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

- Qi Jianming
^{1, 3}, - Li Yezhou
^{2}and - Yuan Wenjun
^{3}Email author

**2012**:6

https://doi.org/10.1186/1687-1847-2012-6

© Jianming et al; licensee Springer. 2012

**Received:**2 July 2011**Accepted:**1 February 2012**Published:**1 February 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.

## Keywords

- Meromorphic functions
- Nevanlinna theory
- Normal family
- Growth order
- Algebraic differential equation

## 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]).

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.

*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

*M*

_{{0}}[

*w*] = 1. We call

*p*(

*r*) :=

*r*

_{1}+ 2

*r*

_{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 $\lambda =2\le 1+\frac{3}{2\times 2-1}=2$. 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*

*where H*(*w*(*z*)) = *w*^{(k)}(*z*) + *b*_{k-1}*w*^{(k-1)}(*z*) + *b*_{k-2}*w*^{(k-2)}(*z*) + ⋯ + *b*_{1}*w*(*z*) + *b*_{0}*and b*_{k-1},..., *b*_{0}*are constants*.

where *m*_{1}, *m*_{2} are the non-negative 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*$\nu =\text{deg}{\left(a\left(z\right)\right)}^{{m}_{2}}$.

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*

_{k-1}

*w*

^{(k-1)}(

*z*)+

*b*

_{k-2}

*w*

^{(k-2)}(

*z*)+⋯+

*b*

_{0}

*and b*

_{k-1}, ...,

*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 up-to-date 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^{+};

*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, ${g}^{\#}\left(\xi \right)=\frac{\left|{g}^{\prime}\left(\xi \right)\right|}{1+{\left|g\left(\xi \right)\right|}^{2}}$ 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

## 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

*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

*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:

*ρ*<

*λ*- 1, we have

*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

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*

_{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

*w*(

*z*) = (

*w*

_{1}(

*z*),

*w*

_{2}(

*z*)) which satisfies the system of equations (1.4) such that

*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

*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:

*ρ*<

*λ*- 1, we have

*z*by

*b*

_{ m }+

*ρ*

_{ m }

*ζ*, then we obtain

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 $\lambda \left({w}_{2}\right)\le 1+\frac{a+{\text{deg}}_{z,\infty}a}{{m}_{1}{m}_{2}\phantom{\rule{0.3em}{0ex}}qk-\text{deg}P\left({w}_{2}\right)}$.

The proof of Theorem 1.9 is complete.

## Declarations

### 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.

## Authors’ Affiliations

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