Let $f(z)$ be a function meromorphic in the complex plane $\mathbb{C}$. We assume that the reader is familiar with the standard notations and results in Nevanlinna's value distribution theory of meromorphic functions (see e.g. [1-3]). We denote the order of $f(z)$ by $\rho(f)$.

As one knows, it was one of the important topics to research the algebraic differential equation of Malmquist type. In 1913, Malmquist [4] gave a result for the first order algebraic differential equations. In 1933, Yosida [5] proved the Malmquist's theorem by using the Nevanlinna theory. In 1970s, Laine [6], Yang [7] and Hille [8] gave a generalization of Malmquist's theorem. Later, Steinmetz [9], Rieth [10] and He-Laine 11] all gave corresponding generalizations of Malmquist's theorem for the first order algebraic differential equations. In 1980, Gackstatter and Laine [12] gave a generalized result of Malmquist's theorem for some certain type of higher order algebraic differential equations. However, Malmquist type theorem for an arbitrary second order algebraic differential equation remains open. For a second order algebraic differential equation

$\label{a-eq6} f''=R(z, f, f'), $

(1.1)

where $R$ is a rational function in $z, f$ and $f', $ a classical and unsolved conjecture is the following.

Conjecture 1.1  (see [3]) If equation (1.1) has a transcendental meromorphic solution, then the equation can be reduced into the form

$ \label{a-eq7} f''=L_{2}(z, f)(f')^{2}+L_{1}(z, f)f'+L_{0}(z, f), $

(1.2)

where $L_{i}(z, f)\ (i=0, 1, 2)$ are rational functions in their variables.

In 2011, Gao, Zhang and Li [13] studied the problem of growth order of solutions of a type of non-linear algebraic differential equations. In 2001, Liao and Yang [14] considered the finite order of growth of the meromorphic solutions of equation (1.2) and obtained the following result.

Theorem A  Let $f$ be a meromorphic solution of equation (1.2). Further assume that $L_{2}(z, f) \not\equiv 0$ in equation (1.2) and has the form

$ L_{2}(z, f)=\frac{P(z, f)}{Q(z, f)} = \frac{a_{n}(z)f^{n}+a_{n-1}(z)f^{n-1}+ \cdots +a_{s}(z)f^{s}}{b_{m}(z)f^{m}+b_{m-1}(z)f^{m-1}+ \cdots +b_{r}(z)f^{r}}, $

where $a_{i}(z), b_{j}(z)\ (s \leq i \leq n, r \leq j \leq m)$ are rational functions. If $m-n < 1 $ or $r-s>1$, then $\rho(f) < \infty$.

Remark  The conditions $m-n < 1 $ and $r-s>1$ in Theorem A cannot be omitted simultaneously. Liao and Yang [14] gave a simple example to show it.

The paper is organized into 3 sections. After introduction some basic concepts and lemmas will be given in Section 2. In Section 3, we will give the main results.

Let $D$ be a domain in $\mathbb{C}$. We say that a family $\cal F$ of meromorphic functions in $D$ is normal, if each sequence $\{f_n\}\subset \cal F$ contains a subsequence which converges locally uniformly by spherical distance to a meromorphic function $g(z)$ in $D$ ($g(z)$ is permitted to be identically infinity). In this paper, we denote the spherical derivative of meromorphic function $f(z)$ by $f^{\sharp}(z), $ where

$ f^{\sharp}(z):=\frac{|f'(z)|}{1+|f(z)|^2} $

and define

$ A(r, f)=\frac{1}{\pi}\displaystyle\int\int_{|z|\leq r}[f^{\sharp}(z)]^2 dxdy. $

For convenience, we still assume that $L_{2}(z, f)=\frac{P(z, f)}{Q(z, f)}$ and rewrite equation (1.2) into

$ \label{a-eq8} Q(z, f)f''= P(z, f)(f')^{2} +M(z, f)f'+N(z, f), $

(2.1)

where $M(z, f)=Q(z, f)L_{1}(z, f), $ $N(z, f)=Q(z, f)L_{0}(z, f), $ $P, Q$ are defined as in Theorem A.

Let $H(z)=\frac{p(z)}{q(z)}$ be a rational function, where $p(z)$ and $q(z)$ are irreducible polynomials in $z$. Define the degree at infinity of $H(z)$ by

$ \deg_{z, \infty} (H) = \deg p(z) - \deg q(z). $

We denote the largest number of the degrees at infinity of all the rational function coefficients in variable $z$ concerning $L(z, f)$ by $\deg_{z, \infty} L(z, f)$. Denoting

$\deg_{z, \infty} a = \max \{\deg_{z, \infty}P(z, f), \deg_{z, \infty}Q(z, f), \deg_{z, \infty}M(z, f), \deg_{z, \infty}N(z, f), 0\}, $

where $P(z, f), Q(z, f)$ are two polynomials in $f$ with rational function coefficients, $M(z, f)$ and $N(z, f)$ are rational functions in variable $z$ and $f$.

The following lemmas will be needed in the proof of our results. Lemma 2.1 is a result of Zalcman concerning normal families.

Lemma 2.1  (see [15]) Let $\cal F$ be a family of meromorphic functions on the unit disc, $\alpha$ is a real number. Then $\cal F$ is not normal on the unit disc if and only if there exist, for each $-1<\alpha<1$,

a) $a$ number $r$, $0<r<1;$

b) $a$ sequence points $\{w_k\}, $ $|w_k|<r;$

c) $a$ sequence $\{f_k \}_{k\in N}\subset \cal F;$

d) $a$ positive sequence $\{\rho_k\}, $ $\rho_k\rightarrow 0$

such that $g_k(\zeta):=\rho_k^{\alpha}f_k(w_k+\rho_k\zeta)$ converges locally uniformly to a nonconstant meromorphic function $g(\zeta)$. In particular, we may choose $w_k$ and $\rho_k$ properly such that

$ \rho_k\leq \frac{2}{[f_k^{\sharp}(w_k)]^{\frac{1}{1+|\alpha|}}}, \ f_k^{\sharp}(w_k)\geq f_k^{\sharp}(0). $

The next lemma is a generalization of the Lemma 2 in [16] of Yuan et al.

Lemma 2.2  Let $f(z)$ be meromorphic in the complex plane, $\rho:=\rho (f)>2$, then for any positive constants $ \varepsilon > 0 $ and $ 0<\lambda<(\frac{\rho -2}{2})\varepsilon, $ there exist points $z_k\rightarrow \infty(k\rightarrow \infty), $ such that

$\mathop {\lim }\limits_{k \to \infty } \frac{{{{({f^\sharp }({z_k}))}^\varepsilon }}}{{|{z_k}{|^\lambda }}} = + \infty .$

Proof   Suppose that the conclusion of Lemma 2.2 is not true, then there exist a positive number $M>0$, such that for arbitrary $z\in \mathbb{C}$, we have

$ \label{a-eq10} (f^{\sharp}(z))^{\varepsilon}\leq M|z|^{\lambda}. $

By (2.2) we can get

$ \begin{array}{rl} \displaystyle A(t, f)=\frac{1}{\pi}\displaystyle\int\int_{|z|\leq t}[f^{\sharp}(z)]^2dxdy \leq\frac{M^\frac{2}{\varepsilon}}{\pi}\displaystyle\int\int_{|z|\leq t}|z|^\frac{2\lambda}{\varepsilon}dxdy =O(|t|^{2+\frac{2\lambda}{\varepsilon}}). \end{array} $

Thus we obtain an estimation of Ahlfors-Shimizu characteristic function

$ \begin{array}{rl} \displaystyle T_{0}(r, f)=\displaystyle\int^r_0\frac{A(t, f)}{t}dt \leq O(|r|^{2+\frac{2\lambda}{\varepsilon}}). \end{array} $

Therefore, the order of $f(z)$ can be estimated as $\rho \leq 2+\frac{2\lambda}{\varepsilon}$, namely, $\lambda \geq (\frac{\rho -2}{2})\varepsilon$. This is a contradiction with the choice of $\lambda$.

Lemma 2.3  (see [17]) Let $f(z)$ be holomorphic in the complex plane, $\sigma > -1$. If $ f^{\sharp}(z)=O(r^{\sigma}), $ then $ T(r, f)=O(r^{\sigma+1}). $

The result of Lemma 2.4 is more sharper than Lemma 2.2 when $f(z)$ is an entire function.

Lemma 2.4  Let $f(z)$ be holomorphic in the complex plane, $\rho:=\rho (f)>1$, then for any positive constants $ \varepsilon > 0 $ and $ 0<\lambda<(\rho -1)\varepsilon, $ there exist points $z_k\rightarrow \infty, $ as $k\rightarrow \infty, $ such that

$ \lim\limits_{k\rightarrow \infty}\frac{(f^{\sharp}(z_k))^{\varepsilon}}{|z_k|^{\lambda}}=+\infty. $

Proof  Suppose that the conclusion of Lemma 2.4 is not true, then there exist a positive number $M>0$, such that for arbitrary $z\in \mathbb{C}$, we have $ (f^{\sharp}(z))^{\varepsilon}\leq M|z|^{\lambda}, $ namely, $ f^{\sharp}(z)=O(r^{\frac{\lambda}{\varepsilon}}). $ By Lemma 2.3, we have

$ T(r, f)=O(r^{\frac{\lambda}{\varepsilon}+1}). $

Therefore the order of $f(z)$ can be estimated as $\rho \leq 1+\frac{\lambda}{\varepsilon}$, namely, $\lambda \geq (\rho -1)\varepsilon$. This is a contradiction with the choice of $\lambda$.

We are now giving our main results as follows.

Theorem 3.1   Let $f$ be a meromorphic solution of equation (2.1). Further assume that $\frac{P(z, f)}{Q(z, f)}\not\equiv 0$ in equation (2.1), $M(z, f)\not\equiv 0, N(z, f)\not\equiv 0$ are birational functions and have following forms

$M(z, f) = \frac{c_{q_{1}}(z)f^{q_{1}}+\cdots + c_{t_{1}}(z)f^{t_{1}}}{d_{q_{2}}(z)f^{q_{2}}+\cdots+d_{t_{2}}(z)f^{t_{2}}}, \quad N(z, f) = \frac{e_{q_{3}}(z)f^{q_{3}}+\cdots + e_{t_{3}}(z)f^{t_{3}}}{u_{q_{4}}(z)f^{q_{4}}+\cdots+u_{t_{4}}(z)f^{t_{4}}}, $

where $c_{j_{1}}(z)\ (t_{1}\leq j_{1}\leq q_{1}), d_{j_{2}}(z)\ (t_{2}\leq j_{2}\leq q_{2}), e_{j_{3}}(z)\ (t_{3}\leq j_{3}\leq q_{3})$ and $u_{j_{4}}(z)\ (t_{4}\leq j_{4}\leq q_{4})$ are rational functions, $c_{t_{1}}(z) \not\equiv 0$, $d_{t_{2}}(z)\not\equiv 0$, $e_{t_{3}}(z)\not\equiv 0$ and $u_{t_{4}}(z)\not\equiv 0, $ then

$\rho(f) \leq 2+ \frac{2(1+\alpha)\deg_{z, \infty} a}{\alpha}, $

where $0 < \alpha < \min \{\frac{1}{|q_{1}-q_{2}-n|}, \frac{2}{|q_{3}-q_{4}-n-1|}, 1\}$ when $m-n <1 $ and $0 < \alpha < \min \{\frac{1}{|1+s+t_{2}-t_{1}|}, $ $\frac{2}{|2+s+t_{4}-t_{3}|}, 1\}$ when $r-s > 1$.

Proof   We assume that $f$ is a meromorphic solution of equation (2.1). Next we discuss into two cases.

Case 1  $m-n<1$. We choose $\alpha$ such that $0 < \alpha < \min \{\frac{1}{|q_{1}-q_{2}-n|}, $ $\frac{2}{|q_{3}-q_{4}-n-1|}, 1\}$ and assume that

$\rho(f) > 2+ \frac{2(1+\alpha)\deg_{z, \infty} a}{\alpha}.$

By Lemma 2.2 we know that for $\frac{\alpha}{1+\alpha}$ and $ 0<\lambda<(\frac{\rho -2}{2})\frac{\alpha}{1+\alpha}, $ there exist points $z_k\rightarrow \infty, $ as $k\rightarrow \infty, $ such that

$ \label{a-eq33} \lim\limits_{k\rightarrow \infty}\frac{(f^{\sharp}(z_k))^{\frac{\alpha}{1+\alpha}}}{|z_k|^{\lambda}}=+\infty. $

(3.1)

This implies that the family $\{f(z_k+z)\}_{k\in\mathrm{N}}$ is not normal at $z=0$. Then by Lemma 2.1, there exist a sequence $\{\beta_{k}\}$ and a positive sequence $\{\rho_k\}$ such that

$ \label{a-eq34} |z_k-\beta_k|<1, \quad \rho_k\rightarrow 0, $

(3.2)

and $g_k(\zeta):=\rho_{k}^{\alpha}f(\beta_k+\rho_k\zeta)$ converges locally uniformly to a nonconstant meromorphic function $g(\zeta)$. In particular, we may choose $\beta_k$ and $\rho_k$, such that

$ \label{a-eq35} \rho_k\leq \frac{2}{f^{\sharp}(\beta_k)^{\frac{1}{1+\alpha}}}, \ f^{\sharp}(\beta_k)\geq f^{\sharp}(z_k). $

(3.3)

According to (3.1), (3.2) and (3.3), we can get the following conclusion.

For positive constant $\alpha$ and any constant $0\leq \lambda<(\frac{\rho -2}{2})\frac{\alpha}{1+\alpha}$, we have

$ \label{a-eq36} \lim\limits_{k\rightarrow \infty}\beta_k^{\lambda}\rho_k^\alpha=0. $

(3.4)

Substituting $\beta_{k}+\rho_{k}\zeta$ for $z$ in (2.1), we have

$\quad \quad Q (\beta_{k}+\rho_{k}\zeta, \frac{g_{k}(\zeta)}{\rho_{k}^\alpha})\frac{g_{k}''(\zeta)}{\rho_{k}^{2+\alpha}}\\ =P(\beta_{k}+\rho_{k}\zeta, \frac{g_{k}(\zeta)}{\rho_{k}^\alpha})\frac{(g_{k}'(\zeta))^2}{\rho_{k}^{2(1+\alpha)}} +M(\beta_{k}+\rho_{k}\zeta, \frac{g_{k}(\zeta)}{\rho_{k}^\alpha})\frac{g_{k}'(\zeta)}{\rho_{k}^{1+\alpha}} +N(\beta_{k}+\rho_{k}\zeta, \frac{g_{k}(\zeta)}{\rho_{k}^\alpha}).$

Noting $0\leq \deg_{z, \infty} a <(\frac{\rho -2}{2})\frac{\alpha}{1+\alpha}$, by (3.4), we have

$ \label{a-eq37}\quad \quad b_{m}(\beta_{k}+\rho_{k}\zeta)g_{k}^{m}(\zeta)g_{k}''(\zeta)\frac{1}{\rho_{k}^{2+(m+1)\alpha}} +o(\frac{1}{\rho_{k}^{2+(m+1)\alpha}})\nonumber\\ =a_{n}(\beta_{k}+\rho_{k}\zeta)g_{k}^{n}(\zeta)g_{k}'(\zeta)^{2}\frac{1} {\rho_{k}^{2(1+\alpha)+n\alpha}}+o(\frac{1} {\rho_{k}^{2(1+\alpha)+n\alpha}})\nonumber\\ \quad \quad +o(\frac{1} {\rho_{k}^{1+2\alpha+(q_{1}-q_{2})\alpha}})+o(\frac{1} {\rho_{k}^{\alpha+(q_{3}-q_{4})\alpha}}). $

(3.5)

Multiplying $\rho_{k}^{2(1+\alpha)+n\alpha}\frac{1}{a_{n}(\beta_{k}+\rho_{k}\zeta)}$ on both sides of (3.5), and noting $m-n<1, $ $1+n\alpha+(q_{2}-q_{1})\alpha>0, $ and $2(1+\alpha)+n\alpha+(q_{4}-q_{3})\alpha-\alpha>0, $ we can conclude from this, by letting $k \rightarrow \infty$, $g^{n}g'^{2}\equiv 0.$ Thus $g$ is a constant, which is a contradiction. Therefore we have

$ \rho(f) \leq 2+ \frac{2(1+\alpha)\deg_{z, \infty} a}{\alpha}. $

Case 2  $r-s>1$. We choose $\alpha$ such that $0 < \alpha < \min \{\frac{1}{|1+s+t_{2}-t_{1}|}, \frac{2}{|2+s+t_{4}-t_{3}|}, 1\}$ and assume that

$ \rho(f) > 2+ \frac{2(1+\alpha)\deg_{z, \infty} a}{\alpha}. $

Then there exist a sequence $\{\beta_{k}\}$ and a positive sequence $\{\rho_{k}\}$ satisfying

$ |z_k-\beta_k|<1, \quad \rho_k\rightarrow 0 $

such that $h_{k}(\zeta)=\rho_{k}^{-\alpha}f(\beta_k+\rho_k\zeta)$ converges locally uniformly to a nonconstant meromorphic function $h(\zeta)$. By similar argument as in Case 1, we can obtain

$h(\zeta)^{s}h'^{2}(\zeta)\equiv 0.$

Hence $h$ is a constant, which is a contradiction. Thus we have completed the proof of Theorem 3.1.

Similarly, from the proof of Theorem 3.1 and Lemma 2.4, we have

Corollary 3.2   Let $f$ be an entire solution of equation (2.1). Further assume that $\frac{P(z, f)}{Q(z, f)}\not\equiv 0$ in equation (2.1), $M(z, f)\not\equiv 0, N(z, f)\not\equiv 0$ are birational functions and have the forms

$M(z, f) = \frac{c_{q_{1}}(z)f^{q_{1}}+\cdots + c_{t_{1}}(z)f^{t_{1}}}{d_{q_{2}}(z)f^{q_{2}}+\cdots+d_{t_{2}}(z)f^{t_{2}}}, \quad N(z, f) = \frac{e_{q_{3}}(z)f^{q_{3}}+\cdots + e_{t_{3}}(z)f^{t_{3}}}{u_{q_{4}}(z)f^{q_{4}}+\cdots+u_{t_{4}}(z)f^{t_{4}}}, $

where $c_{j_{1}}(z)\ (t_{1}\leq j_{1}\leq q_{1}), d_{j_{2}}(z)\ (t_{2}\leq j_{2}\leq q_{2}), e_{j_{3}}(z)\ (t_{3}\leq j_{3}\leq q_{3})$ and $u_{j_{4}}(z)\ (t_{4}\leq j_{4}\leq q_{4})$ are rational functions, $c_{t_{1}}(z) \not\equiv 0$, $d_{t_{2}}(z)\not\equiv 0$, $e_{t_{3}}(z)\not\equiv 0$, $u_{t_{4}}(z)\not\equiv 0, $ then

$\rho(f) \leq 1+ \frac{1+\alpha}{\alpha}\deg_{z, \infty} a, $

where $0 < \alpha < \min \big\{\frac{1}{|q_{1}-q_{2}-n|}, \frac{2}{|q_{3}-q_{4}-n-1|}, 1\big\}$ when $m-n < 1 $ and $0 < \alpha < \min \big\{\frac{1}{|1+s+t_{2}-t_{1}|}, $ $\frac{2}{|2+s+t_{4}-t_{3}|}, 1\big\}$ when $r-s > 1$.

Remark  In Theorem 3.1 and Corollary 3.2, if $m-n < 1, M(z, f)\equiv 0$ and $N (z, f)\not\equiv 0, $ then for arbitrary $0 < \alpha < \min \{\frac{2}{|q_{3}-q_{4}-n-1|}, 1\}$, the results of Theorem 3.1 and Corollary 3.2 are also true. Similarly, if $m-n < 1, M(z, f)\not\equiv 0$ and $N(z, f)\equiv 0, $ then we may choose any $0 < \alpha < \min \{\frac{1}{|q_{1}-q_{2}-n|}, 1\}$. If $r-s>1, M(z, f)\equiv 0$ and $N (z, f)\not\equiv 0$, then we may choose any $ 0 < \alpha < \min \{\frac{2}{|2+s+t_{4}-t_{3}|}, 1\}$. If $r-s>1, M(z, f)\not\equiv 0$ and $N(z, f)\equiv 0$, then we may choose any $0 < \alpha < \min \{\frac{1}{|1+s+t_{2}-t_{1}|}, 1\}$. If $M(z, f)= N(z, f) \equiv 0, $ $m-n < 1$ or $r-s>1$, then we may choose any $0 < \alpha < 1$.

Example  There exists the entire function $f(z)= e^{z^{n}} (n\geq1)$ such that it is of order $n$ and satisfies the following second-order differential equation

$ \label{a-eq9} f''=\frac{f^{2}+f-1}{f^{2}}(f')^{2}-nz^{n-1}ff'+n(n-1)z^{n-2}f+n^{2}z^{2(n-1)}, $

(3.6)

where $\deg_{z, \infty} a=2(n-1)$ and $0<\alpha<1, $ then the order of any meromorphic solution $f$ of equation $(3.6)$ can be estimated as $\rho(f)\leq 2+\frac{4(1+\alpha)(n-1)}{\alpha}$ and the order of any entire solution $f$ of equation $(3.6)$ can be estimated as $\rho(f)\leq 1+\frac{2(1+\alpha)(n-1)}{\alpha}$ by Theorem 3.1 and Corollary 3.2, respectively. In particular, the estimation of growth order of entire solution is sharp when $n=1$.