Growth of meromorphic solutions of linear difference equations without dominating coefficients

Abstract

This paper is devoted to studying the growth of meromorphic solutions of difference equation

$$ P_{n}{(z)}f(z+n)+P_{n-1}f(z+n-1)+\cdots +P_{1}{(z)}f(z+1)+P_{0}{(z)}f(z)=0, $$

where the coefficients $P_{j}$ ($j=0,\ldots ,n$) are meromorphic functions. With some additional conditions on coefficients, we obtain precise estimates of the growth of meromorphic solutions of such an equation.

1 Introduction and main results

In this article, we shall consider the linear higher order difference equation

$$ P_{n}{(z)}f(z+n)+P_{n-1}{(z)}f(z+n-1)+\cdots +P_{1}{(z)}f(z+1)+P _{0}{(z)}f(z)=0, $$

(1)

where $P_{0}(z),\ldots ,P_{n}(z)$ are meromorphic functions. Nevanlinna theory is an important and basic tool in our discussion; for references see [9, 12, 18]. For a meromorphic function $f(z)$, we denote by $T(r,f)$, $N(r,f)$ the characteristic function and the counting function of $f(z)$, respectively. In particular, we define the order and the lower order of a meromorphic function $f(z)$ by

$$ \sigma (f)=\limsup_{r\rightarrow \infty }\frac{\log ^{+}T(r,f)}{ \log r} \quad \mbox{and} \quad \mu (f)=\liminf_{r\rightarrow \infty }\frac{\log ^{+}T(r,f)}{\log r}, $$

while

$$ \lambda (f)=\limsup_{r\rightarrow \infty }\frac{\log ^{+}N(r, \frac{1}{f})}{\log r} \quad \mbox{and} \quad \delta (a,f)= \liminf_{r\to \infty }\frac{m(r,\frac{1}{f-a})}{T(r,f)}=1- \limsup _{r\to \infty }\frac{N(r,\frac{1}{f-a})}{T(r,f)} $$

stand for the exponents of convergence of zero sequence of f and the deficiency of f at the point a, respectively. Let $\alpha (z)$ be a meromorphic function. We say that $\alpha (z)$ is a small function with respect to $f(z)$, if $T(r,\alpha (z))=o(T(r,f))$, possible outside of a set E with finite linear measure.

In [5], Chiang and Feng considered the growth of transcendental entire solutions of linear higher order difference equations.

Theorem A

Let $P_{j}(z)$ ($j=0,1,\ldots ,n$) be polynomials. If there exists an integer l, $0\le l\le n$, such that

$$ \deg (P_{l})> \max_{0\leq j \leq n,j\neq l}\bigl\{ \deg (P_{j})\bigr\} . $$

(2)

Then every meromorphic solution $f(z)$ of Eq. (1) satisfies $\sigma (f)\geq 1$.

The condition (2) shows the degree of $P_{l}$ is larger than the other coefficients which means the polynomial $P_{l}$ is dominating coefficient. Chen [3] weakens the condition (2), and obtained.

Theorem B

Let $P_{j}(z)$ ($j=0,\ldots ,n$) be polynomials such that $P_{n}P_{0}\neq 0$ and

$$ \deg \{P_{n}+\cdots +P_{0}\}=\max \{\deg P_{j}:j=0,\ldots ,n\}\geq 1. $$

(3)

If $f(z)$ is a meromorphic solution of Eq. (1), then $\sigma (f) \geq 1$, and $f(z)$ assumes every finite value $a\in \mathbb{C}$ except zero infinitely often and $\lambda (f-a)=\sigma (f)$.

The above theorems deal with the case that the coefficients of Eq. (1) are polynomials, and there exists a dominating coefficient. Then a natural question arises: can we estimate the growth of the Eq. (1) provided the coefficients are entire functions or meromorphic functions. To answer this question, Chiang and Feng [5] obtained the following.

Theorem C

Let $P_{0}(z),\ldots ,P_{n}(z)$ be entire functions such that there exists an integer l, $0\le l\le n$, such that

$$ \sigma (P_{l})>\max \bigl\{ \sigma (P_{j}):0 \le j\le n, j\ne l\bigr\} . $$

(4)

If $f(z)$ is a nontrivial meromorphic solution of Eq. (1), then $\sigma (f)\ge \sigma (P_{l})+1$.

Clearly, since the order of $P_{l}$ is larger than that of the others, $P_{l}$ in Theorem C is a dominating coefficient. But if no such coefficient exists, Laine and Yang [13] also proved a similar conclusion in Theorem D below.

Definition 1.1

([9])

The type of a meromorphic function $f(z)$ of order σ ($0<\sigma <\infty$) is defined by

$$ \tau (f)={ \lim_{r \to \infty }\sup }\frac{T(r,f)}{r ^{\sigma }}. $$

Theorem D

([13])

Let $P_{0}(z),\ldots ,P_{n}(z)$ be entire functions of finite order so that among those having the maximal order $\sigma :=\max \{\sigma (P_{j}):0\le j\le n \}$, exactly one has its type strictly greater than the others. Then, for any meromorphic solution of Eq. (1), we have $\sigma (f)\ge \sigma +1$.

The condition about type in Theorem D still means the growth of some coefficient is faster than the others, although there exist several coefficients have the same order. This implies there still exists dominating coefficient. For a linear difference equation with meromorphic coefficients, Chen [4] proved the following result.

Theorem E

Let $P_{0}(z),\ldots ,P_{n}(z)$ be meromorphic functions such that there exists an integer l, $0\le l\le n$, such that $\sigma (P_{l})>\max \{\sigma (P_{j}):0 \le j\le n, j\ne l\}$ and $\delta (\infty , P_{l})>0$. If $f(z)$ is a nontrivial meromorphic solution of Eq. (1), then $\sigma (f)\ge \sigma (P_{l})+1$.

Laine and Yang [13] posed the following question, which it is natural to ask.

Question 1.1

Do all meromorphic solution f of Eq. (1) satisfy $\sigma (f)\ge 1+\max \{\sigma (P_{j}):0\le j\le n\}$, even if there is no dominating coefficient?

In [11], some examples are given to illustrate that the Laine–Yang conjecture is not true in whole. In this paper, we consider this question and shall give estimates of growth of solutions of Eq. (1), and there is no dominating coefficient.

Theorem 1.1

Let $P_{j}(z)$ ($j=0,\ldots ,n$) be meromorphic functions. If $P_{1}(z)$ has a finite deficient value a, and $\sigma (P_{j})<\sigma (P_{0})<\frac{1}{2}$, $j\neq 0,1$, then every finite order meromorphic solution $f(z)$ of Eq. (1) satisfies $\sigma (f)\geq \mu (P_{0})-\max \{\sigma (P_{j}),j\neq 0,1\}+1$.

Edrei and Fuchs proved that the number of deficient values cannot be infinite when the zeros and poles of a meromorphic function distributed near some curves. We first give some definitions and results in order to relate their results with Eq. (1).

Let

$$\begin{aligned}& \varOmega (\theta _{1},\theta _{2},r)=\bigl\{ z:\theta _{1}< \arg z< \theta _{2}, \vert z \vert < r\bigr\} , \\& \overline{\varOmega }(\theta _{1},\theta _{2},r)=\bigl\{ z: \theta _{1}\leq \arg z \leq \theta _{2}, \vert z \vert < r \bigr\} . \end{aligned}$$

Definition 1.2

([6, 7])

Let $f(z)$ be a meromorphic function with finite order of growth ($0<\sigma (f)< \infty$). A ray $\arg z=\theta$ starting from the origin is called a zero-pole accumulation ray of $f(z)$, if, for any given real number $\varepsilon >0$, the following equality holds:

$$ \varlimsup _{r\rightarrow \infty }\frac{\log n \{\overline{\varOmega }(\theta -\varepsilon ,\theta +\varepsilon ,r),f=0 \}+\log n \{\overline{\varOmega }(\theta -\varepsilon ,\theta +\varepsilon ,r),f=\infty \}}{\log r}=\sigma (f). $$

Zhang [18] proved that if $f(z)$ is a meromorphic function with order $\sigma =\sigma (f)$, $0<\sigma <\infty$ and $f(z)$ has q zero-pole accumulation rays and p deficient values other than 0 and ∞, then $p\leq q$. We say $f(z)\in \mathit{EF}$ if $p=q\geq 1$, and the set of such $f(z)$ is called an Edrei–Fuchs set. This means that $f(z)$ is a positive finite order meromorphic function with p zero-pole accumulation rays and p nonzero finite deficient values.

Now we are in the position to give a result concerning an Edrei–Fuchs set.

Theorem 1.2

Let $P_{j}(z)$ ($j=0,\ldots ,n$) be meromorphic functions. If $P_{0}(z)$ has a deficient value ∞, $P_{1}(z)\in \mathit{EF}$, and $\sigma (P_{j})<\sigma (P_{0})$ for $j\neq 0,1$, then every meromorphic solution $f(z)$ of Eq. (1) satisfies $\sigma (f)\geq \sigma (P_{0})-\max \{\sigma (P_{j}),j\neq 0,1\}+1$.

Remark 1.1

In Theorems 1.1 and 1.2, we require one coefficient in the equation to have one or several finite deficient value, but we do not know which one of $P_{0}$ and $P_{1}$ has faster growth, and this means there does not exist a dominating coefficient.

Recently, some results about the connections between complex dynamics with linear differential equations have been obtained; see [10, 14]. In the following, we shall consider Question 1.1 from the dynamical system point of view. Some notations of complex dynamics are needed; see [10].

Let $f:\mathbb{C}\mapsto \widehat{\mathbb{C}}$ be a transcendental meromorphic function. Denote by $f^{n}$, $n\in \mathbb{N}$, the nth iterate of f, that is, $f^{1}=f,\ldots ,f^{n}=f\circ (f^{n-1})$. The Fatou set of f is denoted by $F(f)$, and the Julia set of f is defined by $J(f)=\widehat{\mathbb{C}}\setminus F(f)$. Let U be a connected component of $F(f)$, then $f^{n}(U)$ is contained in a component of $F(f)$, denoted by $U_{n}$. We say U is a wandering domain, if $U_{n}\cap U_{m}=\emptyset$. Furthermore, if U is wandering, and all $U_{n}$ are multiply-connected and surround 0 and the Euclidean distance $\operatorname{dist}(0,U_{n})\rightarrow \infty$ as $n\rightarrow \infty$, then U is called a Baker wandering domain. The reader can refer to [2, 19] for more details.

Definition 1.3

([17])

Let $f(z)$ be a meromorphic function of order σ, where $0<\sigma <\infty$. A ray $\arg z=\theta _{0}$ ($0\leq \theta _{0}<2\pi$) is named a Borel direction of order σ of $f(z)$, if for any positive number ε, the inequality

$$ \varlimsup _{r\rightarrow \infty }\frac{\log n (r, \theta _{0}-\varepsilon ,\theta _{0}+\varepsilon ,f=a)}{\log r}\geq \mu $$

holds for any finite complex value a, with possibly one exceptional value, where $n(r,\theta _{0}-\varepsilon ,\theta _{0}+\varepsilon ,f=a)$ denotes the number of zeros of $f(z)-a$ in the region $(|z|\leq r) \cap (\theta _{0}-\varepsilon \leq \arg z\leq \theta _{0}+\varepsilon )$, multiple zeros being counted with their multiplicities.

Suppose that $f(z)$ is an entire function of finite lower order $\mu >0$. Let $q\ (<\infty )$ denote the number of Borel directions and p denote the number of finite deficient values of $f(z)$. Yang in [17] proved that $p\leq \frac{q}{2}$. We will say that an entire function $f(z)$ is extremal for Yang’s inequality if $f(z)$ satisfies the assumptions of Definition 1.3 with $p=\frac{q}{2}$.

Next, we consider Eq. (1) from the dynamical system point of view, and we can establish the following result.

Theorem 1.3

Let $P_{j}(z)$ ($j=0,\ldots ,n$) be entire functions. Suppose that $P_{1}(z)$ is extremal for Yang’s inequality, and $F(P_{0})$ has a Baker wandering domain. If $\sigma (P_{j})<\sigma (P_{0})$ for $j\neq 0,1$, then every meromorphic solution $f(z)$ of Eq. (1) satisfies $\sigma (f)\geq \sigma (P_{0})-\max \{\sigma (P_{j}),j\neq 0,1\}+1$.

2 Auxiliary results

In order to prove the theorems, we need some lemmas.

Lemma 2.1

([1])

Let $f(z)$ be a transcendental meromorphic function, $0\leq \mu (f) <1$. Then, for every $\alpha \in (\mu (f), 1)$, there exists a set $E \subset [0, \infty )$ such that $\overline{\log \operatorname{dens}}\,E \geq 1-\frac{\mu (f)}{\alpha }$, where

$$ E=\bigl\{ r\in [0,\infty ): m(r)>M(r)\cos \pi \alpha \bigr\} , $$

$m(r)=\inf_{|z|=r}{\log |f(z)|}$, and $M(r)=\sup_{|z|=r}{\log |f(z)|}$.

Lemma 2.2

([8])

Let $f(z)$ be a non-constant meromorphic function with finite order σ. For any given $\varepsilon >0$ and $0< l<\frac{1}{2}$, there exist a constant $K(\sigma , \varepsilon )$ and a set $E(\varepsilon )\subset [0, \infty )$ that satisfy the lower logarithmic density of $\underline{\log \operatorname{dens}}\,E(\varepsilon )\geq 1-\varepsilon$, such that, for $r\in E(\varepsilon )$ and each interval J with a length of l, we have

$$ r \int _{J}{ \biggl\vert \frac{f'(re^{i\theta })}{f(re^{i\theta })} \biggr\vert }\, d \theta < K(\sigma , \varepsilon ) \biggl(l\log \frac{1}{l}\biggr)T(r, f). $$

Lemma 2.3

([5])

Let $\eta _{1}$, $\eta _{2}$ be two arbitrary complex numbers, and let $f(z)$ be a meromorphic function of finite order σ. Let $\varepsilon >0$ be given, then there exists a subset $E_{1}\subset \mathbf{R}$ with finite logarithmic measure such that, for all $r\notin E_{1}\cup [0,1]$, we have

$$ \exp \bigl\{ -r^{\sigma -1+\varepsilon }\bigr\} \leq \biggl\vert \frac{f(z+\eta _{1})}{f(z+ \eta _{2})} \biggr\vert \leq \exp \bigl\{ r^{\sigma -1+\varepsilon }\bigr\} . $$

Lemma 2.4

([5])

Let $f(z)$ be a meromorphic function of order $\sigma (f)=\sigma <\infty$. Then, for any given $\varepsilon >0$, there is a set $E\subset (1, \infty )$ that has finite linear measure $m\,E$ and finite logarithmic measure $\mathit{lm}\,E$, such that, for all z satisfying $|z|=r\notin E\cup [0, 1]$,

$$ \exp \bigl\{ -r^{\sigma +\varepsilon }\bigr\} \leq \bigl\vert f(z) \bigr\vert \leq \exp \bigl\{ r^{\sigma + \varepsilon }\bigr\} . $$

Lemma 2.5

([19])

Let $f(z)$ be a transcendental meromorphic function with at most finitely many poles. If $f(z)$ has a Baker wandering domain, there exist a constant $0< d<1$ and two sequences $\{r_{n}\}$ and $\{R_{n}\}$ of positive numbers with $r_{n}\rightarrow \infty$ and $\frac{R_{n}}{r_{n}}\rightarrow \infty$ ($n\rightarrow \infty$) such that

$$ \log \bigl\vert f(z) \bigr\vert \geq d\log M(r, f), $$

where $|z|=r$, $G=\bigcup_{n=1}{\{r: r_{n}\leq r\leq R_{n}\}}$.

Lemma 2.6

([15])

Let $f(z)$ be a meromorphic function of order $0<\sigma <\infty$ having p finite deficient values $a_{1}, a_{2}, \ldots , a_{p}$ ($p\geq 1$) and let $g(z)$ be a meromorphic function with finite order having a deficient value ∞. Suppose that $\beta >1$ and $0<\eta <\sigma (f)$ are two constants. Then there exists a sequence $\{t_{n}\}$ such that

$$ \lim_{n \to \infty }\frac{t_{n}}{T(t_{n}, f)}=0. $$

(5)

Moreover, for every sufficiently large n, there is a set $F_{n} \subset [t_{n}, (\beta +1)t_{n})$ with $m(F_{n})\leq \frac{(\beta -1)t _{n}}{4}$ such that, for all $R\in [t_{n}, \beta t_{n}]\setminus F _{n}$, the arguments θ sets $E_{v}(R)$ ($v=1, 2, \ldots , p$) and $E_{\infty }(R)$ satisfy the following inequalities:

$$ m\bigl(E_{v}(R)\bigr)=m\biggl(\biggl\{ \theta \in [0, 2 \pi )\Big|\log \frac{1}{|f(Re^{i\theta })-a _{v}|}\geq \frac{\delta _{0}}{4}T(R, f)\biggr\} \biggr)\geq M_{1}>0 $$

(6)

and

$$ m\bigl(E_{\infty }(R)\bigr)=m\biggl(\biggl\{ \theta \in [0, 2 \pi )\Big|\log \bigl\vert g\bigl(Re^{i\theta }\bigr) \bigr\vert \geq \frac{\delta _{1}}{4}T(R, g)\biggr\} \biggr)\geq M_{2}>0, $$

(7)

where $M_{1}$, $M_{2}$ are two positive constants depending only on f, g, $\delta _{0}= \min_{0\leq v\leq p}\delta (a _{v}, f)$, $\delta _{1}=\delta (\infty , g)$, β and η.

Lemma 2.7

([15])

Let $f(z)\in \mathit{EF}$, then, for any given $\varepsilon >0$ (sufficiently small) and $\beta >1$, when n is sufficiently large, there exists a sequence of angular regions $\overline{\varOmega }({\theta _{k}}_{v}, {{\theta _{k}}_{v}}+1, t_{n}, \beta t_{n})$, $n=1, 2, 3, \ldots$ , $v=1, 2, \ldots , p$, such that, for every $1\leq v\leq p$, the inequality

$$ \log \frac{1}{|f(z)-a_{v}|}>\log \frac{4}{d} $$

(8)

holds for $z\in \overline{\varOmega }({\theta _{k}}_{v}, {{\theta _{k}} _{v}}+1, t_{n}, \beta t_{n})\setminus \bigcup_{v=1}^{p}(\gamma _{v})_{n}$, where $\bigcup_{v=1}^{p}(\gamma _{v})_{n}$ is defined by Lemma 2.8 with the sum of total radius not exceeding $\frac{p}{8}\varepsilon t_{n}$ and $t_{n}$, $\beta t_{n}$, $d= \min_{1\leq v \neq v'}\{|a _{v}-a_{v}'|\}$ and $a_{v}$ are deficient values of $f(z)$.

Lemma 2.8

([16])

Suppose that $f(z)$ is extremal for Yang’s inequality, i.e., $f(z)$ is an entire function of lower order $\mu <+ \infty$ and it satisfies $p=\frac{q}{2}$ where p ($1\leq p<+\infty$) denotes the number of finite deficient values and q denotes the number of Borel directions of order ≥μ of $f(z)$. Then, for every deficient value $a_{i}$ ($i=1, 2, \ldots , p$), there exists a corresponding angular domain $\varOmega (\theta _{k_{i}}, \theta _{k_{i}+1})$ such that for every $\varepsilon >0$ the inequality

$$ \log \frac{1}{|f(z)-a_{i}|}>A\bigl(\theta _{k_{i}}, \theta _{k_{i}+1}, \varepsilon , \delta (a_{i}, f)\bigr)T\bigl( \vert z \vert , f \bigr) $$

holds for $z\in \varOmega (\theta _{k_{i}}+\varepsilon , \theta _{k_{i}+1}- \varepsilon , r_{\varepsilon }, +\infty )$, where $A(\theta _{k_{i}}, \theta _{k_{i}+1}, \varepsilon , \delta (a_{i}, f))$ is a positive constant depending only on $\theta _{k_{i}}$, $\theta _{k_{i}+1}$, ε, $\delta (a_{i},f)$.

3 Proof of Theorem 1.1

We divide our proof into two steps.

Step 1. In this step, we use the idea in [20] with some changes. According to Lemma 2.1, for any given constant $\varepsilon >0$, there exists a set $E_{1}=E_{1}\{r\in [0, \infty ): m(r)>M(r)\cos \pi \alpha _{0}\}$ with $\overline{\log \operatorname{dens}}\,E_{1}\geq 1- \frac{\mu (P_{0})}{\alpha _{0}}$, where $m(r)=\inf_{|z|=r}{\log |P_{0}(z)|}$, and $M(r)=\sup_{|z|=r}{\log |P _{0}(z)|}$. Therefore, there exists some constant $r_{0}$ such that, for $r\in E_{1}\setminus [0, r_{0}]$, we have

$$ \bigl\vert P_{0}(z) \bigr\vert >\exp \bigl\{ r^{\mu (P_{0})-\varepsilon }\bigr\} . $$

(9)

Suppose $P_{1}(z)$ has a finite deficient value a, and $\delta (a, P _{1}(z))= 2\delta >0$. Then by the definition of deficiency, there exits a constant $r_{1}$ such that, for each $r>r_{1}>r_{0}$, $m(r, \frac{1}{P _{1}-a})\geq \delta T(r, P_{1})$ holds. Hence, for $r>r_{1}$, there exists $z_{r}$ with $|z_{r}|=r$ such that

$$ \log \bigl\vert P_{1}(z_{r})-a \bigr\vert \le -\delta T(r,P_{1}). $$

(10)

Set $0<\varepsilon _{0}<1-\frac{\mu (P_{0})}{\alpha _{0}}$. Applying Lemma 2.2 to $P_{1}(z)-a$, we can choose sufficiently small $l_{0}$ such that $K(\sigma (P_{1}), \varepsilon _{0})l_{0}\log \frac{1}{l_{0}}<\frac{ \delta }{4}$, then, for every interval J with a length of $l_{0}$ and all $r>r_{1}>r_{0}$, $r\in E(\varepsilon _{0})$, we have

$$ r \int _{J}{ \biggl\vert \frac{P_{1}'(re^{i\theta })}{P_{1}(re^{i\theta })-a} \biggr\vert }\, d \theta < \frac{\delta }{2}T(r, P_{1}), $$

(11)

where $E(\varepsilon _{0})$ is the set of lower logarithmic density $\underline{\log \operatorname{dens}}\,E(\varepsilon _{0})\geq {1-\varepsilon _{0}}$ determined by Lemma 2.2. Let $z_{r}=re^{i\theta }$ and $\phi _{0}=\frac{l _{0}}{4}$. It follows from (10)–(11) that, for all $|z_{r}|=r\in E( \varepsilon _{0})\setminus [0, r_{1}]$ and $\theta \in [\theta _{r}- \phi _{0}, \theta _{r}+\phi _{0}]$,

$$\begin{aligned} \log \bigl\vert P_{1}\bigl(re^{i\theta }\bigr)-a \bigr\vert =& \log \bigl\vert P_{1}\bigl(re^{i\theta _{r}}\bigr)-a \bigr\vert + \int _{\theta _{r}}^{\theta }\frac{d\log \vert P_{1}(re^{it})-a \vert }{dt} \,dt \\ \leq& -\delta T(r,P_{1})+ \int _{\theta _{r}}^{\theta } \biggl\vert \frac{P'_{1}(re^{it})}{P'_{1}(re^{it})-a} \biggr\vert \,dt \\ \leq& -\frac{\delta }{2}T(r,P_{1})\leq 0. \end{aligned}$$

Therefore, for $|z_{r}|=r\in E(\varepsilon _{0})\setminus [0,r_{1}]$ and $\theta \in [\theta _{r}-\phi _{0}, \theta _{r}+\phi _{0}]$, we get

$$ \bigl\vert P_{1}\bigl(re^{i\theta }\bigr)-a \bigr\vert \leq 1. $$

(12)

Set $E=E(\varepsilon _{0})\cap E_{1}$. Clearly,

$$ \overline{\log \operatorname{dens}}\,E+\underline{\log \operatorname{dens}} \bigl(E(\varepsilon _{0})\setminus E_{1}\bigr)\geq \underline{\log \operatorname{dens}}\,E(\varepsilon _{0}). $$

Hence,

$$ \overline{\log \operatorname{dens}}\,E\geq 1-\varepsilon _{0}- \underline{\log \operatorname{dens}}\,E _{1}^{c}. $$

Since $\overline{\log \operatorname{dens}}\,E_{1}+\underline{\log \operatorname{dens}}\,E_{1}^{c}=1$ and $\overline{\log \operatorname{dens}}\,E_{1}\geq 1-\frac{\mu (P_{0})}{\alpha _{0}}$, we have $\underline{\log \operatorname{dens}}\,E_{1}^{c}\le \frac{\mu (P_{0})}{\alpha _{0}}$. Thus,

$$ \overline{\log \operatorname{dens}}\,E\geq {1-\varepsilon _{0}- \frac{\mu (P_{0})}{\alpha _{0}}}>0. $$

Therefore, from (9) and (12), for any sequence $\{r_{n}\}\subset E$ and $\theta _{n}\in [\theta _{r}-\phi _{0},\theta _{r}+\phi _{0}]$,

$$ \begin{aligned} & \bigl\vert P_{0}(z_{n}) \bigr\vert >\exp \bigl\{ r_{n}^{\mu (P_{0})-\varepsilon }\bigr\} , \\ & \bigl\vert P_{1}\bigl(r_{n}e^{i\theta _{n}}\bigr)-a \bigr\vert \leq 1, \end{aligned} $$

(13)

where $z_{n}=r_{n}e^{i\theta _{n}}$.

Step 2. Now suppose that $f(z)$ is a non-trivial meromorphic solution of Eq. (1) with $\sigma (f)< \mu (P_{0})-\max \{\sigma (P_{j}),j \neq 0,1\}+1$. We shall seek a contradiction. Set $\sigma =\sigma (f)$, $\alpha =\max \{\sigma (P_{j}),j\neq 0,1\}$. From Eq. (1),

$$ \bigl\vert P_{0}(z) \bigr\vert \leq \bigl\vert P_{n}(z) \bigr\vert \biggl\vert \frac{f(z+n)}{f(z)} \biggr\vert + \cdots + \vert P _{2} \vert \biggl\vert \frac{f(z+2)}{f(z)} \biggr\vert +\bigl( \bigl\vert P_{1}(z)-a \bigr\vert + \vert a \vert \bigr) \biggl\vert \frac{f(z+1)}{f(z)} \biggr\vert . $$

(14)

We may choose ε so that $0<4\varepsilon < \mu (P_{0})- \alpha +1-\sigma$. By using Lemma 2.3, for ε there exists a set $E_{2}\subset (1, \infty )$ with finite logarithmic measure $\mathit{lm}\,E_{2}<\infty$ such that, for all $|z|=r\notin E_{2}\cup [0, r_{1}]$, we have

$$ \biggl\vert \frac{f(z+j)}{f(z)} \biggr\vert \leq \exp \bigl\{ r^{\sigma -1+\varepsilon }\bigr\} ,\quad j=1, \ldots , n. $$

(15)

Again by Lemma 2.4, there exists a set $E_{3}\subset (1, \infty )$ with logarithmic measure $\mathit{lm}\,E_{3}<\infty$ such that, for all $|z|=r\notin E_{3}\cup [0, r_{1}]$, we have

$$ \bigl\vert P_{j}(z) \bigr\vert \leq \exp \bigl\{ r^{\alpha +\varepsilon }\bigr\} , \quad j=2, \ldots , n. $$

(16)

It follows from (9), (12), (14), (15) and (16) that there exist $r_{n}\subset E\setminus E_{2}\cup E_{3}$ and $\theta _{n}\in [ \theta _{r_{n}}-\phi _{0}, \theta _{r_{n}}+\phi _{0}]$ such that, for $z_{n}=r_{n}e^{i\theta _{n}}$, we have

$$\begin{aligned} \exp \bigl\{ r_{n}^{\mu (P_{0})-\varepsilon }\bigr\} &\leq \bigl( \bigl\vert P_{n}(z) \bigr\vert +\cdots + \bigl\vert P_{2}(z) \bigr\vert + \bigl\vert P_{1}(z)-a \bigr\vert + \vert a \vert \bigr)\exp \bigl\{ r_{n}^{\sigma -1+\varepsilon }\bigr\} \\ &\leq \bigl((n-1)\exp \bigl\{ r_{n}^{\alpha +\varepsilon }\bigr\} +1+ \vert \alpha \vert \bigr)\exp \bigl\{ r_{n} ^{\sigma -1+\varepsilon }\bigr\} . \end{aligned}$$

Clearly $\mu (P_{0})-\varepsilon \leq \sigma +\alpha -1+3\varepsilon$, and hence $\mu (P_{0})-\sigma -\alpha +1\leq 4\varepsilon$. We have a contradiction.

4 Proof of Theorem 1.2

Suppose that $f(z)\neq 0$ is a meromorphic solution of Eq. (1) with $\sigma (f)<\sigma (P_{0})-\max \{\sigma (P_{j}), j\neq 0, 1\}+1$. We shall seek a contradiction. Set $\sigma =\sigma (f)<\infty$, $\alpha =\max \{\sigma (P_{j}), j\neq 0, 1\}<\infty$. From Eq. (1), we have

$$ \bigl\vert P_{0}(z) \bigr\vert \leq \bigl\vert P_{n}(z) \bigr\vert \biggl\vert \frac{f(z+n)}{f(z)} \biggr\vert + \cdots + \vert P _{2} \vert \biggl\vert \frac{f(z+2)}{f(z)} \biggr\vert + \vert P_{1} \vert \biggl\vert \frac{f(z+1)}{f(z)} \biggr\vert . $$

(17)

First, we prove $\sigma (P_{0})<\infty$. If $\sigma (P_{0})=\infty$, by the difference logarithmic derivative lemma, we get

$$\begin{aligned} T\bigl(r,P_{0}(z)\bigr)&=T\biggl(r,P_{n}(z) \frac{f(z+n)}{f(z)}+ \cdots +P _{2}(z)\frac{f(z+2)}{f(z)}+P_{1}(z) \frac{f(z+1)}{f(z)}\biggr) \\ &\leq \sum_{j\neq 0} T\bigl(r,P_{j}(z)\bigr)+ \sum_{j\neq 0} N\bigl(r,f(z+j)\bigr)+N\biggl(r, \frac{1}{f(z)}\biggr)+O\bigl(T(r,f)\bigr), \end{aligned}$$

possible outside of a set H satisfying $\limsup \frac{\int _{H\cap [1,r)} \frac{dt}{t}}{\log r}=0$. Clearly, this is a contradiction since $\sigma (P_{j})<\infty$ for $j\ne 0$ and $\sigma (f)<\infty$. Hence, $\sigma (P_{0})<\infty$.

Choose ε such that $0<2\varepsilon <\sigma (P_{0})- \alpha -\sigma +1$. Applying Lemma 2.3 to $f(z)$, there exist a positive constant $r_{1}$ and a set $E_{2}\subset [0, \infty )$ with $m(E_{2})<\infty$ such that, for all z satisfying $|z|=r\notin E _{2}\cup [0,r_{1}]$, we have

$$ \biggl\vert \frac{f(z+j)}{f(z)} \biggr\vert \leq \exp \bigl\{ r^{\sigma -1+\varepsilon }\bigr\} ,\quad j=1, \ldots ,n. $$

(18)

It follows from Lemma 2.4 that there exists a set $E_{3}\subset (0, \infty )$ with logarithmic measure $\mathit{lm}\,E_{3}<\infty$ such that, for all $|z|=r\notin E_{3}\cup [0, 1]$, we have

$$ \bigl\vert P_{j}(z) \bigr\vert \leq \exp \bigl\{ r^{\alpha +\varepsilon }\bigr\} ,\quad j\ne 0,1. $$

(19)

Since $P_{1}(z)\in \mathit{EF}$, we can assume that $P_{1}(z)$ has p finite deficient values $a_{1}, a_{2}, \ldots , a_{p}$ with $\delta (a_{v},P_{1})>0$ for $1\leq v\leq p$, and $P_{1}$ has p zero-pole accumulation rays $\arg z=\theta _{k}$ $(0\leq \theta _{1}< \theta _{2}<\cdots <\theta _{p}, \theta _{p+1}=\theta _{1}+2 \pi )$. Set $w= \min_{1\leq k\leq p-1}(\theta _{k+1}- \theta _{k})$.

Applying Lemma 2.6 to $P_{0}(z)$, $P_{1}(z)$, there exists a sequence of closed intervals $\{[t_{n}, \beta t_{n}]\}$ with $t_{n}\rightarrow \infty$, $t_{n+1}>\beta t_{n}$ and a set $F_{n}\subset [t_{n},( \beta +1)t_{n}]$ with $m(F_{n})\leq \frac{(\beta -1)t_{n}}{4}$ and a sequence $R_{n}\in [t_{n}, \beta t_{n}]\setminus F_{n}$ such that $P_{0}(z)$, $P_{1}(z)$ satisfy

$$ m\bigl(E_{\infty }(R_{n})\bigr)=m\biggl(\biggl\{ \theta \in [0, 2\pi )\Big|\log \bigl\vert P_{0} \bigl(R_{n}e ^{i\theta }\bigr) \bigr\vert \geq \frac{\delta _{1}}{4}T(R_{n}, P_{0})\biggr\} \biggr)\geq M_{2}>0 $$

(20)

and

$$ m\bigl(E_{v}(R_{n})\bigr)=m\biggl(\biggl\{ \theta \in [0, 2\pi )\Big|\log \frac{1}{|P_{1}(R_{n}e ^{i\theta })-a_{v}|}\geq \frac{\delta _{0}}{4}T(R_{n}, P_{1})\biggr\} \biggr)\geq M _{1}>0, $$

(21)

where $\theta \in [0, 2\pi )$ and $M_{1}$, $M_{2}$ are two positive constants depending only on $P_{0}$, $P_{1}$, $\delta _{1}=\delta ( \infty , P_{0})$, $\delta _{0}= \min_{0\leq v\leq p} \delta (a_{v}, P_{1})$, β and η.

Now we choose a sufficiently small $\varepsilon _{0}$ so that $0<\varepsilon _{0}<\min \{\frac{M_{1}}{8p},\frac{w}{2}, \frac{M_{2}}{8p}, \frac{\beta -1}{2p}\}$. It follows from Lemma 2.8 that the following inequalities:

$$ \log \frac{1}{|P_{1}(r_{n}e^{i\varphi })-a_{v}|}>\log \frac{4}{d},\quad v=1, 2, \ldots , p, $$

(22)

hold for $n\geq n_{1}>n_{0}$ and $r_{n}e^{i\varphi }\in \bigcup_{v=1} ^{p}\overline{\varOmega }({{\theta _{k}}_{v}}_{+1}+2\varepsilon _{0}, {{\theta _{k}}_{v}}-2\varepsilon _{0}, t_{n}, \beta t_{n})\setminus \bigcup_{v=1}^{p}(\gamma _{v})_{n}$, where $\bigcup_{v=1}^{p}{(\gamma _{v})}_{n}$ are some disks with the sum of total radius not exceeding $\frac{p}{8}\varepsilon _{0}t_{n}<\frac{\beta -1}{16}t_{n}$, $d= \min_{1\le v,v'\le p, v \neq v'}\{|a_{v}-a_{v}'|\}$ and $a_{v}$, $a_{v}'$ are deficient values of $P_{1}(z)$.

It is easy to choose $R^{*}_{n}\in [t_{n}, \beta t_{n}]\setminus (F _{n}\cup E_{2}\cup E_{3}\cup [0, 1])$ such that, for $n\geq n_{0}$,

$$ \bigl\{ z: \vert z \vert =R^{*}_{n}\bigr\} \cap \Biggl\{ \bigcup_{v=1}^{p}{(\gamma _{v})}_{n}\Biggr\} = \varPhi . $$

On the other hand, from Lemma 2.6, for such a sequence $\{R^{*}_{n}\}$ and sufficiently large n, we also have

$$ m\bigl(E_{\infty }\bigl(R^{*}_{n}\bigr) \bigr)=m\biggl(\biggl\{ \theta \in [0, 2\pi )\Big|\log \bigl\vert P_{0}\bigl(R^{*} _{n}e^{i\theta }\bigr) \bigr\vert \geq \frac{\delta _{1}}{4}T\bigl(R^{*}_{n}, P_{0}\bigr)\biggr\} \biggr) \geq M_{2}>0. $$

(23)

Clearly, by the choice of $\varepsilon _{0}$, there exists some $k_{v}$, $1\leq k_{v}\leq p$, such that

$$ m\bigl(E_{\infty }\bigl(R^{*}_{n}\bigr) \bigr)\cap [{{\theta _{k}}_{v}}+2\varepsilon _{0}, {{{\theta _{k}}_{v}}}_{+1}-2 \varepsilon _{0}]\geq \frac{M_{2}}{2p}>0. $$

(24)

Therefore, from (20), (22) and (24), we can choose $\theta _{n}\in E _{\infty }(R^{*}_{n})\cap [{{\theta _{k}}_{v}}+2\varepsilon _{0}, {{{\theta _{k}}_{v}}}_{+1}-2\varepsilon _{0}]$ such that

$$ \log \bigl\vert P_{0}\bigl(R^{*}_{n}e^{i\theta _{n}} \bigr) \bigr\vert \geq \frac{\delta _{1}}{4}T\bigl(R ^{*}_{n}, P_{0}\bigr) $$

(25)

and

$$ \log \frac{1}{|P_{1}(R^{*}_{n}e^{i\theta _{n}})-a_{v}|}>\log \frac{4}{d}, \quad v=1, 2, \ldots , p, $$

(26)

hold simultaneously, where $z_{n}=r_{n}e^{i\theta _{n}}$, $\beta >1$, $d= \min_{1\leq v \neq v'}\{|a_{v}-a_{v}'|\}$ and $a_{v}$, $a_{v}'$ are deficient values of $P_{1}(z)$.

Rewrite (17) as

$$ \bigl\vert P_{0}(z) \bigr\vert \leq \bigl\vert P_{n}(z) \bigr\vert \biggl\vert \frac{f(z+n)}{f(z)} \biggr\vert + \cdots + \vert P _{2} \vert \biggl\vert \frac{f(z+2)}{f(z)} \biggr\vert +\bigl( \vert P_{1}-a_{v} \vert + \vert a_{v} \vert \bigr) \biggl\vert \frac{f(z+1)}{f(z)} \biggr\vert . $$

(27)

It follows from (18), (19), (25) and (26) that

$$\begin{aligned} \exp \biggl\{ \frac{\delta _{1}}{4}T\bigl(R^{*}_{n},P_{0} \bigr)\biggr\} \leq& (n-1)\exp \bigl\{ {\bigl(R^{*} _{n} \bigr)}^{\alpha +\varepsilon }\bigr\} \exp \bigl\{ {\bigl(R^{*}_{n} \bigr)}^{\sigma -1+\varepsilon }\bigr\} \\ &{}+\biggl(\frac{d}{4}+ \vert a_{v} \vert \biggr)\exp \bigl\{ \bigl(R^{*}_{n} \bigr)^{\sigma -1+\varepsilon }\bigr\} . \end{aligned}$$

Clearly $\sigma (P_{0})\leq \alpha +\sigma -1+2\varepsilon$, and hence $\sigma (f)\geq \sigma (P_{0})-\max \{\sigma (P_{j}), j\neq 0, 1\}+1+2 \varepsilon$. We have a contradiction.

5 Proof of Theorem 1.3

Suppose that $f\not \equiv 0$ is a meromorphic solution of Eq. (1) with $\sigma (f)<\sigma (P_{0})-\max \{\sigma (P_{j}), j\neq 0 ,1\}+1$.

Since $P_{1}(z)$ is extremal for Yang’s inequality, we can assume that $a_{i}$ ($i=1, 2, \ldots , 2p$) are all the finite deficient values of $P_{1}(z)$. By Lemma 2.8, for every deficient value $a_{i}$, there exists a corresponding angular domain $\varOmega (\theta _{j},\theta _{j+1})$ such that for every $\varepsilon >0$ the inequality

$$ \log \frac{1}{ \vert P_{1}(z)-a_{i} \vert }>A\bigl(\theta _{j}, \theta _{j+1}, \varepsilon , \delta (a_{i}, P_{1})\bigr)T\bigl( \vert z \vert , P_{1}\bigr) $$

holds for $z\in \varOmega (\theta _{j}+\varepsilon , \theta _{j+1}-\varepsilon , r_{\varepsilon }, +\infty )$, where $\varepsilon >0$ and $A(\theta _{j}, \theta _{j+1}, \varepsilon , \delta (a_{i}, P_{1}))$ is a positive constant depending only on $\theta _{j}$, $\theta _{j+1}$, ε and $\delta (a_{i}, f)$. Set $C=A(\theta _{j}, \theta _{j+1}, \varepsilon , \delta (a_{i}, P_{1}))$ for brevity. Thus

$$ \bigl\vert P_{1}(z)-a_{1} \bigr\vert < \exp \bigl\{ -CT\bigl( \vert z \vert , P_{1}\bigr)\bigr\} . $$

(28)

Note that $P_{0}(z)$ is a transcendental entire function with a Baker wandering domain. According to Lemma 2.5, there exists one integer $d<1$ and two positive sequences $\{r_{n}\}$, $\{R_{n}\}$, such that $\frac{R_{n}}{r_{n}}\rightarrow \infty$ as $r_{n}\rightarrow \infty$, $n\rightarrow \infty$ and $|z|=r$, $r\in G$, and we obtain

$$ \log \bigl\vert P_{0}(z) \bigr\vert \geq d\log M(r, P_{0}), $$

where $G=\bigcup_{n=1}\{r: r_{n}\leq r\leq R_{n}\}$. Thus

$$ \bigl\vert P_{0}(z) \bigr\vert \geq M(r, P_{0})^{d}. $$

(29)

Let $0<2\varepsilon <\sigma (P_{0})-\sigma (f)-\alpha +1$. By Lemma 2.3, there exists a set $E_{2}\subset [0, \infty )$ with finite linear measure and a constant $\eta >0$ such that, for all z satisfying $|z|=r\notin E_{2}\cup [0, 1]$, (18) holds. From Lemma 2.4, there still exists a set $E_{3}\subset (0, \infty )$ with finite logarithmic measure such that, for all $|z|=r\notin E_{3}\cup [0, 1]$, (19) holds.

Hence, for $z=re^{i\theta }$, $r\in G\setminus (E_{2}\cup E_{3}\cup [0, 1])$ and $\theta \in (\theta _{j}, \theta _{j+1})$, we get

$$ \bigl\vert P_{0}(z) \bigr\vert \geq M(r, P_{0})^{d} $$

and

$$ \bigl\vert P_{1}(z)-a_{1} \bigr\vert < \exp \bigl\{ -CT\bigl( \vert z \vert , P_{1}\bigr)\bigr\} . $$

(30)

Rewriting Eq. (1), we have

$$ \bigl\vert P_{0}(z) \bigr\vert \leq \bigl\vert P_{n}(z) \bigr\vert \biggl\vert \frac{f(z+n)}{f(z)} \biggr\vert + \cdots + \bigl\vert P _{2}(z) \bigr\vert \biggl\vert \frac{f(z+2)}{f(z)} \biggr\vert +\bigl( \bigl\vert P_{1}(z)-a \bigr\vert + \vert a \vert \bigr) \biggl\vert \frac{f(z+1)}{f(z)} \biggr\vert . $$

(31)

Therefore, we deduce from (23), (24) and (28) that

$$ M(r,P_{0})^{d}\leq (n-1)\exp \bigl\{ r^{\alpha +\varepsilon }\bigr\} \exp \bigl\{ r^{\sigma (f)-1+\varepsilon }\bigr\} +\bigl(\exp \bigl\{ -CT\bigl( \vert z \vert ,P_{1}\bigr)\bigr\} + \vert a \vert \bigr)\exp \bigl\{ r^{\sigma (f)-1+ \varepsilon }\bigr\} . $$

Clearly $\sigma (P_{0})\leq \sigma +\alpha -1+2\varepsilon$, that is, $2\varepsilon \geq \sigma (P_{0})-\sigma (f)-\alpha +1$. We have a contradiction.