Proof of lemma 2.3. Since $\alpha \not =1$, then by Steinmetz's result [Reference Steinmetz30] for exponential polynomials, we have $T(r,\,b_1e^{p_1}+b_2e^{p_2})=K(1+o(1))r^k$ for some nonzero constant $K$ depending only on $\alpha$. Recall that the coefficients of equation (2.1) are combinations of functions in $\mathcal {S}$. By the lemma on the logarithmic derivative, we deduce from equation (2.1) that

(2.6)\begin{equation} \begin{aligned} T\left(r,b_1e^{p_1}+b_2e^{p_2}\right) & =m\left(r,b_1e^{p_1}+b_2e^{p_2}\right)\\ & =m\left(r,f^n+P(z,f)\right)\leq nm(r,f)+O(\log r). \end{aligned} \end{equation}

Therefore, $f$ is transcendental and $T(r,\,f)\geq K_1 r^k$ for some positive constant $K_1$. On the other hand, by the lemma on the logarithmic derivative we also have from equation (2.1) that

(2.7)\begin{equation} \begin{aligned} nT(r,f) & =T\left(r,f^n\right)=m\left(r,f^n\right)=m\left(r,b_1e^{p_1}+b_2e^{p_2}-P(z,f)\right)\\ & \leq m\left(r,b_1e^{p_1}+b_2e^{p_2}\right)+m\left(r,P(z,f)\right)+O(1)\\ & \leq K(1+o(1))r^k+(n-1)m(r,f)+O(\log r), \end{aligned} \end{equation}

which yields that $T(r,\,f)\leq K_2 r^k$ for some positive constant $K_2$. This together with $T(r,\,f)\geq K_1 r^k$ yields $\sigma (f)=k$. Then by definition of $\mathcal {S}$ and looking at the proof of [Reference Zhang33, theorem 2.1], we see that $\alpha$ is real. Now, $-1\leq \alpha <0$ or $0<\alpha <1$.

Recall that $\theta _1=-\pi /(2k)$ and from (2.2) that $\delta (p_1,\,\theta )=\cos k\theta$ and $\delta (p_2,\,\theta )=\alpha \cos k\theta$. When $\alpha <0$, we see that $\delta (p_1,\,\theta )$ and $\delta (p_2,\,\theta )$ have opposite signs for each $\theta$ in the sectors $S_j$ defined in (2.3) for $p_1$ and $\delta (p_1,\,\theta )>0$ for $\theta$ in the sectors $S_{j}$ where $j\in J_1$; when $\alpha >0$, we see that $\delta (p_1,\,\theta )>0$ and $\delta (p_2,\,\theta )>0$ simultaneously for each $\theta$ in the sectors $S_{j}$ where $j\in J_1$ and $\delta (p_1,\,\theta )<0$ and $\delta (p_2,\,\theta )<0$ simultaneously for each $\theta$ in the sectors $S_{j}$ where $j\in J_2$. Then we see that the assertion (1) and the assertion (2) for the case that $\delta (p_1,\,\theta )>0$ can be obtained by directly following the proof of [Reference Zhang, Gao and Zhang34, lemma 2.5].

Now we consider the growth behaviour of $f(z)$ along the ray $z=re^{i\theta }$ such that $\delta (p_1,\,\theta )<0$ when $0<\alpha <1$. Let $\varepsilon >0$ be given. By [Reference Gundersen12, corollary 1], there exists a constant $r_0=r_0(\theta )>1$ such that for all $z$ on the ray $z=re^{i\theta }$ which does not meet $\tilde {R}$ when $r\geq r_0$, and for all positive integers $j$,

(2.8)\begin{equation} \left|\frac{f^{(j)}(re^{i\theta})}{f(re^{i\theta})}\right|\leq r^{j(k-1+\varepsilon)}. \end{equation}

Since all coefficients of $P(z,\,f)$ are combinations of functions in $\mathcal {S}$, then for each coefficient of $P(z,\,f)$, say $a_l$, by [Reference Gundersen12, corollary 1], we also have, along the ray $z=re^{i\theta }$, that

(2.9)\begin{equation} \left|a_l(re^{i\theta})\right|\leq r^{M}, \end{equation}

for sufficiently large $r$ and some large integer $M$. Recalling from the introduction that $P(z,\,f)=\sum ^{m}_{l=1}a_{l}f^{n_{l0}}(f')^{n_{l1}}\cdots (f^{(s)})^{n_{ls}}$, where $m$ is an integer and $n_{l0}+n_{l1}+\cdots n_{ls}\leq n-1$, we may write

(2.10)\begin{equation} P(z,f)=\sum^{m}_{l=1}\hat{a}_{l}f^{n_{l0}+n_{l1}+\cdots+n_{ls}}, \end{equation}

with the new coefficients $\hat {a}_l=a_l(f'/f)^{n_{l1}}\cdots (f^{(s)}/f)^{n_{ls}}$, where $n_{l0},\,\cdots,\,n_{ls}$ are nonnegative integers. Note that the greatest order of the derivatives of $f$ in $P(z,\,f)$ is equal to $s\geq 0$. Suppose now that $|f(r_je^{i\theta })|\geq r_j^{N}$ for some infinite sequence $z_j=r_je^{i\theta }$ and some large $N\geq M+s(k-1+\varepsilon )$. Then, from (2.1), (2.8), (2.9) and (2.10) we have

(2.11)\begin{equation} \begin{aligned} & \left|b_1(r_je^{i\theta})e^{p_1(r_je^{i\theta})}+b_2(r_je^{i\theta})e^{p_2(r_je^{i\theta})}\right|\\ & \quad=\left|f(r_je^{i\theta})^n\right|\left|1+\frac{P(r_je^{i\theta},f(r_je^{i\theta}))}{f(r_je^{i\theta})^n}\right|\geq(1-o(1))r^{nN}, \end{aligned} \end{equation}

which is impossible when $r_j$ is large since $b_1(r_je^{i\theta })e^{p_1(r_je^{i\theta })}+b_2(r_je^{i\theta })e^{p_2(r_je^{i\theta })}\to 0$ as $z_j\to \infty$. Therefore, along the ray $z=re^{i\theta }$ such that $\delta (p_1,\,\theta )<0$ we must have $|f(re^{i\theta })|\leq r^{N}$ for all large $r$ and some integer $N$. Thus our second assertion follows.

Proof of theorem 2.1. For simplicity, we denote $P=P(z,\,f)$. By taking the derivatives on both sides of (2.1) and eliminating $e^{p_2}$ and $e^{p_1}$ from (2.1) and the resulting equation, respectively, we get the following two equations:

(2.12)\begin{align} b_2B_2f^n-nb_2f^{n-1}f'+b_2B_2 P-b_2P'& =A_1{e^{p_1}}, \end{align}

(2.13)\begin{align} b_1B_1f^n-nb_1f^{n-1}f'+b_1B_1 P-b_1P'& ={-}A_1{e^{p_2}}, \end{align}

where $B_1=b_1'/b_1+p_1'$, $B_2=b_2'/b_2+p_2'$ and $A_1=b_1b_2(B_2-B_1)$. Note that $B_1B_2A_1\not \equiv 0$. By differentiating on both sides of (2.12) and then eliminating $e^{p_1}$ from (2.12) and the resulting equation, we get

(2.14)\begin{equation} h_1 f^n+h_2f^{n-1}f'+h_3f^{n-2}(f')^2+h_4f^{n-1}f''+P_1=0, \end{equation}

where $h_1=b_2B_2(A_1'+p_1'A_1)-(b_2B_2)'A_1$, $h_2=-n b_2A_1(p_1'+p_2')-nb_2A_1'$, $h_3=n(n-1)b_2A_1$, $h_4=nb_2A_1$, and $P_1=(A_1'+p_1'A_1)(b_2B_2P-b_2P')-A_1(b_2B_2P-b_2P')'$ is a differential polynomial in $f$ of degree $\leq n-1$. By lemma 2.3 and our assumption, $\alpha$ is a nonzero real number such that $-1\leq \alpha < 1$. Below we consider the two cases where $-1\leq \alpha < 0$ and $0<\alpha <1$, respectively.

Case 1: $-1\leq \alpha < 0$. We multiply both sides of equations (2.12) and (2.13) and obtain

(2.15)\begin{equation} g_1 f^{2n}+g_2f^{2n-1}f'+g_3f^{2n-2}(f')^2+P_2={-}A_1^2e^{p_1+p_2}, \end{equation}

where $g_1=b_1b_2B_1B_2$, $g_2=-nb_1b_2(B_1+B_2)$, $g_3=n^2b_1b_2$ and $P_2=b_1b_2(B_2f^n-nf^{n-1}f')(B_1P-P')+b_1b_2(B_1f^n-nf^{n-1}f')$ $(B_2P-P')+b_1b_2(B_1P-P')(B_2P-P')$ is a differential polynomial in $f$ of degree $\leq 2n-1$. By eliminating $(f')^2$ from (2.14) and (2.15), we get

(2.16)\begin{equation} f^{2n-1}\left[(g_3h_1-h_3g_1)f+(g_3h_2-h_3g_2)f'+g_3h_4f''\right]+P_3=h_3A_1^2e^{p_1+p_2}, \end{equation}

where $P_3=g_3f^nP_1-h_3P_2$ is a differential polynomial in $f$ of degree $\leq 2n-1$. For simplicity, we denote

(2.17)\begin{equation} \varphi=\frac{h_3A_1^2}{g_3h_4}\frac{e^{p_1+p_2}}{f^{2n-1}}-\frac{1}{g_3h_4}\frac{P_3}{f^{2n-1}}. \end{equation}

Recalling $B_1=b_1'/b_1+p_1'$ and $B_2=b_2'/b_2+p_2'$, we get from equation (2.16) that

(2.18)\begin{equation} f''+H_1f'+H_2f=\varphi, \end{equation}

where

(2.19)\begin{equation} \begin{aligned} H_1 & =\frac{h_2}{h_4}-\frac{g_2h_3}{g_3h_4}={-}\left[\frac{1}{n}(p_1'+p_2')-\frac{n-1}{n}\left(\frac{b_1'}{b_1}+\frac{b_2'}{b_2}\right)+\frac{A_1'}{A_1}\right],\\ H_2 & =\frac{h_1}{h_4}-\frac{g_1h_3}{g_3h_4}=\frac{1}{n}\left[B_2\left(\frac{A_1'}{A_1}-\frac{b_1'}{b_1}\right)-\frac{(b_2B_2)'}{b_2}\right]+\frac{1}{n^2}B_1B_2. \end{aligned} \end{equation}

Now we prove that $\varphi$ is a rational function. Recall that $b_1,\,b_2,\,p_1,\,p_2$ are all polynomials and $B_1=b_1'/b_1+p_1'$, $B_2=b_2'/b_2+p_2'$ and $A_1=b_1b_2(B_2-B_1)$. Since $f$ is entire, we see that $\varphi$ has only finitely many poles. By lemma 2.3, $\sigma (f)=k$. By the lemma on the logarithmic derivative, we deduce from (2.18) that

(2.20)\begin{equation} T(r,\varphi)=m(r,\varphi)+O(\log r)\leq m(r,f)+O(\log r)=T(r,f)+O(\log r). \end{equation}

Therefore, $\sigma (\varphi )\leq k$. Now let $\theta \in [0,\,2\pi )$ be such that $\delta (p_1,\,\theta )\not =0$ and $z=re^{i\theta }$ is a ray that meets only finitely discs in $\tilde {R}$. Since $\alpha <0$, then by lemma 2.3 (1) we see that in both cases that $\delta (p_1,\,\theta )>0$ and $\delta (p_1,\,\theta )<0$ we always have $|e^{p_1(re^{i\theta })+p_2(re^{i\theta })}/f(re^{i\theta })^{2n-1}|\to 0$ as $r\to \infty$ along the ray $z=re^{i\theta }$. Together with [Reference Gundersen12, corollary 1] we see from (2.17) that there is some integer $N$ such that $|\varphi (re^{i\theta })|\leq r^{N}$ for all large $r$. Then by the Phragmén–Lindelöf theorem we see that $|\varphi |\leq r^{N}$ uniformly in each $\overline {S}_{j,\epsilon }$, $j=1,\,2,\,\cdots,\,2k$, for some integer $N=N(j)$. Since $\epsilon$ can be arbitrarily small, then by the Phragmén–Lindelöf theorem again we conclude that $\varphi$ is a rational function. From now on we fix one large $N$.

Recall that $B_2=b_2'/b_2+p_2'$. Denote $F_1=f'-(B_1/n)f$. Then by simple computations we obtain from (2.18) that

(2.21)\begin{equation} F_1'-\left(\frac{1}{n}p_2'-\frac{b_1'}{b_1}-\frac{n-1}{n}\frac{b_2'}{b_2}+\frac{A_1'}{A_1}\right)F_1=\varphi. \end{equation}

Denote $\xi _1=p_2'/n-b_1'/b_1-(n-1)b_2'/nb_2+A_1'/A_1$. Then the general solution of the homogeneous equation $F_1'-\xi _1 F_1=0$ is defined on a finite-sheeted Riemann surface and is of the form $F_1=C_2b_2^{1/n}A_1/(b_1b_2)e^{p_2/n}$, where $C_2$ is a constant and $b_2^{1/n}$ is in general an algebraic function (see [Reference Katajamäki21] for the theory of algebroid functions). Suppose that $\Gamma _2$ is a particular solution of $F_1'-\xi _1F_1=\varphi$. We may write the meromorphic solution of this equation as $F_1=C_2b_2^{1/n}A_1/(b_1b_2)e^{p_2/n}+\Gamma _2$. By an elementary series expansion analysis around the zeros of $b_2$, we conclude that $\Gamma _2/b_2^{1/n}$ is a meromorphic function. This implies that $b_2$ is an $n$-square of some polynomial. Then by lemma 2.2 we integrate the equation (2.21) along the ray $z=re^{i\theta }$ in $S_2$ such that $\delta (p_2,\,\theta )>0$ and obtain

(2.22)\begin{equation} F_1=f'-\frac{1}{n}B_1f=\frac{c_2}{n}\frac{b_2^{1/n}A_1}{b_1b_2}e^{p_2/n}+\Gamma_2, \end{equation}

where

(2.23)\begin{equation} \Gamma_2=\frac{A_1b_2^{1/n}}{b_1b_2}e^{p_2/n}\int_{0}^{z} e^{{-}p_2/n}\frac{b_1b_2}{A_1b_2^{1/n}} \varphi \,{\rm d}t-a_{2,2}\frac{A_1b_2^{1/n}}{b_1b_2}e^{p_2/n}, \end{equation}

where $a_{2,2}=a_{2,2}(\theta )$ is a constant such that $|\Gamma _2|=O(r^N)$ along the ray $z=re^{i\theta }$ in $S_2$. Now, for $z\in S_{j,\epsilon }$ where $j\in J_2$, we have $\delta (p_2,\,\theta )>0$ and so $\Gamma _2=(c_2d_{2,j}/n)b_2^{1/n}A_1/(b_1b_2)e^{p_2/n}+\gamma _{2,j}$, where $d_{2,j}$ are some constants related to a sector $S_{j,\epsilon }$ and $|\gamma _{2,j}|=O(r^N)$ uniformly in $\overline {S}_{j,\epsilon }$. Of course, for $j=2$, we have $d_{2,2}=0$. Furthermore, $|\Gamma _2|=O(r^N)$ uniformly in $\overline {S}_{j,\epsilon }$ where $j\in J_1$. We then define $d_{2,j}=0$ for $j\in J_1$.

Similarly, denoting that $\xi _2=p_1'/n-b_2'/b_2-(n-1)b_1'/nb_1+A_1'/A_1$ we also have $F_2'-\xi _2F_2=\varphi$ and it follows by integration that $F_2=-(c_1/n)b_1^{1/n}A_1/(b_1b_2)e^{p_1/n}+\Gamma _1$, where $\Gamma _1=-(c_1d_{1,j}/n)b_1^{1/n}A_1/(b_1b_2)e^{p_1/n}+\gamma _{1,j}$, where $d_{l,j}$ are some constants related to a sector $S_{j,\epsilon }$ and $|\gamma _{1,j}|=O(r^N)$ uniformly in $\overline {S}_{j,\epsilon }$ for $j\in J_1$. Of course, for $j=1$, we have $d_{1,1}=0$. Furthermore, $|\Gamma _1|=O(r^N)$ uniformly in $\overline {S}_{j,\epsilon }$ where $j\in J_2$. We then define $d_{1,j}=0$ for $j\in J_2$.

Denoting $B=n/(B_2-B_1)$, we have $f=B(F_1-F_2)$. Together with the relation $A_1=b_1b_2(B_2-B_1)$, we have $f=c_1b_1^{1/n}e^{p_1/n}+c_2b_2^{1/n}e^{p_2/n}+\eta$ with an entire function $\eta =B(\Gamma _2-\Gamma _1)$. We see that $\eta =c_2d_{2,j}b_2^{1/n}e^{p_2/n}+B(\gamma _{2,j}-\gamma _{1,j})$ when $j\in J_1$ and $\eta =c_1d_{1,j}b_1^{1/n}e^{p_1/n}+B(\gamma _{2,j}-\gamma _{1,j})$ when $j\in J_2$.

Now we determine $d_{1,j}$ and $d_{2,j}$. By [Reference Gundersen12, corollary 1], we may suppose that along the ray $z=re^{i\theta }$ we have $|f^{(j)}(re^{i\theta })/f(re^{i\theta })|=r^{j(k-1+\varepsilon )}$ for all $j>0$ for all sufficiently large $r$ and thus write $P$ in the form in (2.10) with the new coefficients $\hat {a}_l=a_l(f'/f)^{n_{l1}}\cdots (f^{(s)}/f)^{n_{ls}}$, where $n_{l1},\,\cdots,\,n_{ls}$ are nonnegative integers. For simplicity, denote $D_{1,j}=c_1+c_1d_{1,j}$. By substituting $f=c_1b_1^{1/n}e^{p_1/n}+c_2b_2^{1/n}e^{p_2/n}+\eta$ into (2.1), we obtain, for $z=re^{i\theta }$ for a $\theta$ in $S_j$ and $j\in J_1$,

(2.24)\begin{equation} \begin{aligned} & \left(D_{1,j}^n-1\right)b_1e^{p_1}+\sum_{k_0=1}^{n-1}\binom{n}{k_0}\left(D_{1,j}b_1^{1/n}\right)^{n-k_0}(c_2b_2^{1/n})^{k_0}e^{[(n-k_0)p_1+k_0p_2]/n}\\ & \quad+(c_2^n-1)b_2e^{p_2}+\sum_{s=1}^{n}\sum_{k_s=0}^{n-s}\alpha_{s,k_s}e^{[(n-s-k_s)p_1+k_sp_2]/n}=0, \end{aligned} \end{equation}

where $\alpha _{s,k_s}$, $s=1,\,\cdots,\,n$, $k_s=0,\,\cdots,\,n-s$, are functions satisfying $|\alpha _{s,k_s}(re^{i\theta })|=O(r^N)$ along the ray $z=re^{i\theta }$. By letting $r\to \infty$ along the above ray $z=re^{i\theta }$ such that $\delta (p_1,\,\theta )>0$ and comparing the growth on both sides of the above equation we conclude that $c_1^n(1+d_{1,j})^n=1$. Since $d_{1,1}=0$, we have $c_1^n=1$ and $d_{1,j}=\mu _{1,j}-1$ for some $\mu _{1,j}$ such that $\mu _{1,j}^n=1$. Similarly, we can prove that $d_{2,j}=\mu _{2,j}-1$ for some $\mu _{2,j}$ such that $\mu _{2,j}^n=1$. In particular, when $k=1$, since $d_{1,1}=d_{2,2}=0$ and $|\eta _j|=O(r^N)$ uniformly in the sectors $\overline {S}_{j,\epsilon }$, $j=1,\,2$ and since $\epsilon$ can be arbitrarily small, by the Phragmén–Lindelöf theorem we conclude that $\eta$ is a polynomial. Thus our first assertion follows.

Case 2: $0<\alpha <1$.

As in the proof of [Reference Zhang33, theorem 2.1], we first define some functions in the following way: We let $m$ be the smallest integer such that $\alpha \leq [(m+1)n-1]/[(m+1)n]$ and ${\iota _0,\,\cdots,\,\iota _m}$ be a finite sequence of functions such that

(2.25)\begin{equation} \begin{aligned} \iota_0 & =\frac{A_1}{nb_1},\\ \iota_j & =({-}1)^{j}\left(\frac{A_1}{nb_1}\right)^{j+1}(jn-1)\cdots(n-1), \quad j=1,2,\cdots,m. \end{aligned} \end{equation}

Recall that $B_1=b_1'/b_1+p_1'$. We also let ${\kappa _0,\,\cdots,\,\kappa _m}$ be a finite sequence of functions defined in the following way:

(2.26)\begin{equation} \begin{aligned} \kappa_0 & =\frac{1}{n}\frac{b_1'}{b_1}+\frac{1}{n}p_1',\\ \kappa_j & =\frac{\iota_{j-1}'}{\iota_{j-1}}-\frac{jn-1}{n}\frac{b_1'}{b_1}+\left[j(\alpha-1)+\frac{1}{n}\right]p_1', \quad j=1,2,\cdots,m. \end{aligned} \end{equation}

Then we define $m+1$ functions $G_0$, $G_1$, $\cdots$, $G_m$ in the way that $G_0=f'-\kappa _0f$, $G_{1}=G_{0}'-\kappa _1G_{0}$, $\cdots$, $G_{m}=G_{m-1}'-\kappa _mG_{m-1}$. Now we have equation (2.13) and it follows that

(2.27)\begin{equation} G_0=f'-\kappa_0f=\iota_0\frac{{e^{p_2}}}{f^{n-1}}+W_0, \end{equation}

where $W_0=-(B_1P-P')/(nf^{n-1})$. Moreover, when $m\geq 1$, by simple computations we obtain

\begin{align*} G_1& =G_0'-\kappa_1G_0=\iota_1\frac{e^{2p_2}}{f^{2n-1}}+W_1,\\ W_1& =W'_0-\kappa_1W_0-(n-1)\iota_0\frac{e^{p_2}}{f^n}W_0, \end{align*}

and by induction we obtain

(2.28)\begin{align} G_{j}& =G_{j-1}'-\kappa_jG_{j-1}=\iota_j\frac{e^{(j+1)p_2}}{f^{(j+1)n-1}}+W_{j}, \quad j=1,\cdots,m, \end{align}

(2.29)\begin{align} W_{j}& =W'_{j-1}-\kappa_jW_{j-1}-(jn-1)\iota_{j-1}\frac{e^{jp_2}}{f^{jn}}W_0, \quad j=1,\cdots,m. \end{align}

For an integer $l\geq 0$, by elementary computations it is easy to show that $W_0^{(l)}=W_{0l}/f^{n+l-1}$, where $W_{0l}=W_{0l}(z,\,f)$ is a differential polynomial in $f$ of degree $\leq n+l-1$, and also that $(e^{p_2}/f^{n})^{(l)}=e^{p_2}W_{1l}/f^{n+l}$, where $W_{1l}=W_{1l}(z,\,f)$ is a differential polynomial in $f$ of degree $\leq n+l$. We see that $W_j$, $1\leq j\leq m$, is formulated in terms of $W_0$ and $e^{p_2}/f^{n}$ and their derivatives. We may write

(2.30)\begin{equation} G_m=\iota_m\frac{e^{(m+1)p_2}}{f^{(m+1)n-1}}+F(W_0,e^{p_2}/f^{n}), \end{equation}

where $F(W_0,\,e^{p_2}/f^{n})$ is a combination of $W_0$ and $e^{p_2}/f^{n}$ and their derivatives with functions being combinations of functions in $\mathcal {S}$. Moreover, from the recursion formula $G_j=G_{j-1}'-\kappa _jG_{j-1}$, $j\geq 1$, and $G_0=f'-\kappa _0f$, we easily deduce that $f$ satisfies the linear differential equation

(2.31)\begin{equation} f^{(m+1)}-\hat{t}_{m}f^{(m)}+\cdots+({-}1)^{m+1}\hat{t}_0f=G_m, \end{equation}

where $\hat {t}_m$, $\hat {t}_{m-1}$, $\cdots$, $\hat {t}_0$ are functions formulated in terms of $\kappa _0$, $\cdots$, $\kappa _m$ and their derivatives.

Now we prove that $G_m$ is a rational function. Recall that $b_1,\,b_2,\,p_1,\,p_2$ are all polynomials. Since $f$ is entire, then by the definitions of $\kappa _0$ and $\kappa _j$ in (2.26), we see that $G_m$ has only finitely many poles. With an application of the lemma on the logarithmic derivative as in previous case, we deduce from (2.31) that $\sigma (G_m)\leq \sigma (f)=k$. Now let $\theta \in [0,\,2\pi )$ be such that $\delta (p_1,\,\theta )\not =0$ and $z=re^{i\theta }$ be a ray that meets only finitely may discs in $\tilde {R}$. By [Reference Gundersen12, corollary 1] and lemma 2.3 (2), we see from (2.31) that there is some integer $N$ such that $|G_m(re^{i\theta })|\leq r^{N}$ for all large $r$ along the ray $z=re^{i\theta }$ such that $\delta (p_1,\,\theta )<0$. On the other hand, by lemma 2.3 (2) there is some integer $N$ such that

1. (1) if $\alpha <[(m+1)n-1]/[(m+1)n]$, then $|e^{(m+1)p_2(re^{i\theta })}/f(re^{i\theta })^{(m+1)n-1}|\to 0$ as $r\to \infty$ along the ray $z=re^{i\theta }$ such that $\delta (p_1,\,\theta )>0$;

2. (2) if $\alpha =[(m+1)n-1]/[(m+1)n]$, then $|e^{(m+1)p_2(re^{i\theta })}/f(re^{i\theta })^{(m+1)n-1}|\leq e^{Nr^{k-1}}$ for all large $r$ along the ray $z=re^{i\theta }$ such that $\delta (p_1,\,\theta )>0$.

Note that $e^{p_2(re^{i\theta })}/f(re^{i\theta })^{n}\to 0$ as $r\to \infty$ along the ray $z=re^{i\theta }$ such that $\delta (p_1,\,\theta )>0$. In case (1), together with [Reference Gundersen12, corollary 1] we see from (2.30) that $|G_m(re^{i\theta })|\leq r^{N}$ for all large $r$ and thus by the Phragmén–Lindelöf theorem we see that $|G_m|\leq r^{N}$ uniformly in each $\overline {S}_{j,\epsilon }$, $j\in J_2$, for some integer $N=N(j)$; in case (2), together with [Reference Gundersen12, corollary 1] we see from (2.30) that $|G_m(re^{i\theta })|\leq e^{Nr^{k-1}}$ for all large $r$ and, since the set of rays $z=re^{i\theta }$ meeting infinitely many discs in $\tilde {R}$ has zero linear measure, then by the Phragmén–Lindelöf theorem we see that $|G_m|\leq e^{Nr^{k-1}}$ uniformly in each $\overline {S}_{j,\epsilon }$, $j\in J_2$, for some integer $N=N(j)$. Since $\epsilon$ can be arbitrarily small, then in either case of (1) and (2) by the Phragmén–Lindelöf theorem again we conclude that $G_m$ is a rational function. From now on we fix one large $N$.

We denote $D_0=b_1^{1/n}$ and $D_j=\iota _{j-1}b_1^{-j}b_1^{1/n}$, $j=1,\,\cdots,\,m$. Now we choose one $\theta$ such that $\delta (p_1,\,\theta )>0$ and let $z=re^{i\theta }\in S_1$. Let $t_0=1/n$, $t_1=(\alpha -1)+1/n$, $\cdots$, $t_m=m(\alpha -1)+1/n$. Similarly as in the proof of [Reference Zhang33, theorem 2.1], we may use lemma 2.2 to integrate the recursion formulas $G_j=G_{j-1}'-\kappa _jG_{j-1}$ from $j=m$ to $j=1$ along the above ray $z=re^{i\theta }$ such that $\delta (p_1,\,\theta )>0$ inductively and finally integrating $G_0=f'-\kappa _0f$ along this ray $z=re^{i\theta }$ to obtain

(2.32)\begin{equation} f=b_1^{1/n}\sum_{i=0}^mc_i\left(\frac{b_2}{b_1}\right)^ie^{t_ip_1}+H_0, \end{equation}

where $c_0$, $\cdots$, $c_m$ are constants and

(2.33)\begin{equation} H_{0}=b_1^{1/n}e^{t_0p_1}\int_{0}^{z}b_1^{{-}1/n}e^{{-}t_0p_1}H_1ds-a_{0}b_1^{1/n}e^{t_0p_1}, \end{equation}

where $a_0=a_{0}(\theta )$ is a constant such that $|H_0|=O(r^N)$ along the ray $z=re^{i\theta }$.

As is shown in the proof of [Reference Zhang33, theorem 2.1], $b_1$ is an $n$-square of some polynomial and we may write the entire solution of (2.1) as $f=\gamma _1\sum _{j=0}^mc_j(b_2/b_1)^je^{t_jp_1}+\eta$, where $\gamma _1$ is a polynomial such that $\gamma _1^n=b_1$ and $\eta$ is a meromorphic function with at most finitely many poles. Then we can integrate $G_j=G_{j-1}'-\kappa _jG_{j-1}$ from $j=m$ to $j=1$ inductively and finally integrate $G_0=f'-\kappa _0f$ to obtain that $H_0$ is a meromorphic function with at most finitely many poles. We choose $\eta =H_0$. Recall that along the ray $z=re^{i\theta }$ such that $\delta (p_1,\,\theta )>0$ and $z=re^{i\theta }\in S_1$, we have $|H_0|=O(r^N)$. Denote $g=b_1^{1/n}\sum _{i=0}^mc_i(b_2/b_1)^ie^{t_ip_1}$. Then

(2.34)\begin{equation} g^n=b_1\sum_{k_0=0}^{mn}C_{k_0}\left(\frac{b_2}{b_1}\right)^{k_0}e^{(k_0t-k_0+1)p_1}, \end{equation}

where

(2.35)\begin{equation} C_{k_0}=\sum_{\substack{j_0+\cdots+j_m=n,\\j_1+\cdots+mj_m=k_0}}\frac{n!}{j_0!j_1!\cdots j_m!}c_0^{j_0}c_1^{j_1}\cdots c_m^{j_m}, \quad k_0=0,1,\cdots,mn. \end{equation}

By [Reference Gundersen12, corollary 1], we may suppose that along the ray $z=re^{i\theta }$ we have $|f(re^{i\theta })^{(j)}/f(re^{i\theta })|=r^{j(k-1+\varepsilon )}$ for all $j>0$ and all sufficiently large $r$. By writing $P$ in the form in (2.10) with the new coefficients $\hat {a}_l=a_l(f'/f)^{n_{l1}}\cdots (f^{(s)}/f)^{n_{ls}}$, where $n_{l1},\,\cdots,\,n_{ls}$ are nonnegative integers, and using [Reference Gundersen12, corollary 1], we see that each term in $P(z,\,f)$ of degree $n-j$, $1\leq j\leq n-1$, equals a linear combination of exponential functions of the form $e^{[nk_{j}(\alpha -1)+n-j]p_1/n}$, $0\leq k_{j}\leq (n-j)m$, with coefficients $\beta _j$ having polynomial growth along the ray $z=re^{i\theta }$. Therefore, by substituting $f=g+H_0$ into (2.1) we obtain by the same arguments in the proof of [Reference Zhang33, theorem 2.1] that $c_0^n=1$ when $m=0$, and $c_0^{n}=1$, $nc_0^{n-1}c_1=1$ and $p_2=\alpha p_1$ when $m=1$ and further that $C_{k_0}\equiv 0$ for all $2\leq k_0\leq m$ when $m\geq 2$.

Now, $b_1^{1/n}$ denotes a polynomial. By the definition of $\iota _j$ and $D_j$, we see that $D_j$ are rational functions. Recall that $G_m$ is a rational function. By lemma 2.2 and looking at the calculations to obtain $H_0$ in (2.33), we have, for $z\in S_{j,\epsilon }$, $j\in J_1$, such that $\delta (p_1,\,\theta )>0$, $H_0=\gamma _1\sum _{l=0}^md_{l,j}(b_2/b_1)^je^{t_jp_1}+\eta _{j}$, where $d_{l,j}$, $l=0,\,\cdots,\,m$, are some constants related to a sector $S_{j,\epsilon }$ and $|\eta _j|=O(r^N)$ uniformly in $\overline {S}_{j,\epsilon }$, $j\in J_1$. Of course, for $j=1$, we have $d_{l,1}=0$ for all $l$. Since $c_0^{n}=1$ when $m=0$, $c_0^{n}=nc_0^{n-1}c_1=1$ when $m=1$, and $c_0^{n}=nc_0^{n-1}c_1=1$ and $C_{k_0}=0$ for all $2\leq k_0\leq m$ when $m\geq 2$, then by simple computations, we deduce that $c_j=s_jc_0$ for some nonzero rational numbers $s_j$, $j=0,\,1,\,\cdots,\,m$. Therefore, by considering the growth of $f$ along the ray $z=re^{i\theta }$ such that $z\in S_{j,\epsilon }$, $j\in J_1$, as for the ray $z=re^{i\theta }\in S_1$, we have $(c_0+d_{0,j})^n=1$ when $m=0$, $(c_0+d_{0,j})^n=n(c_0+d_{0,j})^{n-1}(c_1+d_{1,j})=1$ when $m=1$ and further that $\hat {C}_{k_0}=\sum _{\substack {j_0+\cdots +j_m=n,\\j_1+\cdots +mj_m=k_0}}\frac {n!}{j_0!j_1!\cdots j_m!}(c_0+d_{0,j})^{j_0}(c_1+d_{1,j})^{j_1}\cdots (c_m+d_{m,j})^{j_m}=0$ for $k_0=2,\,\cdots,\,m$ when $m\geq 2$. Therefore, for each $j\in J_1$, there is a $\mu _j$ satisfying $\mu _j^n=1$ such that $c_l+a_{l,j}=\mu _j c_l$ for all $l$. Note that $\mu _{1}=1$. Also, we have $|\eta |=O(r^N)$ uniformly in the sectors $\overline {S}_{j,\epsilon }$, $j\in J_2$. In conclusion, we may write $\eta =\gamma _1\sum _{l=0}^m(\mu _j-1)c_{j}(b_2/b_1)^je^{[jn(\alpha -1)+1]p_1/n}+\eta _{j}$, where $\mu _j$ are the $n$-th roots of $1$ such that $\mu _j=1$, $j=\{1\}\cup \in J_2$, and $|\eta _j|=O(r^N)$ uniformly in the sector $\overline {S}_{j,\epsilon }$. In particular, when $k=1$, since $\epsilon$ can be arbitrarily small, then by the Phragmén–Lindelöf theorem we conclude that $\eta$ is a rational function. This completes the proof.

Proof of theorem 3.1. Let $f$ be a nontrivial solution of equation (3.1) such that $\lambda (f)<\infty$. By Hadamard's factorization theorem we may write $f=\kappa e^{h}$, where $h$ is an entire function and $\kappa$ is the canonical product from the zeros of $f$ satisfying $\rho (\kappa )=\lambda (\kappa )<\infty$. Denoting $g=h'$, then from (3.1) we have

(3.2)\begin{equation} g^2+g'+2\frac{\kappa'}{\kappa}g+\frac{\kappa''}{\kappa}=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}. \end{equation}

Below we consider the two cases where $0<\alpha \leq 1/2$ and $(2m-1)/(2m)< \alpha \leq [2(m+1)-1]/[2(m+1)]$, $m\geq 1$, respectively.

Case 1: $0<\alpha \leq 1/2$.

By theorem 2.1, we may write $g=\gamma _1e^{p_1/2}+\eta$, where $\gamma _1$ is a polynomial such that $\gamma _1^2=b_1$ and $\eta$ is an entire function such that $|\eta |=O(r^N)$ uniformly in $\overline {S}_{1,\epsilon }$ and $\overline {S}_{2,\epsilon }$. By substituting this expression into equation (3.2), we obtain

(3.3)\begin{equation} 2\gamma_1\left(\frac{\kappa'}{\kappa}+\frac{1}{2}\frac{\gamma_1'}{\gamma_1}+\frac{p_1'}{4}+\eta\right)e^{p_1/2}-b_2e^{p_2}+\frac{\kappa''}{\kappa}+2\eta\frac{\kappa'}{\kappa}+\eta^2+\eta'=0. \end{equation}

Suppose that $0<\alpha <1/2$. We define

(3.4)\begin{equation} w=\kappa \gamma_1^{1/2}e^{p_1/4+\int_{z_0}^{z} \eta \,{\rm d}t}, \end{equation}

where $z_0$ is chosen so that $|z_0|$ is large. Then $w$ is analytic outside a finite disc centred at $0$ and satisfy

(3.5)\begin{equation} \frac{w'}{w}=\frac{\kappa'}{\kappa}+\frac{1}{2}\frac{\gamma_1'}{\gamma_1}+\frac{p_1'}{4}+\eta. \end{equation}

Dividing by $2\gamma _1e^{p_1/2}$ on both sides of equation (3.3) and then considering the growth of $w'/w$ along the ray $z=re^{i(\theta _2-\epsilon )}$ such that $w$ has no zero around the neighbourhood of the ray $z=re^{i(\theta _2-\epsilon )}$, we have by [Reference Gundersen12, corollary 1] that $|w'(re^{i\theta })/w(re^{i\theta })|=O(r^{-2})$ as $r\to \infty$. By integration, we obtain that $w(re^{i(\theta _2-\epsilon )})\to a$ as $r\to \infty$ along the ray $z=re^{i(\theta _2-\epsilon )}$ for some nonzero constant $a=a(\theta _2,\,\epsilon )$. On the other hand, by applying lemma 3.2 to equation (3.1) we have $|f(re^{i(\theta _2+\epsilon )})|=O(r)$ along the ray $z=re^{i(\theta _2+\epsilon )}$. Recalling that $f=\kappa e^{h}$ and $g=h'=\gamma _1e^{p_1/2}+\eta$, we may write

(3.6)\begin{equation} w=fe^{{-}h}\gamma_1^{1/2}e^{p_1/4+\int \eta dz}=f\gamma_1^{1/2}e^{p_1/4-\int_{z_0}^z \gamma_1e^{p_1/2}\,{\rm d}t}. \end{equation}

Since $\delta (p_1,\,\theta _2+\epsilon )<0$ and thus along the ray $z=re^{i(\theta _2+\epsilon )}$ we have $\int _{z_0}^z \gamma _1e^{p_1/2}\,{\rm d}t\to c$ for some constant $c=c(\theta _2,\,\epsilon )$, we see from (3.6) that $w$ defined in (3.4) satisfies $w(re^{i(\theta +\epsilon )})\to 0$ as $r\to \infty$. Denote

(3.7)\begin{equation} S_{\epsilon}=\{re^{i\theta}:\theta_2-\epsilon\leq \theta\leq \theta_2+\epsilon\}. \end{equation}

By choosing $\epsilon$ to be small and applying the Phragmén–Lindelöf theorem to $w$ defined in (3.4) in the sector in (3.7), we get $a=0$, a contradiction. Therefore, we must have $\alpha =1/2$ when $b_3\equiv 0$.

Now, if $k=1$, then obviously $p_2=p_1/2$ since we have assumed $p_1(0)=p_2(0)=0$. If $k>1$, then by theorem 2.1 we have $g=\mu _j\gamma _1e^{p_1/2}+\eta _j$, where $\mu _j^n=1$, $\gamma _1$ is a polynomial such that $\gamma _1^2=b_1$ and $\eta _j$ is an entire function such that $|\eta _j|=O(r^N)$ uniformly in $\overline {S}_{j,\epsilon }$. Note that $\eta _j$ has finite order. Denoting $p_3=p_2-p_1/2$, we rewrite equation (3.3) as

(3.8)\begin{equation} \left[b_2e^{p_3}-2\mu_j\gamma_1\left(\frac{\kappa'}{\kappa}+\frac{1}{2}\frac{\gamma_1'}{\gamma_1}+\frac{p_1'}{4}+\eta_j\right)\right]e^{p_1/2}=\frac{\kappa''}{\kappa}+2\eta_j\frac{\kappa'}{\kappa}+\eta_j^2+\eta_j'. \end{equation}

If $p_2\not \equiv p_1/2$, then $p_3$ is a nonconstant polynomial such that $\deg (p_3)\leq \deg (p_2)-1$. By the definition of $S_j$ in (2.3), we may choose a $\theta \in [0,\,2\pi )$ so that the ray $z=re^{i\theta }$ meets only finitely discs in $\tilde {R}$ and also that $\log |e^{p_1/2}|$ and $\log |e^{p_2-p_1/2}|$ both increase along the ray $z=re^{i\theta }$. By [Reference Gundersen12, corollary 1] we see that $\kappa '/\kappa +\gamma _1'/2\gamma _1+p_1'/4+\eta _j$ and $\kappa ''/\kappa +2\eta _j\kappa '/\kappa +\eta _j^2+\eta _j'-b_3$ both have polynomial growth along the ray $z=re^{i\theta }$. Then by comparing the growth on both sides of equation (3.8) along the ray $z=re^{i\theta }$, we get a contradiction. Therefore, we must have $p_2\equiv p_1/2$ when $\alpha =1/2$.

Case 2: $(2m-1)/(2m)<\alpha \leq [2(m+1)-1]/[2(m+1)]$, $m\geq 1$.

In this case, by theorem 2.1 we already have $p_2=\alpha p_1$ and we may write $g=\gamma _1\sum _{j=0}^mc_j(b_2/b_1)^je^{[2j(\alpha -1)+1]p_1/2}+\eta$, where $m\geq 1$, $\gamma _1$ is a polynomial such that $\gamma _1^2=b_1$ and $\eta$ is a mermorphic function with at most finitely many poles such that $|\eta |=O(r^N)$ uniformly in $\overline {S}_{1,\epsilon }$ and $\overline {S}_{2,\epsilon }$. By substituting this expression into equation (3.2), we obtain

(3.9)\begin{equation} \begin{aligned} & 2\gamma_1\sum_{j=0}^mc_j\left(\frac{b_2}{b_1}\right)^{j}\left[\frac{\kappa'}{\kappa}+\frac{1}{2}\frac{\gamma_1'}{\gamma_1}+j\frac{(b_2/b_1)'}{b_2/b_1}+\frac{2j(\alpha-1)+1}{4}p_1'+\eta\right]e^{L_jp_1}\\ & \gamma_1^2\sum_{k_0=m+1}^{2m}C_{k_0}\left(\frac{b_2}{b_1}\right)^{k_0}e^{M_{k_0}p_1}+\frac{\kappa''}{\kappa}+2\eta\frac{\kappa'}{\kappa}+\eta^2+\eta'=0, \end{aligned} \end{equation}

where $L_j=[2j(\alpha -1)+1]/2$, $M_{k_0}=k_0\alpha -k_0+1$ and the coefficients $C_{k_0}=\sum _{\substack {j_0+\cdots +j_m=2,\\j_1+\cdots +mj_m=k_0}}\frac {2!}{j_0!j_1!\cdots j_m!}c_0^{j_0}c_1^{j_1}\cdots c_m^{j_m}$, $k_0=m+1,\,\cdots,\,2\,m$. Suppose that $\alpha <[2(m+1)-1]/2(m+1)$. Then, for $k_0=m+1+j$, $j=0,\,1,\,\cdots,\,m-1$, we have $L_{j+1}< k_0\alpha -\alpha +1< L_j$. As in previous case, we also define the function $w$ in (3.4), where $z_0$ is chosen so that $|z_0|$ is large and $w$ is analytic outside a finite disc centred at $0$. It follows that $w'/w$ has the form in (3.5). Similarly as in previous case, we first divide by $2c_0\gamma _1e^{p_1/2}$ on both sides of equation (3.9) and conclude that $w(re^{i(\theta _2-\epsilon )})\to a$ as $r\to \infty$ along the ray $z=re^{i(\theta _2-\epsilon )}$ for some nonzero constant $a=a(\theta _2,\,\epsilon )$; then we use the expression $g=h'=\gamma _1\sum _{j=0}^mc_j(b_2/b_1)^je^{[2j(\alpha -1)+1]p_1/2}+\eta$ to derive from (3.4) that $w(re^{i(\theta +\epsilon )})\to 0$ as $r\to \infty$ along the ray $z=re^{i(\theta _2+\epsilon )}$. An application of the Phragmén–Lindelöf theorem to $w$ in the sector in (3.7) then yields a contradiction. We omit those details. Therefore, we must have $\alpha =[2(m+1)-1]/2(m+1)$. We complete the proof.

Proof of lemma 4.2. By Wittich [Reference Wittich32], we have $w=e^{cz}(a_0+a_1e^{z}+\cdots +a_ke^{kz})$. By taking the derivatives of $w$ and then dividing $w'$ and $w''$ by $w$, respectively, we get

(4.4)\begin{align} \frac{w'}{w}& =\frac{\sum_{j=0}^k(c+j)a_je^{jz}}{\sum_{j=0}^ka_je^{jz}}, \end{align}

(4.5)\begin{align} \frac{w''}{w}& =\frac{\sum_{j=0}^k(c^2+2jc+j^2)a_je^{jz}}{\sum_{j=0}^ka_je^{jz}}. \end{align}

We write equation (4.3) as $Q(e^z)=-w''/w-P(e^z)w'/w$. Since $w$ is of finite order, then an application of the lemma on the logarithmic derivative yields $\deg (Q(z))m(r,\,e^z)\leq \deg (P(z))m(r,\,e^z)+O(\log r)$, i.e., $[\deg (Q(z))-\deg (P(z))]T(r,\,e^z)\leq O(\log r)$. Therefore, $\deg (Q(z))\leq \deg (P(z))$ and thus $P(z)$ is nonconstant. Together with equations (4.4) and (4.5), we rewrite equation (4.3) as

(4.6)\begin{equation} \frac{\sum_{j=0}^k(c^2+2jc+j^2)a_je^{jz}}{\sum_{j=0}^ka_je^{jz}}+\frac{\sum_{j=0}^k(c+j)a_je^{jz}}{\sum_{j=0}^ka_je^{jz}}P(e^z)+Q(e^z)=0. \end{equation}

Since along a ray $z=re^{i\theta }$ such that $\cos \theta <0$, we have $e^{z}\to 0$ as $r\to \infty$, then by letting $r\to \infty$ along the ray $z=re^{i\theta }$, we obtain from equation (4.6) that $c^2+cP(0)+Q(0)=0$. This completes the proof.

Proof of theorem 4.1. We first suppose that $f$ is a nontrivial solution such that $\lambda (f)<\infty$ of equation (4.1) and use the expressions in (1.3) and (1.4) to write $f(z)=\Psi (x)=x^{c}\psi (x)e^{\chi (x)}$, where $x=e^{z/h}$, $h=1$ or $h=2$. From the proof of [Reference Chiang and Yu11, theorem 1.2], we know that in the broken plane $\mathbb {C}^{-}$ equation (4.13) admits a solution of the form

(4.20)\begin{equation} u=\exp\left(\int \omega {\rm d}x\right), \end{equation}

where $\omega :=\chi '+\psi '/\psi +(2hc+1)/(2x)$ is rational function in the complex plane $\mathbb {C}$. By using Kovacic's algorithm in [Reference Kovacic22] and giving the same discussions as in the proof of [Reference Chiang and Yu11, theorem 3.1] to equation (4.13) for the two cases $b_3\not =1/4$ and $b_3=1/4$, respectively, we conclude that $l$ must be even. It follows that $f$ has the form in (1.4) and thus $p=1$ in (4.10) and $\alpha =1$ in (4.8). We write $f=\kappa e^{h}$, where $\kappa$ and $h'$ are both polynomials in $\zeta =e^z$ such that $\kappa (0)\not =0$. We may also write $f=\kappa _ce^{h_c}$, where $\kappa _c=\kappa e^{cz}$ and $h_c'=h'-c$. Then, denoting $g_c=h_c'$, we have from (4.1) that

(4.21)\begin{equation} g_c^2+g'_c+2\frac{\kappa'_c}{\kappa_c}g_c+\frac{\kappa''_c}{\kappa_c}=e^{lz}+b_2e^{sz}+b_3. \end{equation}

By lemma 4.2 and the expression in (1.4), we see that the constant $c$ in (1.4) satisfies $c^2=b_3$.

Now, for the solution in (4.19), by lemma 4.3 we have $\rho _1+\rho _2=1$ and $\rho _1\rho _2=1/4-b_3$, which yield $(\rho _1-\rho _2)^2=4b_3$. Then $\rho _1-\rho _2=-2c$ and it follows that $\rho _1=(1-2c)/2$ and $\rho _2=(1+2c)/2$. Thus the solutions in (4.19) can be written as

(4.22)\begin{equation} f=(e^{z})^{c}\left[\left(E_1+E_2d\log e^z\right)(e^{z})^{{-}2c}\sum_{j=0}^{\infty}a_j(e^{z})^{j}+E_2\sum_{j=0}^{\infty}b_j(e^{z})^{j}\right], \end{equation}

when $\rho _1-\rho _2\not =0$, or

(4.23)\begin{equation} f=\left(E_1+E_2\log e^z\right)\sum_{j=0}^{\infty}a_j(e^{z})^{j}+E_2\sum_{j=0}^{\infty}b_j(e^{z})^{j}, \end{equation}

when $\rho _1-\rho _2=0$. Note that $d=0$ in (4.22) when $\rho _1-\rho _2$ is not an integer. On the other hand, for the solution $f=\kappa e^{h}$, we may write the expression in (1.4) in the form $f=e^{cz}\sum _{j=0}^{\infty }d_je^{jz}$. By comparing this series with the one in (4.22) or in (4.23), we conclude that the logarithmic term in $u_2$ does not occur. This implies that $d=0$ or $E_2=0$ in (4.22) and $E_2=0$ in (4.23) when $f=\kappa e^{h}$.

With these preparations, we now suppose that $f_1$ and $f_2$ are two linearly independent solutions of (4.1) such that $\max \{\lambda (f_1),\,\lambda (f_2)\}<\infty$. We write $l=2(m+1)$ for some integer $m\geq 0$ and also write $s=2(m+1)-t$ for some integer $t\geq 1$.

Let $q$ be the smallest integer such that $s/l\leq [2(q+1)-1]/[2(q+1)]$. Since $l$ and $s$ are relatively prime, we see that the equality holds only when $q=m$. For each of $f_1$ and $f_2$, denoted by $f$, we may write $f=\kappa _ce^{h_c}$, where $\kappa _c=\kappa e^{cz}$. Then, denoting $g_c=h_c'$, we have equation (4.21). By theorem 2.1, $g_c=h_c'=\sum _{j=0}^{q}c_je^{(m+1-jt)z}$, where $c_0,\,c_1,\,\cdots,\,c_q$ are constants such that $c_0^2=1$ and $c_1,\,\cdots,\,c_q$ satisfy certain relations. In both of the two cases where $q=1$ and $q\geq 2$, we have $2c_0c_1=b_2$ and, by simple computations, that,

(4.24)\begin{equation} c_j=\frac{t_jc_1^{j}}{({-}2c_0)^{j-1}}, \quad j=1,\cdots,q, \end{equation}

where $t_j$ are positive integers such that $t_1<\cdots < t_q$, and further that

(4.25)\begin{equation} C_{q+j}=\sum_{\substack{j_0+\cdots+j_q=2,\\j_1+\cdots+qj_q=q+j}}\frac{2}{j_0!\cdots j_q!}c_0^{j_0}\cdots c_q^{j_q}=\frac{T_{q+j}c_1^{q+j}}{({-}2c_0)^{q+j-2}}, \quad j=1,\cdots,q, \end{equation}

where $T_{q+j}$ are positive integers such that $T_{q+1}<\cdots < T_{2q}$. By substituting $g_c=\sum _{j=0}^{q}c_je^{(m+1-jt)z}$ into (4.21) together with theorem 2.1, we get

(4.26)\begin{equation} \frac{\kappa_c''}{\kappa_c}+2c_0e^{(m+1)z}\frac{\kappa_c'}{\kappa_c}+c_0(m+1)e^{(m+1)z}-b_2e^{sz}-b_3=0, \end{equation}

when $q=0$, and

(4.27)\begin{equation} \begin{aligned} & \frac{\kappa_c''}{\kappa_c}+2\left(\sum_{j=0}^{q}c_je^{(m+1-jt)z}\right)\frac{\kappa_c'}{\kappa_c}-b_3\\ & \quad+\sum_{j=0}^{q}\left[C_{k_j}e^{[m+1-(q+1)t]z}+(m+1-jt)c_j\right]e^{(m+1-jt)z}=0, \end{aligned} \end{equation}

when $q\geq 1$, where $C_{k_j}=C_{q+1+j}$ and $C_{2q+1}=0$. By substituting equations (4.4) and (4.5) for $\kappa _{c}=e^{cz}(\sum _{i=1}^ka_ie^{iz})$, $a_0a_k\not =0$, into (4.26) or (4.27) and noting that $b_3=c^2$, we finally get

(4.28)\begin{equation} \sum_{i=0}^k\left[(2ic+i^2)a_ie^{iz}+c_0(2c+2i+m+1)a_ie^{(m+1+i)z}-b_2a_ie^{(i+s)z}\right]=0, \end{equation}

when $q=0$, and

(4.29)\begin{equation} \begin{aligned} & \sum_{i=0}^k(2ic+i^2)a_ie^{iz}+2\left(\sum_{i=0}^k(c+i)a_ie^{iz}\right)\left(\sum_{j=0}^{q}c_je^{(m+1-jt)z}\right)\\ & \quad+\left(\sum_{i=0}^ka_ie^{iz}\right)\left\{\sum_{j=0}^{q}[C_{k_j}e^{[m+1-(q+1)t]z}+(m+1-jt)c_j]e^{(m+1-jt)z}\right\}=0, \end{aligned} \end{equation}

when $q\geq 1$. Note that the inequality $(2q-1)/(2q)< s/l\leq [2(q+1)-1]/[2(q+1)]$ implies $qt< m+1\leq (q+1)t$, where the equality holds when $q=m$. The left-hand side of equations (4.28) and (4.29) are polynomials in $e^{z}$ of degree $k+m+1$ and thus all coefficients of these two polynomials vanish. When $s=2(m+1)-t<2m+1$, we have $q< m$ and $t\geq 2$. By looking at the highest-degree term in the resulting polynomial and noting that $a_k\not =0$, we find

(4.30)\begin{equation} m+2c+2k+1=0. \end{equation}

Similarly, when $s=2m+1$, we have $q=m$ and $t=1$ and find

(4.31)\begin{equation} C_{m+1}+(m+2c+2k+1)c_0=0. \end{equation}

Let $c_+$ or $c_{-}$ be any square root of $b_3$. We may write $f_1=\kappa _{1c}e^{h_{1c}}$ and $f_2=\kappa _{2c}e^{h_{2c}}$, where $\kappa _{1c}=\kappa _1e^{c_+z}$ and $\kappa _{2c}=\kappa _2e^{c_{-}z}$ and $\kappa _{1}$ and $\kappa _{2}$ are two polynomials in $e^z$ of degrees $k_1$ and $k_2$, respectively. Moreover, $h_{1c}$ satisfies $h_{1c}'=\sum _{j=0}^{q}c_je^{(m+1-jt)z}$. Since $c_0^2=1$ and since $2c_0c_1=b_2$ when $q\ge 1$, we easily deduce from (4.24) that $h_{2c}=\pm h_{1c}$. Recall the elementary Wronskian determinant: $f_1'f_2-f_1f_2'=D$, where $D$ is a nonzero constant (see [Reference Laine23]). Then we have $(f_1/f_2)'=D/f_2^2$. If $h_{2c}=h_{1c}$, then $f_1/f_2$ is of finite order while $f_1^2$ is of infinite order, a contradiction. Therefore, $h_{2c}=-h_{1c}$. We may suppose that $c_+=c$. When $s<2m+1$, from (4.30) we deduce that $c_+=c_{-}=c$, which implies that $-2c=m+1+2k_1$ is a positive integer and hence $k_1=k_2$; when $s=2m+1$, using (4.25) and the relation $2c_0c_1=b_2$ we deduce from (4.31) that $c_++c_{-}+m+1+k_1+k_2=0$, which implies that $c_+=c_{-}=c$ and hence $-2c=m+1+k_1+k_2$ is a positive integer. Now $\rho _1-\rho _2=-2c$ is a positive integer. Together with (4.19) and previous preparations, we conclude that equation (4.13) admits in the broken plane $\mathbb {C}^{-}$ two linearly independent solutions of the form $u_1=x^{\rho _1}v_1$ and $u_2=x^{\rho _2}v_2$, where $v_1$ and $v_2$ are two entire functions such that $v_1(0)\not =0$ and $v_2(0)\not =0$, so that

(4.32)\begin{equation} \begin{aligned} f_1 & =x^{c}\kappa_{11}e^{h_{11}}=x^{{-}1/2}\left(D_1x^{\rho_1}v_1+D_2x^{\rho_2}v_2\right),\\ f_2 & =x^{c}\kappa_{12}e^{{-}h_{11}}=x^{{-}1/2}\left(D_3x^{\rho_1}v_1+D_4x^{\rho_2}v_2\right), \end{aligned} \end{equation}

where $D_j$ are constants, $h_{11}=\sum _{j=0}^{q}\frac {c_j}{m+1-jt}x^{m+1-jt}$, and $\kappa _{11}$ and $\kappa _{12}$ are two polynomials of degrees $k_1$ and $k_2$, respectively, such that $\kappa _{11}(0)\not =0$ and $\kappa _{12}(0)\not =0$. Noting $\rho _1=(1-2c)/2$ and $\rho _2=(1+2c)/2$, we see from (4.22) that $D_2D_4\not =0$. Obviously, $E:=D_1D_4-D_2D_3\not =0$. From equation (4.32) we get

(4.33)\begin{equation} u_1=x^{\rho_1}v_1=\frac{1}{E}x^{1/2}x^{c}\left(D_4\kappa_{11}e^{h_{11}}-D_2\kappa_{12}e^{{-}h_{11}}\right). \end{equation}

Since $v_1$ is an entire function with $v_1(0)\not =0$, we see from (4.33) that the function $w:=D_4\kappa _{11}e^{2h_{11}}-D_2\kappa _{12}$ has a zero of order $-2c$ at the point $z=0$ and so $w(0)=w'(0)=\cdots =w^{(-2c-1)}(0)=0$. Denote

(4.34)\begin{equation} \begin{aligned} \kappa_{11} & =a_{1,0}+a_{1,1}x+\cdots+a_{1,k_1}x^{k_1},\\ \kappa_{12} & =a_{2,0}+a_{2,1}x+\cdots+a_{2,k_2}x^{k_2}, \end{aligned} \end{equation}

where $a_{1,0},\,a_{2,0},\,a_{1,k_1},\,a_{1,k_2}\not =0$. $w(0)=0$ implies that $D_4a_{1,0}=D_2a_{2,0}$. Supposing that $a_{1,0}=a_{2,0}=1$, we have $D_4=D_2$. Below we consider the case when $m\geq 1$.

Consider first the case when $s/l<1/2$. Since $m\geq 1$, by theorem 1.1 and (4.30), we see that $k_1=k_2\geq 1$. Now $w^{(m+1)}(0)=0$ implies that $a_{1,m+1}+2(m!)c_0=a_{2,m+1}$. Here $a_{1,m+1}=0$ if $m+1>k_1$ and so is for $a_{2,m+1}$. Obviously, $m+1\leq k_1$. For each of $f_1$ and $f_2$, denoted by $f$, we may write $f=\kappa _ce^{h_c}$. Then we have equation (4.28). Note that $1\leq s=2(m+1)-t\leq m$. The left-hand side of equation (4.28) is a polynomial in $e^{z}$ of degree $m+1+k$ and thus all coefficients of this polynomial vanish. Denoting $a_{-m-1}=\cdots =a_{-2}=a_{-1}=0$ and $a_{k+1}=\cdots =a_{k+m}=0$, we obtain from equation (4.28) that $c_0(2c+2k+m+1)a_k=0$, which yields (4.30), and

(4.35)\begin{equation} c_0(2c+2i-m-1)a_{i-m-1}=b_2a_{i-s}-(2ic+i^2)a_{i}, \quad i=0,\cdots,k+m. \end{equation}

By substituting $2c=-2k-m-1$ into the equations in (4.35) we get

(4.36)\begin{equation} 2c_0(k-i+m+1)a_{i-m-1}={-}b_2a_{i-s}+(2ic+i^2)a_i, \quad i=0,\cdots,k+m. \end{equation}

Since $1\leq s\leq m$, by letting $i=1,\,2,\,\cdots,\,m$, we see that $a_1/a_0=K_1$, $\cdots$, $a_m/a_0=K_m$ for some constants $K_1,\,\cdots,\,K_m$ independent from $c_0$. Then by letting $i=m+1$ in (4.36) together with the relation $a_{1,m+1}+2(m!)c_0=a_{2,m+1}$, we get $2k_1/(m+1)(2c+m+1)+2(m!)=-2k_2/(m+1)(2c+m+1)$. Since $k_1=k_2$ and $-2c=2k_1+m+1$, we get $(m+1)!=1$, which is impossible when $m\geq 1$.