal{F} be a family of meromorphic functions defined in D. \mathcal{F} is said to be normal in D, in the sense of Montel, if each sequence \left\{{f}_{n}\right\}\subset \mathcal{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
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 }∈ ℕ_{0} = ℕ ∪ {0} for j = 1, 2,..., m, and put r = (r_{1}, r_{2},..., r_{ m }). Define M_{ r }[w](z) by
with the convention that M_{{0}}[w] = 1. We call p(r) := r_{1} + 2r_{2} + ⋯ + mr_{ m }the weight of M_{ r }[w]. A differential polynomial P[w] is an expression of the form
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∈l}p(r). deg_{z,∞}a_{ r }denotes the degree at infinity in variable z concerning a_{ r }(z, w). deg_{z,∞}a := max_{r∈l}max{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
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
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
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
we know deg_{z,∞}a = 3, deg P[w] = 2, So \lambda =2\le 1+\frac{3}{2\times 21}=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
where H(w(z)) = w^{(k)}(z) + b_{k1}w^{(k1)}(z) + b_{k2}w^{(k2)}(z) + ⋯ + b_{1}w(z) + b_{0}and b_{k1},..., b_{0}are constants.
In 2009, Gu et al. [11] investigated the growth order of solutions of a type systems of algebraic differential equations of the form
where m_{1}, m_{2} are the nonnegative integer, a(z) is a polynomial, P[w_{2}] is defined by (1.2).
They obtained the following result.
Theorem 1.7. Let w = (w_{1}, w_{2}) be the meromorphic solution vector of a type systems of algebraic differential equations of the form (1.3), if m_{1}m_{2} > deg P(w_{2}), then the growth orders λ(w_{ i }) of w_{ i }(z) for i = 1,2 satisfy
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
where Q(z) is a polynomial with degree q.
They obtained the following result.
Theorem 1.8. Let w = (w_{1}, w_{2}) be a meromorphic solution of a type systems of algebraic differential equations of the form (1.4), if m_{1}m_{2}qk > deg P(w_{2}), and all zeros of w_{2}(z) have multiplicity at least k (k ∈ ℕ), then the growth orders λ(w_{ i }) of w_{ i }(z) for i = 1,2 satisfy
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 = (w_{1}, w_{2}) be an entire solution of a type systems of algebraic differential equations of the form (1.4), if m_{1}m_{2}qk > deg P(w_{2}), and all zeros of w_{2}(z) have multiplicity at least k (k ∈ ℕ), then the growth orders λ(w_{ i }) of w_{ i }(z) for i = 1, 2 satisfy
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 = (w_{1}, w_{2}) be an entire solution of a type systems of algebraic differential equations of the form
where H(w(z)) = w^{(k)}(z)+b_{k1}w^{(k1)}(z)+b_{k2}w^{(k2)}(z)+⋯+b_{0}and b_{k1}, ..., b_{0}are constants. If m_{1}m_{2}qk > deg P(w_{2}), and all zeros of w_{2}(z) have multiplicity at least k (k ∈ ℕ), then the growth orders λ(w_{ i }) of w_{ i }(z) for i = 1, 2 satisfy
where\nu =\text{deg}{\left(a\left(z\right)\right)}^{{m}_{2}}.
Example 2 Set w_{1}(z) = e^{z}+ c, w_{2}(z) = e^{z}satisfy a type systems of algebraic differential equations of the form
where c is a constant, m_{1} = 1, m_{2} = 5, ν = 0, deg_{z,∞}a = 0, and deg P(w_{2}) = 2. The (1.6) satisfies the m_{1}m_{2} = 5 > 2 = deg P(w_{2}). So λ(w_{1}) = λ(w_{2}) = 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 uptodate local version, which is due to Pang and Zaclman [13].
Lemma 2.1[13, 14] Let \mathcal{F} be a family of meromorphic (analytic) functions in the unit disc Δ with the property that for each f\in \mathcal{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 \mathcal{F} and f = 0. If \mathcal{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 \mathcal{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 nonconstant 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
Here, as usual, {g}^{\#}\left(\xi \right)=\frac{\left{g}^{\prime}\left(\xi \right)\right}{1+{\leftg\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
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
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∈ℕ}is not normal at z = 0. By Lemma 2.1, there exist sequences {b_{ m }} and {ρ_{ m }} such that
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
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
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
where
Hence we deduce that
Therefore
Because 0\le \rho =\frac{{\text{deg}}_{z,\infty}{a}_{r}}{nqkp\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
Therefore we have
If w_{2} is a rational function, then w_{1} must be a rational function, so that the conclusion of Theorem 2 is right. If w_{2} is a transcendental meromorphic function, by the systems of algebraic differential equations (1.3), then we have
Suppose that the conclusion of Theorem 2 is not true, then there exists an entire vector w(z) = (w_{1}(z),w_{2}(z)) which satisfies the system of equations (1.4) such that
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∈ℕ}is not normal at z = 0. By Lemma 2.1, there exist sequences {b_{ m }} and {ρ_{ m }} such that
and g_{ m }(ζ) := w_{2,m}(b_{ m } a_{ m }+ ρ_{ m }ζ) = w_{2}(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
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
In the differential equation (3.6) we now replace z by b_{ m }+ ρ_{ m }ζ, then we obtain
where
Namely
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}qkp\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}\phantom{\rule{0.3em}{0ex}}qk\text{deg}P\left({w}_{2}\right)}.
The proof of Theorem 1.9 is complete.
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Acknowledgements
The authors wish to thank the referees and editor for their very helpful comments and useful suggestions. This study was partially supported by Leading Academic Discipline Project (10XKJ01) and Key Development Project (12C104) of Shanghai Dianji University also was partially supported by NSFC of China (11101048 and 10771220), Doctorial Point Fund of National Education Ministry of China (200810780002). The MS ID is 6597357865695822.
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JQ carried out the main part of this manuscript. YL and WY participated discussion and corrected the main theorem. All authors read and approved the final manuscript.
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Jianming, Q., Yezhou, L. & Wenjun, Y. Further results of the estimate of growth of entire solutions of some classes of algebraic differential equations. Adv Differ Equ 2012, 6 (2012). https://doi.org/10.1186/1687184720126
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DOI: https://doi.org/10.1186/1687184720126