On $q$-analogue of meromorphic multivalent functions in lemniscate of Bernoulli domain

1.   Introduction

The content of this paper is related to the value distribution theory and complex dynamic theory in complex analysis, the relevant literatures are available at ^[9,10,25,29]. On the complex plane, let $g$ be an entire function, its order and lower order denote by $\sigma(g)$ and $\mu(g)$ respectively, see reference ^[25] for details.

In Nevanlinna theory, for an entire function $g$, $T(r, g)$ represents its characteristic function and $M(r, g)$ represents its maximum modulus in the domain $|z| < r$. For these two quantities, they have a certain magnitude relationship. It's well known that, if $0 < r < R < +\infty$, then $M(r, g)$ and $T(r, g)$ satisfy the inequalities (see ^[10])

If one takes $R$ as a multiple of $r$ and $r$ goes to infinity, these two quantities are very similar. In order to study the exact relationship between these two quantities, Petrenko ^[18] introduced the deviation of entire function $g$ at $\infty$ and defined it as follows.

Later they were called Petrenko's deviation. Particularly, $\beta^-(\infty, g) < \beta^+(\infty, g)$ is likely to hold, as stated in [18, § 4]. As described in ^[6], for any increasing function $\phi(r)$ which is convex for $\log r$, there exists an entire function $g$ that satisfies the following similarity relation

From the above formula (1.3), we know that there should be many entire functions satisfying $\beta^+(\infty, g) = \beta^-(\infty, g)$.

In the following we will give some specific examples of functions with Petrenko's deviations. For any entire function $g$ with finite lower order $\mu$, let's define a variable $\mathfrak{B}(\mu)$ with respect to $\mu$,

It is shown in [18, Theorem 1] that Petrenko's deviation of $g$ satisfies

Both inequalities in (1.5) can hold strictly, see details in ^[6,18]. For example, the well known Airy integral function $Ai(z)$, which is a solution of $g''(z)-zg(z) = 0$, has lower order $3/2$. It's Petrenko's deviation satisfies

If the upper deviation $\beta^+(\infty, g)$ and the lower deviation $\beta^-(\infty, g)$ are equal and finite, then there exists a constant $\nu\in(0, 1]$ such that $T(r, f)$ and $\log M(r, f)$ satisfy

as $r(\not\in E)\rightarrow \infty$, where $E\subset(0, +\infty)$ is an exceptional set. For the function $g(z) = e^z$ that we're most familiar with, $T(r, e^z) = \frac{r}{\pi}$ and $\log M(r, e^z) = r$ satisfies (1.6) with $\nu = 1/\pi$. Generally, the exponential polynomials can also satisfy (1.6) for suitable $\nu$ as $r(\not\in E)\rightarrow \infty$, where $E\subset(0, +\infty)$ is a set of zero density ^[17].

Another example is entire function with Fabry gaps. For an entire function represented by a power series, $g(z) = \sum_{n = 0}^{\infty}a_nz^{\lambda_n}$, if it satisfies the gaps condition $\frac{\lambda_n}{n}\rightarrow \infty$ as $n\rightarrow \infty$, we call it having Fabry gaps, see ^[3]. The maximum modulus of such a function is similar to its minimum modulus, that is,

as $r(\not\in E)\rightarrow \infty$, where $E\subset(0, +\infty)$ is a set of zero density and $L(r, g) = \min_{|z| = r}|g(z)|$ is the minimum modulus, see ^[8]. The orders of this kinds of functions are positive [11, p.651]. Consequently, entire functions with Fabry gaps satisfy (1.6) with $\nu = 1$ as $r(\not\in E)\rightarrow \infty$.

In 2021, Heittokangas and Zemirni ^[12] investigated the zeros of solutions of second order complex differential equation

in which the coefficient $A(z)$ satisfies (1.6). In fact, they found the lower bound of the zero convergence exponent of solution to this equation is closely related to Petrenko's deviation of $A(z)$. In the same article, they also proved the growth of every nontrivial solution to

is infinite under the hypothesise that $A(z)$ is associated with Petrenko's deviation and $B(z)$ is non-transcendental growth in some angular domains.

For the sake of the following statement, we give the condition that an entire function $g(z)$ has an angular domain in which it is non-transcendental growth. Set

and

More specifically, the authors of paper ^[12] was under the condition that

to prove the infinite growth of solutions of (1.9).

The following content is some introduction about Julia set and the results of its limiting directions. The Julia set is one of two important subjects of the research of complex dynamics of transcendental meromorphic functions, see more details in ^[4,28]. Below we briefly recall the known results on the limiting direction of Julia set of meromorphic functions. To prevent confusion, we use $f^{\circ n} (n\in \mathbb{N})$ to denote the $n$-th iteration of meromorphic function $f$. The sequence of functions $\{f^{\circ n}\} (n\in \mathbb{N})$ forms a normal family in some domains of the complex plane, and such domains are called the Fatou set, denoted as $F(f)$. Its complement on the complex plane is Julia set, denoted as $J(f)$. The focus of our research is $J(f)$, it's well known that $J(f)$ a nonempty closed set. For transcendental entire functions, the distribution of $J(f)$ in the complex plane is so complicated that it cannot be constrained in a finite number of straight lines, see Baker ^[1]. However, this is not true for transcendental meromorphic functions, such as the Julia set of $\tan z$ being the whole real axis.

Therefore, the research direction began to focus on the distribution of $J(f)$ on rays, and Qiao introduced this concept in ^[19]. To explain this concept, let's first give a notation for an angular domain,

So, for any $\theta\in[0, 2\pi)$ and any small $\varepsilon > 0$, if the set $\Omega(\theta-\varepsilon, \theta+\varepsilon)\cap J(f)$ is unbounded, then the ray $\arg z = \theta$ is called a limiting direction of $J(f)$. To see how many such rays there are, the researchers introduced a set called $\Delta(f)$, which contains all the limiting direction of $J(f)$. $\Delta(f)$ has some properties, such as it is closed and measurable, so one can find a lower bound on the measure of $\Delta(f)$. The research on this aspect shows that when $f$ has finite lower order $\mu(f)$, the lower bound of meas$\Delta(f)$ is closely related to $\mu(f)$, see ^[19,30]. So what happens if $\mu(f)$ is infinite? Later, Huang et al. ^[13,14] studied the limiting directions of $J(f)$ of nontrivial solutions to some complex differential equations, these solutions have infinite order or lower order, more results have followed, such as ^{[21,22,23,26]}. Below we list a few results that are useful for this article.

In ^[13] the authors obtained the lower bound of limiting directions of $J(f)$ of nontrivial solutions to linear complex differential equations

Their result showed that the lower bound is related to the order of $A(z)$. In more depth, the authors of paper ^[21] investigated the common limiting directions of $J(f)$ of nontrivial solutions of high order linear complex differential equation

where $A_0$ is the dominated coefficient function comparing to other coefficients functions. The common limiting direction of $J(f)$ which just mentioned is defined as follows:

In the above definition, when $n\geq0$, $f^{(n)}$ denotes $n$ times of the ordinary derivative of $f$, when $n < 0$, $f^{(n)}$ means $|n|$ times integral operation.

Combining the concept of Petrenko's deviation with the results of limiting directions of Julia set of solutions to complex differential equations, the authors of article ^[27] studied the lower bound of common limiting directions of Julia set of solutions to some complex differential equations with coefficient function related to Petrenko's deviation, the two conclusions are stated below.

Theorem 1.1. ^[27] Suppose that $f$ is a nontrivial solution to the complex differential equation (1.9), $A(z)$ is an entire function and has non-transcendental growth angular domain such that $\xi(A) > 0$ and $B(z)$ is a transcendental entire function and has a Petrenko's deviation with $\beta^{-}(\infty, B) < \frac{1}{1-\xi(A)}$, then the common limiting direction of Julia set of $f$ satisfies

Theorem 1.2. ^[27] Suppose that the coefficient $A(z)$ of equation (1.12) is a transcendental entire function and satisfies (1.6) as $r\rightarrow \infty$ outside a set $G$ with $\log dens(G) < 1$, then the common limiting direction of Julia set of every nontrivial solution $f$ to (1.12) satisfies

Moreover, let $f_1, f_2, \cdots, f_n$ be a solution base of equation (1.12) and their product be $E: = f_1\cdot f_2\cdots f_n$, then the common limiting direction of Julia set of $E$ satisfies

While studying the limiting directions of Julia set of solutions to complex differential equations, the limiting direction of Julia set for the solution of difference equation is also concerned, for examples ^[5,7,16,24]. In order to state the following Theorem 1.3 which is proved in ^[5], we give some denotations at first. We set

where $f(z+c_i)$ are some different shift forms of the complex function $f(z)$ with $c_i (i = 1, \cdots, m)$ being distinct complex numbers, these shifts are closely related to the difference of $f$. To be specific, $\overline{P(z, f)}$ is a polynomial in $m$ variables with degree less than $d$,

where $a_{\lambda}$ are constant coefficients and are nonzeros, every element of $\Lambda$ is a multi-indices and has the form $\lambda = (k_1, \cdots, k_m), k_i\in \mathbb{N}$, and the degree of $\overline{P(z, f)}$ is $\max_{\lambda\in \Lambda}\{\sum_{i = 1}^m k_i\} < d$. In addition, the common limiting directions of Julia sets of distinct shifts of $f$ is defined by

where $n$ is a finite positive integer, and $\eta_i(i = 1, 2\cdots, n)$ are distinct complex numbers. Chen et al. ^[5] obtained the following result about the limiting directions of Julia set of solutions to some complex difference equations.

Theorem 1.3. ^[5] Suppose that $\overline{P_i(z, f)}$ $(i = 0, 1, \cdots, n)$ are distinct polynomials of degree less than $d$, and the coefficients $A_i(z)$ are entire functions. Assume that $A_0$ is transcendental with finite lower order, and it's dominant compared to the other coefficients, that is, $T(r, A_j) = o(T(r, A_0))$ as $r\rightarrow \infty$, $j = 1, 2, \cdots, n$. For any nontrivial entire solution $f$ of

we have meas$(\overline{L(f)})\geq\min\{2\pi, \pi/\mu(A_0)\}$. Furthermore, if $\sigma(A_0) > \max\{\sigma(A_1), \cdots, \sigma(A_n)\}$, then $\overline{L(f)}$ contains a closed interval $[v_1, v_2]$ with $v_2-v_1\geq \min\{2\pi, \pi/\sigma(A_0)\}$.

Motivated by Theorems 1.2 and 1.3, we study the limiting directions of Julia sets of solutions to complex difference equations with a coefficient related to Petrenko's deviation and obtain the following theorem as our first conclusion of this article.

Theorem 1.4. Suppose that $\overline{P_j(z, f)} (j = 1, 2, \cdots, n)$ are distinct polynomials with degree less than $d$, and $A_0$ is a transcendental entire function with finite lower order $\mu(A_0)$ and satisfies (1.6) as $r(\not\in G)\rightarrow \infty$ with $\log dens(G) < 1$. The other coefficients $A_i(z)(i = 1, 2, \cdots, n)$ are distinct entire functions satisfying

Then the common limiting direction of Julia set of every nontrivial entire solution $f$ to (1.14) satisfies meas$(\overline{L(f)})\geq2\pi\nu$.

In the paper ^[16], the authors studied a nonlinear differential-difference equation, it belongs to the general form of Tumura-Clunie equation. This equation is shown as below.

where the coefficient functions $A(z), h(z)$ are entire, and $\widetilde{P(z, f)}$ denotes a polynomial in the derivatives of the shifts of $f$,

where $c_i$ are distinct complex numbers and $a_j(z)$ are polynomials. deg$\widetilde{P(z, f)} = \max_{1\leq j\leq s}\sum_{i = 0}^l n_{ij}$ denotes the degree of $\widetilde{P(z, f)}$, where $n_{ij}$ are positive integers. For more details, see [16, Theorems 1.4 and 1.6]. This inspired us to consider a more general nonlinear complex difference equation than (1.15), combined with the coefficient having Petrenko's deviation. In fact, we obtain the following theorem as our second conclusion of this article.

Theorem 1.5. Suppose that $A_0(z)$ is a transcendental entire function and has Petrenko's deviation with (1.6). $A_1(z)$ is an entire function and non-transcendental growth in some angulars such that $\xi(A_1) > 0$. Moreover, $\beta^{-}(\infty, A_0) < \frac{1}{1-\xi(A_1)}$. Then the measure of common limiting direction of Julia set of every nontrivial solution $f$ of equation

satisfies

where $\widetilde{L(f)}: = \bigcap_{n\in\mathbb{Z}}\Delta(f^{(n)}(z+\eta))$ and $\eta$ is any complex number.

The third conclusion of this article is about the dynamic properties of solutions to nonlinear complex differential equations. Let's first recall the definition of a differential polynomial for an entire function $f$. This differential polynomial $P(z, f)$ consists of adding a finite number of differential monomials together, that is

where $a_j(z)$ are coefficient functions and all of them are meromorphic, and $n_{0j}, n_{1j}, \ldots, n_{kj}$ are non-negative integer powers. The minimum degree of $P(z, f)$ is denoted by

Wang et al. ^[23] observed the limiting directions of Julia set of solutions to the following Eq (1.18), they found when $F$ grows fast in a certain direction, that direction is the limiting direction of Julia set of solution $f$.

Theorem 1.6. ^[23] Let $n, s$ be integers. Suppose that the coefficient $F(z)$ is a transcendental entire function with finite lower order, $P(z, f)$ is a differential polynomial with respect to $f$, its minimum degree satisfies $\gamma_P\geq s$ and its coefficients $a_j(j = 1, 2, \cdots, l)$ are entire functions with $\sigma(a_j) < \mu(F)$. Then, for every nontrivial transcendental entire solution $f$ of the differential equation

we have $meas(L(f))\geq \min\{2\pi, \pi/\mu(F)\}$.

We improve the above Theorem 1.6 by introducing Petrenko's deviation for one coefficient of the equation and assuming the existence of angular domain of non-transcendental growth for the other coefficient. We state our result as follow.

Theorem 1.7. Let $P_1(z, f), P_2(z, f)$ be two differential polynomials with respect to $f$ with $\min\{\gamma_{P_1}, \gamma_{P_2}\}\geq s$, where the coefficients $a_{1j}, a_{2j}(j = 1, 2, \cdots, l)$ are entire functions such that $\max\{\sigma(a_{1j}), \sigma(a_{2j})\} < \mu(A_0)$. Suppose that the coefficient $A_0(z)$ is a transcendental entire function with finite lower order and satisfies (1.6), $A_1(z)$ is an entire function such that $\xi(A_1) > 0$ and $\beta^{-}(\infty, A_0) < \frac{1}{1-\xi(A_1)}$. $s$ is an integer. Let $f$ be a nontrivial transcendental entire solution of differential equation

Then the measure of common limiting direction of Julia set of $f$ satisfying

2.   Preliminary lemmas

The following lemma is crucial, which describes that when $f$ maps an angular domain to a hyperbolic domain with some special property, $f$ grows by no more than a polynomial in a slightly smaller angular domain. In order to better express this lemma, let us first give some explanations and notations. $\widehat{\mathbb{C}}$ denotes the extended complex plane and $U$ is a set contained in $\widehat{\mathbb{C}}$. If $\widehat{\mathbb{C}}\backslash U$ contains at least three points, then we say $U$ is a hyperbolic domain. For any $a\in\mathbb{C}\backslash U$, we define the quantity $C_{U}(a): = \inf\{\lambda_U(z)|z-a|: \forall z\in U\}$, where $\lambda_U(z)$ is the hyperbolic density on $U$. In particular, $C_U(a)\geq 1/2$ when every component of $U$ is simply connected, see ^[30].

Lemma 2.1. ([30,Lemma 2.2]) Suppose that $f(z)$ is an analytic mapping and $f:\Omega(r_0, \theta_1, \theta_2)\rightarrow U$, where $\Omega(r_0, \theta_1, \theta_2): = \{re^{i\theta}:\theta\in(\theta_1, \theta_2), r\geq r_0\}$, and $U$ is a hyperbolic domain. If there exists a point $a\in \partial U\backslash \{\infty\}$ such that $C_U(a) > 0$, then there exists a constant $d > 0$ such that, for sufficiently small $\varepsilon > 0$, we have

Because in the process of proving our theorems, we need to use the Nevanlinna theory in the angular domain, so here we list the notations related to it, which are given in accordance with the literatures ^[9,29]. Suppose that in the closed angular domain $\overline{\Omega}(\alpha, \beta)$, $g(z)$ is meromorphic. Set $w = \pi/(\beta-\alpha)$, and $b_n = |b_n|e^{i\beta_n}$ are poles of $g(z)$ on $\overline{\Omega}(\alpha, \beta)$ considering their multiplicities. The three main basic notations are

The sum of these three basic notations is the characteristic function in $\Omega(\alpha, \beta)$, that is,

Moreover, its growth is defined as

The following result is a lemma for the estimate of the first two basic notations above of the logarithmic derivative of a meromorphic function $f$ in an angular domain, which is related to the logarithm of the characteristic function of $f$ in a larger angular domain.

Lemma 2.2. ([29, Theorem 2.5.1]) Let $f(z)$ be a meromorphic function on $\Omega(\alpha-\varepsilon, \beta+\varepsilon)$ for $\varepsilon > 0$ and $0 < \alpha < \beta\leq2\pi$. Then

for $r > 1$ possibly except a set with finite linear measure.

Further, the estimate of the modulus of the logarithmic derivative of the analytic function in the angular domain is obtained from the following lemma. Let's start with a definition of $R$ set, see ^[15]. $B(z_n, r_n)$ are a list of disks, that is, $B(z_n, r_n) = \{z:|z-z_n| < r_n\}$. If $\sum_{n = 1}^{\infty}r_n < \infty$ and $z_n\rightarrow \infty$, then $\cup_{n = 1}^{\infty}B(z_n, r_n)$ is called an $R$-set. Obviously, the linear measure of set $\{|z|:z\in\cup_{n = 1}^{\infty}B(z_n, r_n)\}$ is finite.

Lemma 2.3. ([13, Lemma 7]) Let $g(z)$ be analytic in $\Omega(r_0, \alpha, \beta)$ with $\sigma_{\alpha, \beta}(g) < \infty$ for some $r_0 > 0$, and $z = re^{i\psi}, r > r_0+1$ and $\alpha\leq \psi\leq\beta$. Suppose that $n(\geq2)$ is an integer, and that choose $\alpha < \alpha_1 < \beta_1 < \beta$. Then, for every $\varepsilon_j\in(0, (\beta_j-\alpha_j)/2)(j = 1, 2, \ldots, n-1)$ outside a set of linear measure zero with

There exist $K > 0$ and $M > 0$ only depending on $g$, $\varepsilon_1, \ldots, \varepsilon_{n-1}$ and $\Omega(\alpha_{n-1}, \beta_{n-1})$, not depending on $z$, such that

and

for all $z\in \Omega(\alpha_{n-1}, \beta_{n-1})$ outside an R-set, where $k = \pi/(\beta-\alpha)$ and $k_{\varepsilon_j} = \pi/(\beta_j-\alpha_j)(j = 1, 2, \ldots, n-1)$.

3.   Proof of Theorem 1.4

By the condition $\lambda = \max\{\sigma(A_1), \cdots, \sigma(A_n)\} < \mu: = \mu(A_0)$, we define a constant $\kappa: = \frac{\lambda+\mu}{2}$, a set

and a set of directions of rays

Noting that $A_0$ is an entire function and recalling the definition of Nevanlinna characteristic function in the complex plane, for some $r_1 > 0$, if $r > r_1$, we obtain

Consequently,

Recall that the definition of lower order, we have

and since $A_0$ has Petrenko's deviation with (1.6) outside $G$, we deduce that, for $r\not\in G$,

Then there exists a sequence $\{r_n\}(\subset(r_1, +\infty)\setminus G)$ with $\{r_n\}\rightarrow \infty$ as $n\rightarrow \infty$ such that

We define

where $n = 1, 2, \cdots$. Obviously, the set $B_{n}$ is measurable with $meas(B_n)\leq 2\pi$, and monotone decreasing as $n\rightarrow \infty$. Their intersection is

so $\widetilde{H}$ is independent of the value $r$. In view of (3.4) and the monotone convergence theorem [20, Theorem 1.19], it's easy to get

Let's use proof by contradiction to prove our conclusion and suppose that $meas(\overline{L(f)}) < 2\pi\nu$. Compared with (3.5), there exists an interval $(\alpha, \beta)\subset[0, 2\pi)$ such that

Therefor, for any $\theta\in (\alpha, \beta)$, the ray $\{z:\arg z = \theta\}$ is not a limiting direction of Julia set of $f(z+\eta_{\theta})$ for some complex number $\eta_{\theta}\in\{\eta_i:i = 1, 2, \cdots, n\}$ depending on $\theta$. Immediately, by the definition of limiting direction of Julia set, for a constant $\xi_{\theta}$ depending on $\theta$, we have an angular domain $\Omega(\theta-\xi_{\theta}, \theta+\xi_{\theta})$ such that

as $r(\not\in G)$ sufficiently large. Then, for a larger $r_{\theta}$, the angular domain $\Omega(r_{\theta}, \theta-\xi_{\theta}, \theta+\xi_{\theta})$ is contained in an unbounded Fatou component, say $U_{\theta}$, of $F(f(z+\eta_{\theta}))$ (see ^[2]). Suppose that $\Gamma\subset \partial U_{\theta}$ is unbounded and connected, then the mapping

is analytic. Obviously, $\mathbb{C}\setminus \Gamma$ is a simply connected hyperbolic domain, then by the explanation before Lemma 2.1, we have $C_{\mathbb{C}\setminus \Gamma}\geq1/2$ for any $a\in \Gamma\setminus\{\infty\}$. From Lemma 2.1, the mapping $f(z+\eta_{\theta})$ in $\Omega(r_{\theta}, \theta-\xi_{\theta}, \theta+\xi_{\theta})$ satisfies

for $z\in\Omega(r_{\theta}, \theta-\xi_{\theta}+\varepsilon, \theta+\xi_{\theta}-\varepsilon)$, where $\varepsilon$ is a small positive number and $d_1$ is a positive number. For the sake of simplicity, set $\overline{\alpha}: = \theta-\xi_{\theta}+\varepsilon$ and $\overline{\beta}: = \theta+\xi_{\theta}-\varepsilon$.

Choosing suitable $r_{\theta}^*(>r_{\theta})$, for $z\in \Omega(r_{\theta}^*, \overline{\alpha}+\varepsilon, \overline{\beta}-\varepsilon)$ we have $z+c_j-\eta_{\theta}\in\Omega(r_{\theta}, \overline{\alpha}, \overline{\beta})$ ($j = 1, 2, \cdots, m$). By (3.7), we obtain

for $z\in \Omega(r_{\theta}^*, \overline{\alpha}+\varepsilon, \overline{\beta}-\varepsilon)$ as $|z|\rightarrow \infty$, where $M_0$ is a positive constant. Repeating the above argument some times, we can obtain

for $z\in \Omega(r_{\theta}^*, \overline{\alpha}+\varepsilon, \overline{\beta}-\varepsilon)$ as $|z|\rightarrow \infty$, where $K$ and $M$ are positive constants. From (1.14), we have

where $z_n = r_ne^{i\theta}\in \Omega(r_{\theta}^*, \alpha^*+\varepsilon, \beta^*-\varepsilon)$ as $r_n(\not\in G)\rightarrow \infty$, $K$ and $M$ are two large enough constants. This is impossible since $\kappa > \lambda$. Hence, we obtain $meas(\overline{L(f)})\geq 2\pi\nu$.

4.   Proof of Theorem 1.5

By the assumption and the definition of Petrenko's deviation, for $\xi(A_1) > 0$, we have

We choose two constants $\varepsilon$ and $d$ satisfying

and

It's easy to get

Set

Since $A_0$ is entire function, we have

In view the definition of Petrenko's deviation, see (1.2), the formula (4.3) can convert into

Noting that $\varepsilon$ and $d$ satisfy (4.1), we can choose an infinite sequence $\{r_n\}$ such that

Then we get a lower bound of the measure of $I_d(r_n)$. Set $D_n: = \bigcup_{n = j}^{\infty}I_d(r_j)$ and $\widetilde{I_d}: = \bigcap_{n = 1}^{\infty}D_n$, so $\widetilde{I_d}$ is independent of $r$. By the same arguments in section 3, we deduce that

Thus, we estimate the difference between $meas(\widetilde{I_d})$ and $2\pi(1-\xi(A_1))$ as follows.

By the arbitrariness of $\varepsilon$ and the condition $\beta^-(\infty, A_0) < \frac{1}{1-\xi(A_1)}$, we have

Now let's prove the conclusion of the theorem by contradiction, on the contrary we assume

Noting the definition of $\Xi(A_1)$, see (1.10), and (4.7), (4.8), we can find an interval $(\alpha, \beta)\subset (0, 2\pi]$ satisfying

Therefore, for any $\theta\in (\alpha, \beta)$, the corresponding ray is not a limiting direction of Julia set of $f^{(n_{\theta})}(z+\eta)$, where $n_{\theta}$ is an integer and depends on $\theta$. By the definition of limiting direction of Julia set, we can take an angular domain $\Omega(\theta-\xi_{\theta}, \theta+\xi_{\theta})$ with $(\theta-\xi_{\theta}, \theta+\xi_{\theta})\subset (\alpha, \beta)$ satisfying

when $r(\not\in G)$ is large enough, $\xi_{\theta}$ is small enough, and $\eta$ is a complex constant.

For a corresponding $r_{\theta}$, we have $\Omega(r_{\theta}, \theta-\xi_{\theta}, \theta+\xi_{\theta})\subset U_{\theta}$, where $U_{\theta}$ is an unbounded Fatou component of $F(f^{(n_{\theta})}(z+\eta))$ (see ^[2]). Choose $\Gamma\subset \partial U_{\theta}$ such that $\Gamma$ is unbounded and connected set, then $\mathbb{C}\setminus \Gamma$ is a simply connected hyperbolic domain. Applying Lemma 2.1 to the analytic mapping

we deduce that

for $r_{\theta}\rightarrow \infty$ and $z\in\Omega(r_{\theta}, \theta-\xi_{\theta}+\varepsilon, \theta+\xi_{\theta}-\varepsilon)$, where $\varepsilon$ is a sufficiently small positive constant and $d_1$ is a positive constant. Moreover, we can take a smaller angular domain $\Omega(r'_{\theta}, \theta-\xi_{\theta}+2\varepsilon, \theta+\xi_{\theta}-2\varepsilon)$ where $r'_{\theta} > r_{\theta}$, such that

for all $z\in\Omega(r'_{\theta}, \theta-\xi_{\theta}+2\varepsilon, \theta+\xi_{\theta}-2\varepsilon)$. For simplicity, denote $\overline{\alpha}: = \theta-\xi_{\theta}+2\varepsilon$ and $\overline{\beta}: = \theta+\xi_{\theta}-2\varepsilon$. Therefore, it yields that

for all $z\in \Omega(r'_{\theta}, \overline{\alpha}, \overline{\beta})$ as $|z| = r\rightarrow \infty$, where $c_i(i = 0, 1, \cdots, l)$ are complex numbers. Now, there are two cases based on the sign of $n_{\theta}$.

If $n_{\theta} > 0$, by integral we get

this integral is path independent since $f(z)$ is entire. By (4.12) and (4.13), we obtain

for $z\in \Omega(r'_{\theta}, \overline{\alpha}, \overline{\beta})$. Repeating the above integral we get

for $z\in \Omega(r'_{\theta}, \overline{\alpha}, \overline{\beta})$.

If $n_{\theta} < 0$. From (4.12), we have

and $\sigma_{\overline{\alpha}, \overline{\beta}}(r, f^{(n_{\theta})}(z+c_i)) = 0$. Noting the fact $n_{\theta} < 0$ and applying Lemma 2.3 to $f^{(n_{\theta})}(z+c_i)$, we obtain

when $z\in \Omega(r'_{\theta}, \overline{\alpha}+\varepsilon', \overline{\beta}-\varepsilon')$ outside an $R$ set, where $K$ and $M$ are two positive constants and $\varepsilon'$ is a small positive constant. Then by (4.12) and (4.16), it's easy to obtain

for all $z\in \Omega(r'_{\theta}, \overline{\alpha}+\varepsilon', \overline{\beta}-\varepsilon')$ outside an $R$-set.

By (4.14) and (4.17), for any $n_\theta\in\mathbb{Z}$ in (4.12), we always can get

in the corresponding angular domain. When $n_{\theta}\geq0$, $\alpha^* = \overline{\alpha}, \beta^* = \overline{\beta}$; and when $n_{\theta} < 0$, $\alpha^* = \overline{\alpha}+\varepsilon', \beta^* = \overline{\beta}-\varepsilon'$. This yields that $\sigma_{\alpha^*, \beta^*}(r, f(z+c_i)) = 0$, then by Lemma 2.3, for every $i = 1, 2, \cdots, l$ and all $z\in \Omega(r'_{\theta}, \alpha^*+\varepsilon, \beta^*-\varepsilon)$ outside an $R$-set, we can take two constants $K_i > 0$ and $M_i > 0$ such that

From (4.17) and (4.18), it's easy to get that, for all $z\in \Omega(\alpha^*+\varepsilon, \beta^*-\varepsilon)$ outside an $R$-set, the modulus of $(f^{(i)}(z+c_i))^{n_{ij}}$ satisfies

where $M$ is an appropriate constant. Recall the representations of $\widetilde{P_i(z, f)}$ we can obtain that

for all $z\in \Omega(\alpha^*+\varepsilon, \beta^*-\varepsilon)$ outside an $R$-set, and an appropriate constant $M$. From (1.16), we deduce that, for a suitable sequence $\{r_n\}$ with $z_n = r_ne^{i\theta}\in\Omega(r'_{\theta}, \alpha^*+\varepsilon, \beta^*-\varepsilon)$,

as $r_n\rightarrow \infty$. This inequality contradicts the transcendence of $A_0$. So our conclusion is valid.

5.   Proof of Theorem 1.7

Since the entire coefficients $A_0(z), A_1(z)$ of differential equation (1.19) have the same properties as in Theorem 1.5, then the fact in (4.9) also holds. Then, for some integer $n_{\theta}$ which depends on $\theta$, every $\theta\in (\alpha, \beta)$ is not a limiting direction of Julia set of $f^{(n_{\theta})}$. Therefore, we can choose a constant $\xi_{\theta}$ which depends on $\theta$, so the interval $(\theta-\xi_{\theta}, \theta+\xi_{\theta})\subset(\alpha, \beta)$, and the corresponding angular domain $\Omega(\theta-\xi_{\theta}, \theta+\xi_{\theta})$ satisfies

for sufficiently large $r(\not\in G)$. Therefore, for a large enough $r_{\theta} > r$, we have $\Omega(r_{\theta}, \theta-\xi_{\theta}, \theta+\xi_{\theta})\subset U_{\theta}$, where $U_{\theta}$ is an unbounded Fatou component of $F(f^{(n_{\theta})})$ (see ^[2]). Taking an unbounded and connected curve $\Gamma\subset \partial U_{\theta}$ and noting that $\mathbb{C}\setminus \Gamma$ is simply connected hyperbolic domain, then the mapping

meets the conditions of Lemma 2.1. Consequently, we have

for all $z\in\Omega(r_{\theta}, \theta-\xi_{\theta}+\varepsilon, \theta+\xi_{\theta}-\varepsilon)$, where $\varepsilon$ is a sufficiently small positive constant and $d_1$ is a positive constant. Denote $\overline{\alpha}: = \theta-\xi_{\theta}+\varepsilon$ and $\overline{\beta}: = \theta+\xi_{\theta}-\varepsilon$ for simplicity.

If $n_{\theta} > 0$, similar as in section 4, we integrate $f^{(n_{\theta})}$ for $n_{\theta}$ times and note the modulus in (4.11), then

for $z\in \Omega(r_{\theta}, \overline{\alpha}, \overline{\beta})$. Thus, the Nevanlinna angular characteristic of $f$ is

If $n_{\theta} < 0$, from [29, p.49] and Lemma 2.2, we get

for $\varepsilon' = \frac{\varepsilon}{|n_{\theta}|}$. By (4.11) we get $S_{\overline{\alpha}, \overline{\beta}}(r, f^{(n_{\theta})}) = O(1)$. Combining these together, we deduce

Repeating the above argument $|n_{\theta}|$ times, we obtain

From (5.4), (5.6) and combining the two cases above, we deduce that $S_{\alpha^*, \beta^*}(r, f) = O(\log r)$, where

By the definition of order in angular domain, see (2.1), we have $\sigma_{\alpha^*, \beta^*}(r, f) = 0$. Applying Lemma 2.3 to $f$ in the angular domain $\Omega(\alpha^*, \beta^*)$, we can find two constants $K > 0$ and $M > 0$ such that

for $n\in \mathbb{N}$ and all $z\in \Omega(\alpha^*+\varepsilon, \beta^*-\varepsilon)$ outside an $R$-set. We rewrite (1.19) as

From (1.10), (4.2), (5.3), (5.8) and (5.9), we deduce that, for a suitable sequence $z_n = r_ne^{i\theta}\in\Omega(r_{\theta}, \alpha^*+\varepsilon, \beta^*-\varepsilon)$,

as $r_n\rightarrow \infty$. Noting the assumption that $\max\{\sigma(a_{1j}), \sigma(a_{2j})\} < \mu(A_0)$ and $A_0$ is transcendental, the above inequality is not valid.

6.   Conclusions

By using the Nevanlinna theory in angular domain, three theorems (Theorems 1.4, 1.5 and 1.7) about the lower bounds on the measure of sets consisting of Julia limiting directions of solutions to three corresponding complex equations were proved. The three equations include the differential or difference of entire functions. The feature of this paper is that the coefficients of these equations associated with Petrenko's deviation. The results of this paper show that the lower bounds mentioned above have close relation with Petrenko's deviation. Meanwhile, the results also extend some conclusions in the related literatures referenced by this paper.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

This work was supported by Natural Science Foundation of Henan (No. 232300420129) and the Key Scientific Research Project of Colleges and Universities in Henan Province (No. 22A110004), China. The author would like to thank the referee for a careful reading of the manuscript and valuable comments.

Conflict of interest

The author declares no conflicts of interest.