Theory and Modern Applications

# 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 ).

Let be the whole complex domain. Let D be a domain in 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 ).

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 . 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  and Barsegian  gave some extensions or different proofs, but the results have not changed. Barsegian  and Bergweiler  have extended Gol'dberg's result to certain algebraic differential equations of higher order. In 2009, Yuan et al. , 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 = {1, 2, 3,...}, r j 0 = {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] := maxrlp(r). degz,∞a r denotes the degree at infinity in variable z concerning a r (z, w). degz,∞a := maxrlmax{degz,∞a r , 0}.

Theorem 1.2. Let w(z) be a meromorphic function in the complex plane, 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.  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 ), P[w] be a polynomial with the form (1.2) and nkq > deg P[w] (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 ), P[w] be a polynomial with the form (1.2) and nkq > deg P[w] (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

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

we know degz,∞a = 3, deg P[w] = 2, So $\lambda =2\le 1+\frac{3}{2×2-1}=2$. 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 ), P[w] be a differential polynomial with constant coefficients in variable w or degz,∞a t ≤ 0(t I) in the (1.2) and nkq > deg P[w] (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 ), P[w] be a polynomial with the form (1.2) and nk > deg P[w] (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) + bk-1w(k-1)(z) + bk-2w(k-2)(z) + + b1w(z) + b0and bk-1,..., b0are constants.

In 2009, Gu et al.  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.  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 ), 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 ), 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)+bk-1w(k-1)(z)+bk-2w(k-2)(z)++b0and bk-1, ..., b0are constants. If m1m2qk > deg P(w2), and all zeros of w2(z) have multiplicity at least k (k ), 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) = ez+ c, w2(z) = ezsatisfy 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, degz,∞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  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 .

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 ≤ αk, there exist

1. 1.

a number r (0,1);

2. 2.

a sequence of complex numbers z n , |z n | < r;

3. 3.

a sequence of functions ${f}_{n}\in ℱ$;

4. 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 , 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

${\rho }_{n}\le \frac{2}{{\left[{f}_{n}^{#}\left({w}_{n}\right)\right]}^{\frac{1}{1+\left|\alpha \right|}}},\phantom{\rule{1em}{0ex}}{f}_{n}^{#}\left({w}_{n}\right)\ge {f}_{n}^{#}\left(0\right).$

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 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)}mis not normal at z = 0. By Lemma 2.1, there exist sequences {b m } and {ρ m } such that

$\left|{a}_{m}-{b}_{m}\right|<1,\phantom{\rule{1em}{0ex}}{\rho }_{m}\to 0,$
(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

${\rho }_{m}\le \frac{2}{{w}^{#}\left({b}_{m}\right)},\phantom{\rule{1em}{0ex}}{w}^{#}\left({b}_{m}\right)\ge {w}^{#}\left({a}_{m}\right).$
(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

Hence we deduce that

$\begin{array}{c}{\rho }_{m}^{-nqk}{\left[{\left({g}_{m}^{\left(k\right)}\right)}^{q}\left(\zeta \right)+{\rho }_{m}^{k}{a}_{q-1}{\left({g}_{m}^{\left(k\right)}\right)}^{q-1}\left(\zeta \right)+\cdots +{\rho }_{m}^{qk}{a}_{0}\right]}^{n}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}=\sum _{r\in I}{a}_{r}\left({b}_{m}+{\rho }_{m}\zeta ,{g}_{m}\left(\zeta \right)\right){\rho }_{m}^{-p\left(r\right)}{M}_{r}\left[{g}_{m}\right]\left(\zeta \right).\end{array}$

Therefore

$\begin{array}{c}{\left[{\left({g}_{m}^{\left(k\right)}\right)}^{q}\left(\zeta \right)+{\rho }_{m}^{k}{a}_{q-1}{\left({g}_{m}^{\left(k\right)}\right)}^{q-1}\left(\zeta \right)+\cdots +{\rho }_{m}^{qk}{a}_{0}\right]}^{n}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}={\sum }_{r\in I}\frac{{a}_{r}\left({b}_{m}+{p}_{m}\zeta ,{g}_{m}\left(\zeta \right)\right)}{{b}_{m}^{{\text{deg}}_{z,\infty }{a}_{r}}}{\left[{b}_{m}^{\frac{{\text{deg}}_{z,\infty }{a}_{r}}{nqk-p\left(r\right)}}{\rho }_{m}\right]}^{nqk-p\left(r\right)}{M}_{r}\left[{g}_{m}\right]\left(\zeta \right).\end{array}$
(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), for every fixed ζ , 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)}mis not normal at z = 0. By Lemma 2.1, there exist sequences {b m } and {ρ m } such that

$\left|{a}_{m}-{b}_{m}\right|<1,\phantom{\rule{1em}{0ex}}{\rho }_{m}\to 0,$
(3.8)

and g m (ζ) := w2,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

${\rho }_{m}\le \frac{2}{{w}_{2}^{#}\left({b}_{m}\right)},\phantom{\rule{1em}{0ex}}{w}_{2}^{#}\left({b}_{m}\right)\ge {w}_{2}^{#}\left({a}_{m}\right).$
(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

$\begin{array}{c}{\left(Q\left({w}_{2}^{\left(k\right)}\left({b}_{m}+{\rho }_{m}\zeta \right)\right)\right)}^{{m}_{1}{m}_{2}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}=\sum _{r\in I}a{\left({b}_{m}+{\rho }_{m}\zeta \right)}^{{m}_{2}}{a}_{r}\left({b}_{m}+{\rho }_{m}\zeta ,{g}_{m}\left(\zeta \right)\right){\rho }_{m}^{-p\left(r\right)}{M}_{r}\left[{g}_{m}\right]\left(\zeta \right).\end{array}$

where

Namely

$\begin{array}{c}{\left[{\left({g}_{m}^{\left(k\right)}\right)}^{q}\left(\zeta \right)+{\rho }_{m}^{k}{a}_{q-1}{\left({g}_{m}^{\left(k\right)}\right)}^{q-1}\left(\zeta \right)+\cdots +{\rho }_{m}^{qk}{a}_{1}{g}_{m}^{\left(k\right)}\left(\zeta \right)\right]}^{{m}_{1}{m}_{2}}\\ \phantom{\rule{1em}{0ex}}=\sum _{r\in I}\frac{a{\left({b}_{m}+{\rho }_{m}\zeta \right)}^{{m}_{2}}{a}_{r}\left({b}_{m}+{\rho }_{m}\zeta ,{g}_{m}\left(\zeta \right)\right)}{{b}_{m}^{a+{\text{deg}}_{z,\infty }{a}_{r}}}{\left\{{b}_{m}^{\frac{a+{\text{deg}}_{z,\infty }{a}_{r}}{{m}_{1}\phantom{\rule{0.3em}{0ex}}{m}_{2\phantom{\rule{0.3em}{0ex}}}qk-p\left(r\right)}}{\rho }_{m}\right\}}^{{m}_{1}{m}_{2}qk-p\left(r\right)}{M}_{r}\left[{g}_{m}\right]\left(\zeta \right).\end{array}$
(3.11)

For every fixed ζ , 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.

## References

1. Hayman WK: Meromorphic Functions. Clarendon Press, Oxford; 1964.

2. He YZ, Xiao XZ: Algebroid Functions and Ordinary Differential Equations. Science Press, Beijing; 1988.

3. Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.

4. Gol'dberg AA: On single-valued solutions of algebraic differential equations. Ukrain Mat Zh 1956, 8: 254–261.

5. Bank S, Kaufman R: On meromorphic solutions of first-order differential equations. Comment Math Helv 1976, 51: 289–299. 10.1007/BF02568158

6. Barsegian G: Estimates of derivatives of meromorphic functions on sets of α -points. J Lond Math Soc 1986, 34(2):534–540.

7. Barsegian G: On a method of study of algebraic differential equations. Bull Hong Kong Math Soc 1998, 2(1):159–164.

8. Bergweiler W: On a theorem of Gol'dberg concerning meromorphic solutions of algebraic differential equations. Complex Var 1998, 37: 93–96. 10.1080/17476939808815124

9. Yuan WJ, Xiao B, Zhang JJ: The general result of Gol'dberg's theorem concerning the growth of meromorphic solutions of algebraic differential equations. Comput Math Appl 2009, 58: 1788–1791. 10.1016/j.camwa.2009.07.092

10. Qi JM, Li YZ, Yuan WJ: Further results of Gol'dberg's theorem concerning the growth of meromorphic solutions of algebraic differential equations. Acta Math Sci (in press, in Chinese)

11. Gu RM, Ding JJ, Yuan WJ: On the estimate of growth order of solutions of a class of systems of algebraic differential equations with higher orders. J Zhanjiang Normal Univ (in Chinese) 2009, 30(6):39–43.

12. Zalcman L: A heuristic principle in complex function theory. Am Math Monthly 1975, 82: 813–817. 10.2307/2319796

13. Pang XC, Zalcman L: Normal families and shared values. Bull Lond Math Soc 2000, 32: 325–331. 10.1112/S002460939900644X

14. Zalcman L: Normal families new perspectives. Bull Am Math Soc 1998, 35: 215–230. 10.1090/S0273-0979-98-00755-1

15. Gu RM, Li ZR, Yuan WJ: The growth of entire solutions of some algebraic differential equations. Georgian Math J 2011, 18(3):489–495.

## 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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Correspondence to Yuan Wenjun.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

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/10.1186/1687-1847-2012-6

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

### Keywords

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