\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 143, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/143\hfil Second-order complex linear differential equations]
{Second-order complex linear differential equations with special functions or extremal 
 functions as coefficients}

\author[X. Wu, J. Long, J. Heittokangas, K. Qiu \hfil EJDE-2015/143\hfilneg]
{Xiubi Wu, Jianren Long, Janne Heittokangas, Ke-e Qiu}

\address{Xiubi Wu \newline
School of Science, Guizhou University, Guiyang 550025,  China}
\email{basicmath@163.com}

\address{Jianren Long (corresponding author)\newline
School of Mathematics and Computer Science, Guizhou  Normal
University, Guiyang 550001, China.\newline
Department of Physics and Mathematics, 
University of Eastern Finland, 
P.O. Box 111, 80101 Joensuu, Finland}
\email{longjianren2004@163.com, jianren.long@uef.fi }

\address{Janne Heittokangas \newline
Department of Physics and Mathematics, 
University of Eastern Finland, 
P.O. Box 111, 80101 Joensuu, Finland}
\email{janne.heittokangas@uef.fi}

\address{Ke-e Qiu \newline
School of Mathematics and Computer Science, Guizhou Normal
Colleage, Guiyang 550018, China}
\email{qke456@sina.com}

\thanks{Submitted November 17, 2014. Published May 22, 2015.}
\subjclass[2010]{34M10, 30D35}
\keywords{Complex differential equation; entire function; infinite order;
\hfill\break\indent Denjoy's conjecture; Yang's inequality}

\begin{abstract}
 The classical problem of finding conditions on the entire coefficients 
 $A(z)$ and $B(z)$ guaranteeing that all nontrivial solutions of 
 $f''+A(z)f'+B(z)f=0$ are of infinite order is discussed. 
 Two distinct approaches are used. In the first approach the coefficient 
 $A(z)$ itself is a solution of a differential equation $w''+P(z)w=0$, 
 where $P(z)$ is a polynomial. This assumption yields stability on 
 the behavior of $A(z)$ via Hille's classical method on asymptotic integration. 
 In this case $A(z)$ is a special function of which the Airy integral 
 is one example. The second approach involves extremal functions. 
 It is assumed that either $A(z)$ is extremal for Yang's inequality 
 or $B(z)$ is extremal for Denjoy's conjecture. A combination of these 
 two approaches is also discussed.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction and main results}


It is well known that if $A(z)$ is an entire function, $B(z)\not\equiv 0$ 
is a transcendental entire function, and $f_1,f_2$ are two linearly 
independent solutions of the equation
\begin{equation}\label{e1.1}
f''+A(z)f'+B(z)f=0,
\end{equation}
then at least one of $f_1, f_2$ must have infinite order. Hence,
``most'' solutions of \eqref{e1.1} have infinite order. On the
other hand, there are equations of the form \eqref{e1.1} that possess
a nontrivial solution of finite order; for example,
$f(z)=e^z$ satisfies $f''+e^{-z}f'-(e^{-z}+1)f=0$. Thus a natural question is:
 What conditions on $A(z)$ and $B(z)$ will guarantee that every nontrivial
solution of \eqref{e1.1} has infinite order?

We denote the order and the lower order of an entire function $f$ by 
$\rho (f)$ and $\mu (f)$, respectively. The standard notation and 
basic results in Nevanlinna theory of meromorphic functions can be 
found in \cite{h,l,y}.

From the work by Gundersen \cite{g.2}, Hellerstein, Miles and Rossi \cite{h.j.j}, 
and Ozawa \cite{o}, we know that if $A(z)$ and $B(z)$ are entire
functions with $\rho(A)<\rho(B)$; or $A(z)$ is a polynomial and
$B(z)$ is transcendental; or $\rho(B)<\rho(A)\leq \frac{1}{2}$, then
every nontrivial solution of \eqref{e1.1} has infinite
order. Therefore, the main problem left to consider is that whether
every nontrivial solution of \eqref{e1.1} has infinite order
if $\rho(A)=\rho(B)$ or if $\rho(A)>\frac{1}{2}$,
$\rho(B)<\rho(A)$. In general, the conclusions are false
for these situations. For example, $f(z)=\exp(P(z))$ satisfies the
equation
\begin{equation}\label{e1.2}
f''+A(z)f'+(-P''-(P')^{2}-A(z)P')f=0,
\end{equation}
where $A(z)$ is an entire function and $P(z)$ is a nonconstant
polynomial. For the case of $\rho(B)<\rho(A)$, there are also some
examples \cite{g.2} showing that a nontrivial solution of \eqref{e1.1}
has finite order.

The problem of finding conditions on $A(z)$ and $B(z)$ under which all 
nontrivial solutions of \eqref{e1.1} are of infinite order has raised 
considerable interest, see, for example, \cite{l}. 
Recently, this problem was studied by using a new idea that a coefficient 
of \eqref{e1.1} is a solution of another equation.

\begin{theorem}[\cite{w.w}]\label{thm1}
Let $A(z)$ be a nontrivial solution of $w''+P(z)w=0$, where 
$P(z)=a_{n}z^{n}+\dots+a_0, a_{n}\neq 0$. Let $B(z)$ be a transcendental
entire function with $\rho(B)<\frac{1}{2}$. Then every nontrivial solution 
of \eqref{e1.1} is of infinite order.
\end{theorem}

From Bank and Laine's result \cite[Theorem 1]{b.l}, we known that 
$\rho(A)=\frac{n+2}{2}$, and hence $\rho(A)>\rho(B)$ in Theorem~\ref{thm1}. 
On the other hand, we known that every nontrivial solution of \eqref{e1.1} 
is of infinite order when $\rho(A)<\rho(B)$ by Gundersen's result 
\cite[Theorem 2]{g.2}.
The fact that $A(z)$ solves an equation of the form $w''+P(z)w=0$ makes 
$A(z)$ a special function. In the particular cases when $P(z)=-z$ 
or $P(z)=-z^{n}$, the solution $A(z)$ is known as the Airy integral or a 
generalization of the Airy integral \cite{g.3}. Another special case is 
the Weber-Hermite function, which is a solution in the case 
$P(z)=\nu+\frac{1}{2}-\frac{z^{2}}{4}$, where $\nu$ is a constant. 
In the case when $P(z)$ is an arbitrary polynomial, Hille's classical method 
on asymptotic integration will become available. 
The consequences are summarized in Lemma~\ref{lem2.1} below. 
This background motivated the second and the fourth author to prove the 
following result.


\begin{theorem}[\cite{l.q}]\label{thm2} 
Let $A(z)$ be given as in Theorem~\ref{thm1}, and let $B(z)$ be a 
transcendental entire function with $\mu(B)<\frac{1}{2}$ and 
$\rho(A)\neq\rho(B)$. Then every nontrivial solution of \eqref{e1.1} 
is of infinite order.
\end{theorem}

Theorem \ref{thm2} is proved by using the $\cos \pi\rho$ theorem due to 
Barry \cite{b}, which does not work for entire functions with lower 
order (or order) not less than $1/2$. Thus we need new ideas when the 
lower order (or order) of the coefficients is not less than $1/2$. 
In the present paper, we will prove the following improvement of
 Theorem~\ref{thm2} by using a modification of the 
Phragm\'{e}n-Lindel\"{o}f principle, as well as Hille's classical results 
on asymptotic integration.

\begin{theorem}\label{th1} 
Let $A(z)$ be given as in Theorem \ref{thm1}, and let $B(z)$ be a 
transcendental entire function with $\mu(B)<\frac{1}{2}+\frac{1}{2(n+1)}$ 
and $\rho(A)\neq\rho(B)$. Then every nontrivial solution of \eqref{e1.1} 
is of infinite order.
\end{theorem}

In 1988, Gundersen proved the following result.

\begin{theorem}[\cite{g.2}]\label{thm3} 
Let $\{\phi_{k}\}$ and $\{\theta_{k}\}$ be two finite collections of real 
numbers that satisfy 
$\phi_1<\theta_1<\phi_2<\theta_2<\dots<\phi_{n}<\theta_{n}<\phi_{n+1}$ 
where $\phi_{n+1}=\phi_1+2\pi$, and set 
$\nu=\max_{1\leq k\leq n}\{\phi_{k+1}-\theta_{k}\}$. 
Suppose that $A(z)$ and $B(z)$ are entire functions such that for some constant 
$\alpha>0$,
\begin{equation}\label{e1.3}
|A(z)|=O(|z|^{\alpha})
\end{equation}
as $z\to \infty$ in $\phi_{k}\leq \arg z\leq\theta_{k}$ for $k=1, 2, \ldots, n$,
and where $B(z)$ is transcendental with $\rho(B)<\pi/\nu$.
Then every nontrivial solution of \eqref{e1.1} is of infinite order.
\end{theorem}

The usual order $\rho(B)$ in Theorem~\ref{thm3} can be replaced with the 
lower order $\mu(B)(\leq\rho(B))$.

\begin{theorem}\label{th2} 
Let $\{\phi_{k}\}$, $\{\theta_{k}\}$, $\nu$ and $A(z)$ be given as in 
Theorem~\ref{thm3}, and let $B(z)$ be transcendental with $\mu(B)<\pi/\nu$.
 Then every nontrivial solution of \eqref{e1.1} is of infinite order.
\end{theorem}

The proof of Theorem~\ref{th2} deviates from that of Theorem~\ref{thm3} 
in the sense that we require a modification of the Phragm\'{e}n-Lindel\"{o}f
 principle, see Lemma~\ref{lem3.2} below. In addition, we make use 
of the $\cos\pi\rho$ theorem, which is not needed in proving Theorem~\ref{thm3}.

We proceed to consider conditions on the coefficients $A(z)$ and $B(z)$ 
involving value distribution instead of just growth. We begin by recalling 
a conjecture due to Denjoy \cite{d} from 1907, verified by Ahlfors \cite{a} in 1930.

\subsection*{Denjoy's Conjecture}
\emph{Let $f$ be an entire function of finite order $\rho$. If $f$ has $k$ 
distinct finite asymptotic values, then $k\leq 2\rho$.}


An entire function $f$ is called extremal for Denjoy's conjecture if it is 
of finite order $\rho$ and has $k=2\rho$ distinct finite asymptotic values. 
These functions are investigated by Ahlfors \cite{a}, Drasin \cite{dr},
Kennedy \cite{k} and Zhang \cite{z}, to mention a few.
An example of a function extremal for Denjoy's conjecture is
\begin{equation} \label{e1.4}
f(z)=\int^z_0\frac{\sin t^{q}}{t^{q}}dt,
\end{equation}
where $q>0$ is an integer. We know that the order of $f$ equals to $q$,
and $f$ has $2q$ distinct finite asymptotic values
\[
a_{l}=e^{\frac{l\pi i}{q}}\int^{\infty}_0\frac{\sin r^{q}}{r^{q}}dr,
\]
where $l=1, 2, \ldots, 2q$, see \cite[p. 210]{z2}.


\begin{theorem}\label{th3} 
Let $A(z)$ be given as in Theorem~\ref{thm1}, and let $B(z)$ be a 
function extremal for Denjoy's conjecture and $\rho(A)\neq\rho(B)$. 
Then every nontrivial solution of \eqref{e1.1} is of infinite order.
\end{theorem}

We recall the definition of Borel direction as follows \cite{y.d}.

\begin{definition} \rm
Let $f$ be a meromorphic function in the finite complex plane $\mathbb{C}$ 
with $0<\mu(f)<\infty$. A ray $\arg z=\theta\in [0, 2\pi)$ from the origin 
is called a Borel direction of order $\geq\mu(f)$ of $f$, if for any positive 
number $\varepsilon>0$ and for any complex number $a\in \mathbb{C}\bigcup\{\infty\}$, 
possibly with two exceptions, the following inequality holds
\begin{equation}\label{e1.5}
\limsup_{r\to \infty}\frac
{\log n(S(\theta-\varepsilon,\theta+\varepsilon, r), a, f)}{\log
r}\geq\mu(f),
\end{equation}
where $n(S(\theta-\varepsilon, \theta+\varepsilon, r), a, f)$
denotes the number of zeros, counting the multiplicities, of $f-a$
in the region $
S(\theta-\varepsilon, \theta+\varepsilon, r)
=\{z:\theta-\varepsilon<\arg z<\theta+\varepsilon, |z|<r\}$.
\end{definition}

The definition of Borel direction of order $\rho(f)$ of $f$ can be found 
in \cite[p. 78]{z2}, it is defined similarly with the only exception 
that ``$\geq\mu(f)$'' in \eqref{e1.5} is to be replaced with ``$=\rho(f)$''.

In the sequel we will require the following result, known as Yang's inequality,
 on general value distribution theory.

\begin{theorem}[\cite{y.d}]\label{thm4} 
Suppose that $f$ is an
entire function of finite lower order $\mu >0$. Let $q<\infty$
denote the number of Borel directions of order $\geq\mu$, and let $p$
denote the number of finite deficient values of $f$. Then $p\leq q/2$.
\end{theorem}

An entire function $f$ is called
extremal for Yang's inequality if $f$ satisfies the
assumptions of Theorem~\ref{thm4} with $p=\frac{q}{2}$. 
These functions were introduced in \cite{w}.
The simplest entire function extremal for Yang's inequality is
$e^z$. A slightly more complicated example is
$f(z)=\int^z_0e^{-t^{n}}dt$, $n\geq 2$ is teger, which has $n$ deficient values
\[
a_{l}=e^{i\frac{2\pi l}{n}}\int^{\infty}_0e^{-t^{n}}dt,\quad l=1,
2,\ldots, n,
\]
and $q=2n$ Borel directions $\arg z=\frac{2k-1}{2n}\pi$, $k=1,
2,\ldots, 2n$, see \cite[pp. 210-211]{y} for more details.


\begin{theorem}[\cite{l.w.z}]\label{thm5} 
Let $A(z)$ be an entire function extremal for Yang's inequality,
 and let $B(z)$ be a transcendental entire function such that $\rho(B)\neq\rho(A)$. 
Then every nontrivial solution of \eqref{e1.1} is of infinite order.
\end{theorem}

Also here the usual order $\rho(B)$ can be replaced with the lower order $\mu(B)$.

\begin{theorem}\label{th4}
Let $A(z)$ be an entire function extremal for Yang's inequality, and let
 $B(z)$ be a transcendental entire function such that $\mu(B)\neq\rho(A)$. 
Then every nontrivial solution of \eqref{e1.1} is of infinite order.
\end{theorem}

Let $\lambda(A)$ be the converge exponent of the zero sequence of $A(z)$. 
By Lemma~\ref{lem2.1} below and by similar reasoning used in proving 
Theorems~\ref{thm5} and \ref{th4}, we can easily obtain the following result.

\begin{theorem}\label{th5} 
Let $A(z)$ be given as in Theorem~\ref{thm1} with $\lambda(A)<\rho(A)$, 
and let $B(z)$ be a transcendental entire function satisfying one of the 
following conditions.
\begin{itemize}
\item [\textnormal{(1)}] $\rho(B)\neq\rho(A)$,
\item [\textnormal{(2)}] $\mu(B)\neq\rho(A)$.
\end{itemize}
Then every nontrivial solution of \eqref{e1.1} is of infinite order.
\end{theorem}


\section{Auxiliary results}

 Let $\alpha<\beta$ be such that
$\beta-\alpha <2\pi$, and let $r>0$. Denote
\begin{align*}
&S(\alpha,\beta)=\{z:\alpha <\arg z< \beta\},\\&
S(\alpha,\beta,r)=\{z:\alpha <\arg z < \beta\}\cap \{z:|z|< r\}.
\end{align*}
Let $\overline{F}$ denote the closure of $F$.
Let $f$ be an entire function of order $\rho(f)\in (0, \infty)$. For simplicity, set
$\rho=\rho(f)$ and $S=S(\alpha,\beta)$. We say that $f$ blows up exponentially in
$\overline{S}$ if for any $\theta\in (\alpha, \beta)$
\begin{equation}\label{e2.1}
\lim_{r\to\infty}\frac{\log\log|f(re^{i\theta})|}{\log r}=\rho
\end{equation}
holds. We also say that $f$ decays to zero exponentially in $\overline{S}$
if for any $\theta\in (\alpha, \beta)$
\begin{equation}\label{e2.2}
\lim_{r\to\infty}\frac{\log\log|f(re^{i\theta})|^{-1}}{\log r}=\rho
\end{equation}
holds.

The following lemma, originally due to Hille \cite[Chapter 7.4]{hi},
see also \cite{g,s}, plays an important
role in proving our results. The method used in proving the lemma
is typically referred to as the method of asymptotic integration.

\begin{lemma}\label{lem2.1}
Let $f$ be a nontrivial solution of $f''+P(z)f=0$, where $P(z)=a_nz^n+\dots +a_0$,
$a_n\neq 0$. Set
$\theta_j=\frac{2j\pi-\arg(a_n)}{n+2}$ and $S_{j}=S(\theta_{j}, \theta_{j+1})$,
where $j=0,1,2,\ldots,n+1$ and $\theta_{n+2}=\theta_0+2\pi$. Then $f$ has the
following properties.
\begin{itemize}
\item [(1)] In each sector $S_{j}$, $f$ either blows up or decays to zero
exponentially.

\item [(2)] If, for some $j$,  $f$ decays to zero in $S_{j}$, then it must
blow up in $S_{j-1}$ and $S_{j+1}$.
However, it is possible for $f$ to blow up in many adjacent sectors.

\item [(3)] If $f$ decays to zero in $S_{j}$, then $f$ has at most finitely
many zeros in any closed sub-sector within $S_{j-1}\cup\overline{S_{j}}
\cup S_{j+1}$.

\item [(4)] If $f$ blows up in $S_{j-1}$ and $S_{j}$, then for each
$\varepsilon>0$, $f$ has infinitely many zeros in each sector
$\overline{S}(\theta_{j}-\varepsilon, \theta_{j}+\varepsilon)$, and
furthermore, as $r\to\infty$,
 $$
n(\overline{S}(\theta_{j}-\varepsilon,\theta_{j}+\varepsilon, r), 0, f)
=(1+o(1))\frac{2\sqrt{|a_n|}}{\pi(n+2)}r^{\frac{n+2}{2}},
$$
where $ n(\overline{S}(\theta_{j}-\varepsilon, \theta_{j}+\varepsilon, r),
0, f)$ is the number of zeros of $f$ in
the region $\overline{S}(\theta_{j}-\varepsilon,\theta_{j}+\varepsilon, r).$
\end{itemize}
\end{lemma}

The Lebesgue linear measure of a set $E\subset [0, \infty)$ is
$\operatorname{m}(E)=\int_{E}dt$, and the logarithmic measure of a set
$F\subset [1, \infty)$ is $\operatorname{m_{l}}(F)=\int_{F}\frac{dt}{t}$.
The upper and lower
logarithmic densities of $F\subset[1, \infty)$ are given, respectively, by
\begin{gather*}
\overline{\operatorname{log\,dens}}(F)=\limsup_{r\to\infty}
\frac{\operatorname{m_{l}}(F\cap[1,r])}{\log r} ,\\
\underline{\operatorname{log\,dens}}(F)=\liminf_{r\to
\infty}\frac{\operatorname{m_{l}}(F\cap[1,r])}{ \log r}.
\end{gather*}

A lemma on logarithmic derivatives due to Gundersen \cite{g.1} plays an important
role in proving our results.

\begin{lemma}\label{lem2.2}
Let $f$ be a transcendental meromorphic function of finite
order $\rho(f)$. Let $\varepsilon>0$ be a given real constant, and let
$k$ and $j$ be integers such that $k>j\geq 0$. Then
there exists a set $E\subset[0,2\pi)$ of linear measure zero,
such that if $\psi_0\in[0,2\pi)-E$,
then there is a constant $R_0=R_0(\psi_0)>0$ such that for all
$z$ satisfying $\arg z=\psi_0$ and $|z|\geq R_0$, we have
\begin{equation} \label{e2.3}
\Big|\frac{f^{(k)}(z)}{f^{(j)}(z)}\Big|\leq |z|^{(k-j)(\rho(f)-1+\varepsilon)}.
\end{equation}
\end{lemma}

The following result is due to Barry \cite{b}.

\begin{lemma}\label{lem2.3}
Let $f$ be an entire
function with $0\leq\mu(f)<1$. Then, for every $\alpha\in
(\mu(f),1)$, $\overline{\operatorname{logdens}}(\{r\in [1,\infty)
: m(r)>M(r)\cos \pi \alpha\})\geq 1-\frac{\mu(f)}{\alpha}$,
where $m(r)=\inf_{|z|=r} \log |f(z)|$, and
$M(r)=\sup_{|z|=r}\log |f(z)|$.
\end{lemma}

The following result was proved in \cite{g.2}.

\begin{lemma}\label{lem2.4}
Let $A(z)$ and $B(z)\not\equiv 0$ be two entire functions such that for real
constants $\alpha, \beta, \theta_1, \theta_2$, where $\alpha>0$, $\beta>0$
and $\theta_1<\theta_2$, we have
\begin{gather}\label{e2.4}
|A(z)|\geq \exp\{(1+o(1))\alpha|z|^{\beta}\},\\
\label{e2.5}
|B(z)|\leq \exp\{o(1)|z|^{\beta}\}
\end{gather}
as $z\to \infty$ in $\overline{S}(\theta_1, \theta_2)=\{z:
 \theta_1\leq \arg z\leq\theta_2\}$. Let $\varepsilon>0$ be a given small constant,
and let $\overline{S}(\theta_1+\varepsilon,
\theta_2-\varepsilon)=\{z: \theta_1+\varepsilon\leq \arg z\leq\theta_2-\varepsilon\}$.

If $f$ is a nontrivial solution of \eqref{e1.1} with $\rho(f)<\infty$, then
the following conclusions hold:
\begin{itemize}
\item [(1)] There exists a constant $b(\neq 0)$ such that $f(z)\to b$ as 
$z\to \infty$ in $\overline{S}(\theta_1+\varepsilon, \theta_2-\varepsilon)$. 
Furthermore,
\begin{equation}\label{e2.6}
|f(z)-b|\leq \exp\{-(1+o(1))\alpha|z|^{\beta}\}
\end{equation}
as $z\to \infty$ in $\overline{S}(\theta_1+\varepsilon, \theta_2-\varepsilon)$.

\item [(2)] For each integer $k > 1$,
\[
|f^{(k)}(z)|\leq \exp\{-(1+o(1))\alpha|z|^{\beta}\}
\]
as $z\to \infty$ in $\overline{S}(\theta_1+\varepsilon, \theta_2-\varepsilon)$.
\end{itemize}
\end{lemma}


\section{Modified Phragm\'{e}n-Lindel\"{o}f principle}

 We recall a result due to Phragm\'{e}n and Lindel\"{o}f \cite[Theorem 7.5]{m}.

\begin{lemma}\label{lem3.1} 
Let $f$ be an analytic function in $D$ and continuous in $\overline{D}$, 
where $D=S(\alpha, \beta)\cap\{z: |z|>r_0\}$, and $\alpha, \beta, r_0$ 
are constants such that $0<\beta-\alpha\leq 2\pi$ and $r_0>0$. Suppose that 
there exists a constant $M>0$ such that $|f(z)|\leq M$ for $z\in \partial D$. If
\[
\liminf_{r\to \infty}\frac{\log\log M(r, D, f)}{\log r}<\frac{\pi}{\beta-\alpha},
\]
where $M(r, D, f)=\max_{\substack{|z|=r\\ z\in D}}|f(z)|$, then 
$|f(z)|\leq M$ for all $z\in D$.
\end{lemma}

Next we introduce a key lemma in which the Phragm\'{e}n-Lindel\"{o}f principle 
is tailored to suit for our purposes.

\begin{lemma}\label{lem3.2} 
Let $f$ be an entire function of lower order $\mu(f)\in [\frac{1}{2}, \infty)$. 
Then there exists a sector $S(\alpha, \beta)=\{z: \alpha<\arg z<\beta\}$ with 
$\beta-\alpha\geq\frac{\pi}{\mu(f)}$, such that
\begin{align*}
\limsup_{r\to \infty}\frac{\log\log |f(re^{i\theta})|}{\log r}\geq \mu(f)
\end{align*}
holds for all the rays $\arg z=\theta \in(\alpha, \beta)$, where 
$0\leq\alpha<\beta\leq 2\pi$.
\end{lemma}

\begin{proof}
 Suppose on the contrary to the assertion that any sector $S(\alpha, \beta)$ 
with $\beta-\alpha\geq\frac{\pi}{\mu(f)}$ there exists at least one 
ray $\arg z=\psi_1\in (\alpha, \beta)$ such that
\begin{align*}
\limsup_{r\to \infty}\frac{\log\log |f(re^{i\psi_1})|}{\log r}=\mu_1<\mu(f),
\end{align*}
where $\mu_1$ is a constant.

Let $\psi_1'=\psi_1+\frac{\pi}{\mu(f)}$. 
From our assumption, there exists at least one ray
$\arg z=\psi_2\in (\psi_1, \psi_1')$, such that
\begin{equation}\label{e3.1}
\limsup_{r\to \infty}\frac{\log\log |f(re^{i\psi_2})|}{\log r}=\mu_2<\mu(f),
\end{equation}
where $\mu_2$ is a constant. Let $\eta_1=\max\{\mu_1, \mu_2\}$,
and let $\lambda_1\in (\eta_1, \mu(f))\cap \mathbb{Q}$ be a constant,
where $\mathbb{Q}$ denotes the set of rational numbers.
Suppose that $\omega_0=z^{\lambda_1}$ is the principal branch of
$\omega=z^{\lambda_1}$. Then $S(\psi_1, \psi_2)$ is mapped onto
a subsector of the right half-plane by transformation
$\zeta=e^{i\theta_1}z^{\lambda_1}$, where $\theta_1\in (0, 2\pi)$
is a constant depending on $\lambda_1, \psi_1$ and $\psi_2$.

Let $H(z)=\frac{f(z)}{\exp(e^{i\theta_1}z^{\lambda_1})}$. Then we obtain
\begin{equation} \label{e3.2}
\lim_{r\to \infty}|H(re^{i\psi_1})|=\lim_{r\to \infty}|H(re^{i\psi_2})|=0
\end{equation}
and
\begin{equation} \label{e3.3}
\begin{aligned}
\liminf_{r\to \infty}\frac{\log\log M(r, S(\psi_1, \psi_2), H)}{\log r} 
&\leq \liminf_{r\to \infty}\frac{\log\log M(r, S(\psi_1, \psi_2), f)}{\log r} \\
&\leq \liminf_{r\to \infty}\frac{\log\log M(r, f)}{\log r}\\
&=\mu(f)<\frac{\pi}{\psi_2-\psi_1}.
\end{aligned}
\end{equation}
By Lemma~\ref{lem3.1}, there exists a constant $M_1>0$ such that
\[
|H(z)|\leq M_1
\]
holds for all $z\in S(\psi_1, \psi_2)$; that is,
\begin{equation}\label{e3.4}
|f(z)|\leq M_1\exp(|z|^{\lambda_1})
\end{equation}
holds for all $z\in S(\psi_1, \psi_2)$.

Let $\psi_2'=\psi_2+\frac{\pi}{\mu(f)}$. From our assumption, there
 exists at least one ray $\arg z=\psi_{3}\in (\psi_2, \psi_2')$ such that
\[
\limsup_{r\to \infty}\frac{\log\log |f(re^{i\psi_{3}})|}{\log r}=\mu_{3}<\mu(f),
\]
where $\mu_{3}$ is a constant. Similarly as above, there exist constants 
$M_2>0, \lambda_2<\mu(f)$ and $\theta_2\in (0, 2\pi)$ such that
\begin{equation} \label{e3.5}
|f(z)|\leq M_2\exp(|z|^{\lambda_2})
\end{equation}
holds for all $z\in S(\psi_2, \psi_{3})$. We proceed in this way until there
exists a ray $\arg z=\psi_{m}$ such that
\begin{equation} \label{e3.6}
\limsup_{r\to \infty}\frac{\log\log |f(re^{i\psi_{m}})|}{\log r}=\mu_{m}<\mu(f)
\end{equation}
and $\psi_1+2\pi-\psi_{m}<\frac{\pi}{\mu(f)}$, where $\mu_{m}$ is a constant.
By the discussion above, there exist constants $M_{m-1}>0, \lambda_{m-1}<\mu(f)$
and $\theta_{m-1}\in (0, 2\pi)$ such that
\begin{equation}\label{e3.7}
|f(z)|\leq M_{m-1}\exp(|z|^{\lambda_{m-1}})
\end{equation}
holds for all $z\in S(\psi_{m-1}, \psi_{m})$.
 By \eqref{e3.4}, \eqref{e3.5}, \eqref{e3.7} and Lemma~\ref{lem3.1}, we have
\begin{equation} \label{e3.8}
|f(z)|\leq M\exp(|z|^{\lambda}) \quad \text{for all }z\in\mathbb{C},
\end{equation}
where $M=\max\{M_1, M_2, \ldots, M_{m-1}\}$ and
$\lambda=\max\{\lambda_1, \lambda_2, \ldots, \lambda_{m-1}\}<\mu(f)$.
By \eqref{e3.8}, we obtain
\begin{align*}
\mu(f)=\liminf_{r\to \infty}\frac{\log\log M(r,f)}{\log r}
\leq \lim_{r\to \infty}\frac{\log\log M+\lambda\log r}{\log r}=\lambda,
\end{align*}
which is a contradiction with the fact $\lambda<\mu(f)$.
This completes the proof.
\end{proof}


\section{Proofs of Theorems~\ref{th1} and \ref{th2}}

 We rely heavily on Phragm\'{e}n-Lindel\"{o}f principle and modified 
Phragm\'{e}n-Lindel\"{o}f principle.


\begin{proof}[Proof of Theorem~\ref{th1}]
 Since the case $\rho(A)<\rho(B)$ is proved in \cite{g.2}, we may assume
 $\rho(A)>\rho(B)$.
Suppose on the contrary to the assertion that there exists a nontrivial
solution $f$ of \eqref{e1.1} with $\rho(f) < \infty$. We aim for a contradiction.
 Set
$\theta_j=\frac{2j\pi-\arg(a_n)}{n+2}$ and $S_j=\{z:\theta_j<\arg z
<\theta_{j+1}\}$, where $j=0,1,2,\ldots,n+1$ and $\theta_{n+2}=\theta_0+2\pi$. 
We consider two cases appearing in Lemma~\ref{lem2.1}.
\smallskip

\noindent\textbf{Case 1:} Suppose that $A(z)$ blows up exponentially 
in each sector $S_j$, where $j=0, 1, \ldots, n+1$; that is, for any 
$\theta\in(\theta_{j}, \theta_{j+1})$, we have
\begin{equation} \label{e4.1}
\lim_{r\to\infty}\frac{\log\log|A(re^{i\theta})|}{\log r}=\rho(A)=\frac{n+2}{2}.
\end{equation}
Then for any given constant $\varepsilon\in (0, \frac{\pi}{4\rho(A)})$ and $\eta\in (0, \frac{\rho(A)-\rho(B)}{4})$, we have
\begin{gather}\label{e4.2}
|A(z)|\geq \exp\{(1+o(1))\alpha|z|^{\frac{n+2}{2}-\eta}\},\\
\label{e4.3}
|B(z)|\leq \exp(|z|^{\rho(B)+\eta})\leq \exp(|z|^{\rho(A)-2\eta})
\leq\exp\{o(1)|z|^{\frac{n+2}{2}-\eta}\}
\end{gather}
as $z\to \infty$ in $S_{j}(\varepsilon)=\{z:\theta_j+\varepsilon<\arg z
<\theta_{j+1}-\varepsilon\}$, $j=0, 1, \ldots, n+1$,
 where $\alpha$ is a positive constant depending on $\varepsilon$.
Combining \eqref{e4.2}, \eqref{e4.3}, and Lemma~\ref{lem2.4}, there exist
corresponding constants $b_{j}\neq 0$ such that
\begin{equation}\label{e4.4}
|f(z)-b_{j}|\leq\exp\{-(1+o(1))\alpha|z|^{\frac{n+2}{2}-\eta}\}
\end{equation}
as $z\to \infty$ in $S_{j}(2\varepsilon)$, $j=0, 1, \ldots, n+1$.
 Therefore, $f$ is bounded in the whole complex plane by the
Phragm\'{e}n-Lindel\"{o}f principle. So $f$ is a nonzero constant in
the whole complex plane by Liouville's theorem. This contradicts with
the fact that equation \eqref{e1.1} doesn't have nonzero constant solutions.
\smallskip

\noindent\textbf{Case 2:} 
There exists at least one sector of the $n+2$ sectors, such that $A(z)$ 
decays to zero exponentially, say 
$S_{j_0}=\{z: \theta_{j_0}<\arg z<\theta_{j_0+1}\}$, $0\leq j_0\leq n+1$.
 That is, for any $\theta\in (\theta_{j_0}, \theta_{j_0+1})$, we have
\begin{equation}\label{e4.5}
\lim_{r\to\infty}\frac{\log\log\frac{1}{|A(re^{i\theta})|}}{\log r}=\frac{n+2}{2}.
\end{equation}

If $\mu(B)<\frac{1}{2}$, the assertion follows by Theorem~\ref{thm2}.

If $\frac{1}{2}\leq \mu(B)<\frac{1}{2}+\frac{1}{2(n+1)}$, then by 
Lemma~\ref{lem3.2}, there exists a sector 
$S(\alpha, \beta)=\{z: \alpha<\arg z<\beta\}$ with
 $\beta-\alpha\geq \frac{\pi}{\mu(B)}>\frac{\pi}{\frac{1}{2}
+\frac{1}{2(n+1)}}=2\pi-\frac{2\pi}{n+2}$, such that
\begin{equation}\label{e4.6}
\limsup_{r\to\infty}\frac{\log\log|B(re^{i\theta})|}{\log r}\geq \mu(B)
\end{equation}
holds for any $\theta\in(\alpha, \beta$). Thus, there exists a subsector 
$S(\alpha', \beta')$, such that for any $\theta\in(\alpha', \beta'$) 
we have \eqref{e4.5} and \eqref{e4.6}.

By Lemma~\ref{lem2.2}, there exists a set $E_1\subset[0,2\pi)$ of
 linear measure zero, such that if $\psi_0\in[0,2\pi)-E_1$,
then there is a constant $R_0=R_0(\psi_0)>1$ such that for all
$z$ satisfying $\arg z=\psi_0$ and $|z|\geq R_0$, we have
\begin{equation}\label{e4.7}
\Big|\frac{f^{(k)}(z)}{f(z)}\Big|\leq |z|^{2\rho(f)},\quad k=1, 2.
\end{equation}
Thus, there exists a sequence of points $z_{n}=r_{n}e^{i\theta}$ with 
$r_{n}\to \infty$ as $n\to\infty$ and $\theta\in (\alpha', \beta')-E_1$, 
such that \eqref{e4.5}, \eqref{e4.6} and \eqref{e4.7} hold.

Combining \eqref{e4.5}, \eqref{e4.6}, \eqref{e4.7} and \eqref{e1.1}, for every $n>n_0$, we have
\begin{equation}\label{e4.8}
\begin{aligned}
\exp(r_{n}^{\mu(B)-\varepsilon})
&\leq|B(r_{n}e^{i\theta})|\\
&\leq\Big|\frac{f''(r_{n}e^{i\theta})}{f(r_{n}e^{i\theta})}\Big|
+|A(r_{n}e^{i\theta})|\Big|\frac{f'(r_{n}e^{i\theta})}{f(r_{n}e^{i\theta})}\Big| \\
&\leq r_{n}^{2\rho(f)}(1+o(1)).
\end{aligned}
\end{equation}
Obviously, this is a contradiction for sufficiently large $n$ and arbitrary 
small $\varepsilon$. Therefore we have $\rho(f)=\infty$ for every nontrivial 
solution $f$ of \eqref{e1.1}. This completes the proof. 
\end{proof}


\begin{proof}[Proof of Theorem~\ref{th2}]
 Assume on the contrary to the assertion that there is a nontrivial
solution $f$ of \eqref{e1.1} with $\rho(f)=\rho < \infty$.
\smallskip

\noindent\textbf{Case 1:} Suppose first that $\mu(B)>0$.
By Lemma~\ref{lem2.2}, there exists a set $E_2\subset[0,2\pi)$ of linear 
measure zero, such that if $\varphi\in [\phi_{k}, \theta_{k}]-E_2$ for 
some $k$, $1\leq k\leq n$, we have
\begin{equation}\label{e4.9}
\Big|\frac{f'(re^{i\varphi})}{f(re^{i\varphi})}\Big|=O(r^{\rho}),\quad
\Big|\frac{f''(re^{i\varphi})}{f(re^{i\varphi})}\Big|=O(r^{2\rho}),
\end{equation}
as $r\to\infty$ along $\arg z=\varphi$.
Combining \eqref{e4.9}, \eqref{e1.1}, and our assumption, we have
\begin{equation}\label{e4.10}
|B(re^{i\varphi})|\leq \Big|\frac{f''(re^{i\varphi})}{f(re^{i\varphi})}\Big|
+|A(re^{i\varphi})|\Big|\frac{f'(re^{i\varphi})}{f(re^{i\varphi})}\Big|=O(r^{\sigma})
\end{equation}
in each $[\phi_{k}, \theta_{k}]-E_2$, $1\leq k\leq n$, as $r\to\infty$, where 
$\sigma=\alpha+2\rho$.

For any given $\varepsilon\in (0, \min\{\frac{\pi}{2\mu(B)}
-\frac{\nu}{2}, \frac{\mu(B)}{4}\})$, for any 
$\varphi'\in (\theta_{k}-\varepsilon, \theta_{k})-E_2$, 
$\varphi''\in (\phi_{k+1}, \phi_{k+1}+\varepsilon)-E_2$, $k=1, 2, \ldots, n$, we obtain
\begin{align*}
|B(re^{i\varphi'})|=O(r^{\sigma}),~ |B(re^{i\varphi''})|=O(r^{\sigma}),
\end{align*}
as $r\to\infty$.

Similarly as in the proof of Lemma~\ref{lem3.2}, let 
$H(z)=\frac{B(z)}{\exp(az^{\lambda})}$, where 
$\lambda\in (0, \mu(B)-4\varepsilon)\cap \mathbb{Q}$, $\mathbb{Q}$ denotes 
the set of rational numbers, $a=e^{i\tau}$, $\tau\in (0, 2\pi)$ is a constant 
depending on $\lambda$, $\varphi'$ and $\varphi''$,  and $z^{\lambda}$ 
denotes the principal branch. Note that
\[
\varphi''-\varphi'<\phi_{k+1}-\theta_{k}+2\varepsilon<\nu+2\varepsilon
<\frac{\pi}{\mu(B)}<\frac{\pi}{\lambda}.
\]
So, $S(\varphi', \varphi'')$ is mapped onto a subsector of the right half-plane 
by $\zeta=az^{\lambda}$. Thus, we obtain
\[
\lim_{r\to \infty}|H(re^{i\varphi'})|=\lim_{r\to \infty}|H(re^{i\varphi''})|=0
\]
and
\begin{align*}
\liminf_{r\to \infty}\frac{\log\log M(r, S(\varphi', \varphi''), H)}{\log r}
&\leq \liminf_{r\to \infty}\frac{\log\log M(r, S(\varphi', \varphi''), B)}{\log r}\\
&\leq \liminf_{r\to \infty}\frac{\log\log M(r, B)}{\log r}\\
&=\mu(B)<\frac{\pi}{\varphi''-\varphi'}.
\end{align*}
By Lemma~\ref{lem3.1}, there exists a constant $M>0$ such that
\[
|H(z)|\leq M
\]
for all $z\in S(\varphi', \varphi'')$; that is,
\begin{equation}\label{e4.11}
|B(z)|\leq M\exp(|z|^{\lambda})
\end{equation}
for all $z\in S(\varphi', \varphi'')$. So we have \eqref{e4.11} in each 
$(\theta_{k}-\varepsilon, \phi_{k+1}+\varepsilon)-E_2$, $1\leq k\leq n$. 
By \eqref{e4.10}, \eqref{e4.11} and Phragm\'{e}n-Lindel\"{o}f principle, we obtain
\[
|B(z)|\leq M\exp(|z|^{\lambda})
\]
for all $z\in \mathbb{C}$. Thus, we have
\[
\mu(B)=\liminf_{r\to \infty}\frac{\log\log M(r,B)}{\log r}
\leq \lim_{r\to \infty}\frac{\log\log M+\lambda\log r}{\log r}=\lambda,
\]
which contradicts with the fact that $\lambda<\mu(B)$.
\smallskip

\noindent\textbf{Case 2:} 
Suppose that $\mu(B)=0$. By using Lemma~\ref{lem2.3}, there exists a 
set $E_{3}\subset[1, \infty)$ with $\overline{\operatorname{log\,dens}}(E_{3})=1$ 
such that for all $z$ satisfying $|z|=r\in E_{3}$, we have
\begin{equation}\label{e4.12}
\log|B(z)|>\frac{\sqrt{2}}{2}\log M(r, B),
\end{equation}
where $M(r, B)=\max_{|z|=r}|B(z)|$.

It follows from \eqref{e1.1}, \eqref{e1.3}, \eqref{e4.9} and \eqref{e4.12} 
that there exists a sequence $(R_{n})$ with $R_{n}\to\infty$ as $n\to\infty$, 
such that
\begin{equation}\label{e4.13}
M(R_{n}, B)^{\sqrt{2}/2}<|B(R_{n}e^{i\varphi})|
\leq R_{n}^{2\rho(f)}(1+R_{n}^{\alpha}),
\end{equation}
as $n\to\infty$, $\varphi\in \cup_{k=1}^{n}[\phi_{k}, \theta_{k}]-E_2$. 
However, $B(z)$ is a transcendental entire function, so that
\[
\liminf_{r\to\infty}\frac{\log M(r, B)}{\log r}=\infty,
\]
which contradicts with \eqref{e4.13}. This completes the proof.
\end{proof}

\section{Proof of Theorem~\ref{th3}}

 We begin by recalling some properties satisfied by entire functions 
that are extremal for Denjoy's conjecture.

\begin{lemma}[{\cite[Theorem 4.11]{z2}}]\label{lem5.1} 
Let $f$ be an entire function extremal for Denjoy's conjecture. 
Then, for any $\theta\in (0, 2\pi)$, either $\Delta(\theta)$ is a Borel 
direction of order $\rho(f)$ of $f$ or there exists a constant 
$\sigma (0<\sigma<\frac{\pi}{4})$, such that
\[
\lim_{\substack{|z|\to\infty\\ z\in (S(\theta-\sigma, \theta+\sigma)-E)}}
\frac{\log\log|f(z)|}{\log r}=\rho(f),
\]
where $\Delta(\theta)$ is a half-line from the origin, $E$ denotes a 
subset of $S(\theta-\sigma, \theta+\sigma)$, and satisfies
\[
\lim_{r\to \infty}\operatorname{m}(S(\theta-\sigma, \theta+\sigma; r, \infty)\cap E)=0,
\]
where $S(\theta-\sigma, \theta+\sigma; r, \infty)
=\{z: \theta-\sigma<\arg z<\theta+\sigma, r<|z|<\infty\}$.
\end{lemma}

\begin{lemma}\label{lem5.2} 
Let $f$ be an entire function of
order $\rho\in (0, \infty)$, and let $S(\phi_1, \phi_2)=\{z:
\phi_1<\arg z< \phi_2\}$ be a sector with
$\phi_2-\phi_1<\frac{\pi}{\rho}$. If there exists a Borel
direction of order $\rho$ of $f$ in $S(\phi_1, \phi_2)$, then for at least one of the two rays $L_{j}: \arg z=\phi_{j} (j=1,
2)$, say $L_2$, we have
\[
\limsup_{r\to \infty}\frac{\log\log|f(re^{i\phi_2})|}{\log r}=\rho.
\]
\end{lemma}

Lemma~\ref{lem5.2} is \cite[Lemma 1]{y.z}, which can be proved by
using a result in \cite[pp. 119-120]{t}.

We may assume $\rho(A)>\rho(B)$ due to Gundersen's result \cite[Theorem 2]{g.2}.
If $A(z)$ blows up exponentially in each sector $S_j$, $j=0, 1, \ldots, n+1$, 
then the assertion follows by the proof of Theorem~\ref{th1}. 
Suppose there exists at least one sector of the $n+2$ sectors, such that $A(z)$ 
decays to zero exponentially, say 
$S_{j_0}=\{z: \theta_{j_0}<\arg z<\theta_{j_0+1}\}$, $0\leq j_0\leq n+1$. 
That is, for any $\theta\in (\theta_{j_0}, \theta_{j_0+1})$, we have
\begin{align*}
\lim_{r\to\infty}\frac{\log\log\frac{1}{|A(re^{i\theta})|}}{\log r}=\frac{n+2}{2}.
\end{align*}

Suppose on the contrary to the assertion that there is a nontrivial solution 
$f$ of \eqref{e1.1} with $\rho(f) < \infty$.
By Lemma~\ref{lem2.2}, there exists a set $E_1\subset[0,2\pi)$ of linear measure 
zero, such that if $\psi_0\in[0,2\pi)-E_1$,
then there is a constant $R_0=R_0(\psi_0)>1$ such that for all
$z$ satisfying $\arg z=\psi_0$ and $|z|\geq R_0$, we have \eqref{e4.7}. 
Next we consider the two cases appearing in Lemma~\ref{lem5.1}.
\smallskip

\noindent\textbf{Case 1:} 
Suppose that the ray $\arg z=\theta$ is a Borel direction of order 
$\rho(B)$ of $B(z)$, where $\theta_{j_0}<\theta<\theta_{j_0+1}$. 
Choose $\phi_1\in (\theta_{j_0}, \theta)-E_1$ and 
$\phi_2\in (\theta, \theta_{j_0+1})-E_1$. Then 
$\phi_2-\phi_1<\frac{\pi}{\rho(A)}<\frac{\pi}{\rho(B)}$. 
By Lemma~\ref{lem5.2}, at least one of two rays $L_1: \arg z=\phi_1$ and 
$L_2: \arg z=\phi_2$, say $L_1$, satisfies
\[
\limsup_{r\to\infty}\frac{\log\log|B(re^{i\phi_1})|}{\log r}=\rho(B).
\]
Thus, there exists a sequence of points $z_{n}=r_{n}e^{i\phi_1}$ with 
$r_{n}\to \infty$ as $n\to\infty$, such that
\begin{gather}\label{e5.2}
\lim_{n\to\infty}\frac{\log\log|B(r_{n}e^{i\phi_1})|}{\log r_{n}}=\rho(B), \\
\label{e5.3}
\lim_{n\to\infty}\frac{\log\log\frac{1}{|A(r_{n}e^{i\phi_1})|}}{\log r_{n}}
=\frac{n+2}{2}, \\
\label{e5.4}
\Big|\frac{f^{(k)}(r_{n}e^{i\phi_1})}{f(r_{n}e^{i\phi_1})}\Big|
\leq r_{n}^{2\rho(f)},\quad k=1, 2.
\end{gather}
Combining \eqref{e5.2}-\eqref{e5.4} and \eqref{e1.1}, we arrive at 
a contradiction as in the proof of Theorem~\ref{th1}. 
Thus, we have that every nontrivial solution $f$ of \eqref{e1.1} 
satisfies $\rho(f)=\infty$.
\smallskip

\noindent\textbf{Case 2:} Suppose that the ray $\arg z=\theta$ 
is not a Borel direction of order $\rho(B)$ of $B(z)$, where 
$\theta_{j_0}<\theta<\theta_{j_0+1}$. By Lemma~\ref{lem5.1}, 
there exists a constant 
$\sigma\in \big(0, \min\{\frac{\theta-\theta_{j_0}}{2}, 
\frac{\theta_{j_0+1}-\theta}{2}, \frac{\pi}{4}\}\big)$, such that
\begin{equation}\label{e5.5}
\lim_{\substack{|z|\to\infty\\ z\in (S(\theta-\sigma,\theta+\sigma)-E_2)}}
\frac{\log\log|B(z)|}{\log r}=\rho(B),
\end{equation}
where $E_2$ denotes a subset of $S(\theta-\sigma, \theta+\sigma)$, and satisfies
\begin{align*}
\lim_{r\to \infty}\operatorname{m}(S(\theta-\sigma, \theta+\sigma;
 r, \infty)\cap E_2)=0.
\end{align*}
Let $\Delta=\{z: \arg z=\psi, \psi\in E_1\}$. We can easily see that there 
exists a sequence of points $z_{n}$ with $z_{n}\to \infty$ as 
$n\to\infty$, $\{z_{n}\}\subset (S(\theta-\sigma, 
\theta+\sigma)-E_2)\cap (S_{j_0}-\Delta)$, such that
\begin{gather}\label{e5.6}
\lim_{n\to\infty}\frac{\log\log|B(z_{n})|}{\log |z_{n}|}=\rho(B), \\
\label{e5.7}
\lim_{n\to\infty}\frac{\log\log\frac{1}{|A(z_{n})|}}{\log |z_{n}|}
 =\frac{n+2}{2}, \\
\label{e5.8}
\Big|\frac{f^{(k)}(z_{n})}{f(z_{n})}\Big|\leq |z_{n}|^{2\rho(f)},\quad k=1, 2.
\end{gather}
Combining \eqref{e5.6}-\eqref{e5.8} and \eqref{e1.1}, we arrive at a 
contradiction as in the proof of Theorem~\ref{th1}. 
Thus, we have $\rho(f)=\infty$ for every nontrivial solution $f$ of \eqref{e1.1}. 
This completes the proof.


\section{Proof of Theorem~\ref{th4}}

We begin by recalling some basic properties satisfied by entire functions 
that are extremal for Yang's inequality. To this end, if $A$ is a 
function extremal for Yang's inequality, then the rays $\arg z=\theta_{k}$,
denote the $q$ distinct Borel directions of order $\geq\mu(A)$ of $A$,
 where $k=1,2,\ldots, q$ and 
$0\leq\theta_1<\theta_2<\dots<\theta_{q}<\theta_{q+1}=\theta_1+2\pi$.

\begin{lemma}[\cite{w}]\label{lem6.1} 
Suppose that $A$ is a function extremal for
Yang's inequality. Then $\mu(A)=\rho(A)$. Moreover, for every
deficient value $a_{i}$, $i=1,2,\ldots, p$, there exists a
corresponding sector domain $S(\theta_{k_{i}},
\theta_{k_{i}+1})=\{z:\theta_{k_{i}} <\arg z< \theta_{k_{i}+1}\}$
 such that for every $\varepsilon>0$ the inequality
\begin{equation}\label{e6.1}
\log\frac{1}{|A(z)-a_{i}|}>C(\theta_{k_{i}}, \theta_{k_{i}+1}, 
\varepsilon, \delta(a_{i}, A))T(|z|, A)
\end{equation}
holds for $z\in S(\theta_{k_{i}}+\varepsilon,
\theta_{k_{i}+1}-\varepsilon; r, +\infty)
=\{z: \theta_{k_{i}}+\varepsilon<\arg z<\theta_{k_{i}+1}-\varepsilon, r<|z|<\infty\}$, 
where
$C(\theta_{k_{i}}, \theta_{k_{i}+1}, \varepsilon, \delta(a_{i}, A))$ 
is a positive constant depending only on
$\theta_{k_{i}}, \theta_{k_{i}+1}, \varepsilon$ and $\delta(a_{i}, A)$.
\end{lemma}

In the sequel, we shall say that $A$ decays to $a_{i}$ exponentially in
 $S(\theta_{k_{i}}, \theta_{k_{i}+1})$, if \eqref{e6.1} holds in 
$S(\theta_{k_{i}}, \theta_{k_{i}+1})$. Note that if $A$ is a function 
extremal for Yang's inequality, then $\mu(A)=\rho(A)$. 
Thus, for these functions, we need only to consider the Borel directions 
of order $\rho(A)$.


\begin{lemma}[\cite{l.w.z}]\label{lem6.2} 
Let $A$ be an entire function extremal for Yang's
inequality. Suppose that there exists 
$\arg z=\theta$ with $\theta_{j}<\theta<\theta_{j+1}$, 
$1\leq j \leq q$, such that
\begin{equation}\label{e6.2}
\limsup_{r\to \infty}\frac{\log\log|A(re^{i\theta})|}{\log r}=\rho(A).
\end{equation}
Then $\theta_{j+1}-\theta_{j} =\frac{\pi}{\rho(A)}$.
\end{lemma}

We state one more auxiliary result that covers one particular case of the 
proof of Theorem~\ref{th4}.

\begin{lemma}[\cite{w.z}]\label{lem6.3} 
Let $A(z)$ be a finite order entire function having a finite deficient value, 
and let $B(z)$ be a transcendental entire function with $\mu(B)<\frac{1}{2}$. 
Then every nontrivial solution of \eqref{e1.1} is of infinite order.
\end{lemma}

By \cite[Theorem 2]{g.2} and Lemma~\ref{lem6.3}, we just need prove the 
case $1/2\leq\mu(B)<\rho(A)$. Suppose on the contrary to the assertion 
that there is a nontrivial solution $f$ of \eqref{e1.1} with 
$\rho(f)=\rho < \infty$. We aim for a contradiction.

Suppose that $a_{i}$, $i=1,2,\ldots, p$, are all the finite deficient values 
of $A(z)$. Thus we have $2p$ sectors
$S_{j}=\{z| \theta_{j}<\arg z<\theta_{j+1} \}$, $j=1, 2,\ldots, 2p$, 
such that  $A(z)$ has the
following properties.
In each sector $S_j$, either there exists some $a_{i}$ such that
\begin{equation}\label{e6.3}
\log\frac{1}{|A(z)-a_{i}|}>C(\theta_{j}, \theta_{j+1},
\varepsilon, \delta(a_{i}, A))T(|z|, A)
\end{equation}
holds for $z\in S(\theta_{j}+\varepsilon,\theta_{j+1}-\varepsilon; r,
+\infty)$, where $C(\theta_{j}, \theta_{j+1}, \varepsilon,
\delta(a_{i}, A))$ is a positive constant depending only on
 $\theta_{j}, \theta_{j+1}, \varepsilon$ and $\delta(a_{i}, A)$,
 or there exists $\arg z=\theta\in (\theta_j, \theta_{j+1})$ such that
\begin{equation}\label{e6.4}
\limsup_{r\to \infty}\frac{\log\log|A(re^{i\theta})|}{\log r}=\rho(A)\,.
\end{equation}
For the sake of simplicity, in the sequel we use $C$ to represent 
$C(\theta_{j}, \theta_{j+1}, \varepsilon, \delta(a_{i}, A))$.
Note that if there exists some  $a_{i}$ such that \eqref{e6.3} holds 
in $S_j$, then there exists $\arg z=\theta$ such that \eqref{e6.4} 
holds in $S_{j-1}$ and $S_{j+1}$. If there exists 
$\theta\in (\theta_j, \theta_{j+1})$ such that \eqref{e6.4} holds, then 
there are $a_{i}$ ($a_{i'}$) such that \eqref{e6.3} holds in $S_{j-1}$ and 
$S_{j+1}$, respectively.

Without loss of generality, we assume that there is a ray $\arg z=\theta$ 
in $S_1$ such that \eqref{e6.4} holds. Therefore, there exists a ray in 
each sector $S_{3}, S_{5}, \ldots, S_{2p-1}$, such that \eqref{e6.4} holds.
 By using Lemma~\ref{lem6.2}, we know that all the sectors have the same 
magnitude $\frac{\pi}{\rho(A)}$.

Note that $B(z)$ is an entire function of lower order $1/2\leq\mu(B)<\rho(A)$.
 By Lemma~\ref{lem3.2} we see that there exists a sector $S(\alpha, \beta)$ 
with $\beta-\alpha\geq \frac{\pi}{\mu(B)}$, $0\leq\alpha<\beta\leq2\pi$, 
such that for all the rays $\arg z=\theta\in(\alpha, \beta)$ we have
\begin{equation}\label{e6.5}
\limsup_{r\to\infty}\frac{\log\log|B(re^{i\theta})|}{\log
r}\geq\mu(B).
\end{equation}
Note that $\rho(A)>\mu(B)$. It is not hard to see that there exists a
sector $S(\alpha', \beta')$, where $\alpha<\alpha'<\beta'<\beta$, 
such that there is an $a_{j_0}$ such that
\begin{equation}\label{e6.6}
\log\frac{1}{|A(re^{i\theta})-a_{j_0}|}>CT(r, A)
\end{equation}
holds for all $\theta\in[\alpha', \beta']$.
By using Lemma~\ref{lem2.2}, there exists a $\theta_0\in[\alpha', \beta']$ 
and $R>1$ such that for $k=1,2,$
\begin{equation}\label{e6.7}
\Big|\frac{f^{(k)}(re^{i\theta_0})}{f(re^{i\theta_0})}\Big|\leq r^{2\rho(f)}
\end{equation}
holds for all $r>R$. Note that \eqref{e6.5} holds for $\theta=\theta_0$.
Thus there is a sequence $(r_{n})$ with $r_{n}\to\infty$ as $n\to\infty$, such that
\begin{equation}\label{e6.8}
|B(r_{n}e^{i\theta_0})|\geq \exp(r_{n}^{\mu(B)-\varepsilon})
\end{equation}
holds for every $\varepsilon\in(0, \mu(B))$. Therefore, we deduce from \eqref{e6.6}-\eqref{e6.8} that
\begin{align*}
\exp(r_{n}^{\mu(B)-\varepsilon})
&\leq |B(r_{n}e^{i\theta_0})|\\
&\leq \Big|\frac{f^{''}(r_{n}e^{i\theta_0})}{f(r_{n}e^{i\theta_0})}\Big|
 +\Big|\frac{f^{'}(r_{n}e^{i\theta_0})}{f(r_{n}e^{i\theta_0})}\Big|
(|A(r_{n}e^{i\theta_0})-a_{j_0}|+|a_{j_0}|)\\
&\leq r_{n}^{2\rho(f)}(1+|a_{j_0}|+\exp(-CT(r_{n},A)))
\end{align*}
holds for all sufficiently large $n$. Obviously, this is a contradiction. 
Hence Theorem~\ref{th4} holds.


\subsection*{Acknowledgements} 
This research work is supported in part by the Academy of Finland, 
project $\sharp$268009, and the Centre for International Mobility of 
Finland (CIMO) (Grant TM-13-8775).

The authors want to thank the anonymous reviewer for his/her valuable suggestions.

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