Linear differential equations with fast growing coefficients in the unit disc

1. Introduction

Consider the linear differential equation

where \(k\geq 1\) is an integer, \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) are analytic functions in the unit disc \(\mathbb{D}=\{z\in \mathbb{C}:|z|< 1\}\) and \(a_{0}(z)\not\equiv 0\). The theory of complex differential equations in the unit disc has been developed since 1980's, see [1]. In the year 2000, Heittokangas in [2] firstly investigated the growth and oscillation theory of Equation (1) when the coefficients \( a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) are analytic functions in the unit disc \(\mathbb{D}\) by introducing the definition of the function spaces. His results also gave some important tools for further investigations on the theory of meromorphic solutions of Equation (1).

In this article, we investigate the growth of solutions of the Equation (1) when the coefficients \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) are analytic in \(\mathbb{D}\), and we deal with the case that the coefficients are fast growing in \(\mathbb{D}\). To define the order of fast-growth of analytic functions, we define inductively for \(r\in \lbrack 0,+\infty ),\) \( \exp _{0}r=r\), \(\exp _{1}r=e^{r}\) and \(\exp _{n+1}r=\exp \left( \exp _{n}r\right) ,\) \(n\in \mathbb{N}\). For all \(r\) sufficiently large, we define \(\log _{0}r=r,\) \(\log _{1}r=\log r\) and \(\log _{n+1}r=\log \left( \log _{n}r\right) ,\) \(n\in \mathbb{N}\). Also, we need to be familiar with the fundamental results and the standard notations of the Nevanlinna's theory on the complex plane \(\mathbb{C}\) and in the unit disc \(\mathbb{D}\), for more details on Nevanlinna theory and its applications in complex differential equations in complex plane and in unit disc, we refer to [2,3, 4, 5, 6, 7].

Before stating our main results, we recall definitions and preliminary remarks concerning meromorphic and analytic functions in \(\mathbb{D}\). For the definitions and more discussions, we refer the reader to [7, 8, 9, 10].

Let \(p\geq 1\) be an integer and \(f\) be a meromorphic function in \(\mathbb{D}\) . Then, the iterated \(p\)-order of \(f\) is defined by \begin{equation*} \rho _{p}(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log _{p}^{+}T(r,f)}{\log \frac{1}{1-r}}, \end{equation*} where \(\log _{1}^{+}r=\log ^{+}r=\max \{\log r;0\}\), \(\log _{p+1}^{+}r=\log ^{+}\left( \log _{p}^{+}r\right) \) and \(T(r,f)\) is the Nevanlinna characteristic function. If \(f\) is analytic in \(\mathbb{D}\), then the iterated \(p\)-order of \(f\) is defined by \begin{equation*} \rho _{M,p}(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log _{p+1}^{+}M(r,f)}{\log \frac{1}{1-r}}, \end{equation*} where \(M(r,f)=\max \left\{ |f(z)|:|z|=r\right\} .\)

Remark 1. For \(p=1\), \(\rho _{1}(f)\) is called order, see [2]. And for \(p=2,\) \(\rho _{2}(f)\) is called hyper-order, see [11].

Remark 2. It follows by [7, page 205] that if \(f\) is an analytic function in \(\mathbb{D}\), then we have the inequalities \begin{equation*} \rho _{1}(f)\leq \rho _{M,1}(f)\leq \rho _{1}(f)+1 \end{equation*} which are the best possible in the sense that there are analytic functions \( g \) and \(h\) such that \(\rho _{1}(g)=\rho _{M,1}(g)\) and \(\rho _{M,1}(h)=\rho _{1}(h)+1\), see [12]. However, it follows by [4,Proposition 2.2.2] {laine} that \(\rho _{p}(f)=\rho _{M,p}(f)\) for \(p\geq 2\).

The iterated \(p\)-type of a meromorphic function \(f\) \ in \(\mathbb{D}\) with \( 0< \rho _{p}(f)< +\infty \) is defined by \begin{equation*} \tau _{p}(f)=\limsup_{r\rightarrow 1^{-}}\left( 1-r\right) ^{\rho _{p}(f)}\log _{p-1}^{+}T(r,f), \end{equation*} and if \(f\) is analytic in \(\mathbb{D}\) with \(0< \rho _{M,p}(f)< +\infty \), then the iterated \(p\)-type is defined by \begin{equation*} \tau _{M,p}(f)=\limsup_{r\rightarrow 1^{-}}\left( 1-r\right) ^{\rho _{M,p}(f)}\log _{p}^{+}M(r,f). \end{equation*}

Remark 3. It follows by [4,Proposition 2.2.2] that \(\tau _{p}(f)=\tau _{M,p}(f)\) for \(p\geq 2\).

2. Basic results

Heittokangas et al. in [10] proved the following results.

Theorem 1. ([10]) Let \(k\in \mathbb{N}\). If the coefficients \(a_{0}(z),a_{1}(z), \dots ,a_{k-1}(z)\) are analytic in \(\mathbb{D}\) such that \(\rho _{M,p}(a_{j})< \rho _{M,p}(a_{0})\) for all \(j=1,\dots ,k-1\), then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\rho _{M,p+1}(f)=\rho _{M,p}(a_{0})\).

Theorem 2.([10]) Let \(k\in \mathbb{N}\). If the coefficients \(a_{0}(z),a_{1}(z), \dots ,a_{k-1}(z)\) are analytic in \(\mathbb{D}\) such that \(\rho _{M,p}(a_{j})\leq \rho _{M,p}(a_{0})\) for all \(j=1,\dots ,k-1\) and \begin{equation*} \sum_{\rho _{M,p}(a_{j})=\rho _{M,p}(a_{0})}\tau _{M,p}(a_{j})< \tau _{M,p}(a_{0}), \end{equation*} then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\rho _{M,p+1}(f)=\rho _{M,p}(a_{0})\).

Hamouda in [13], gave an improvement of Theorem 2 as follows.

Theorem 3.([13]) Let \(k\in \mathbb{N}\). If the coefficients \(a_{0}(z),a_{1}(z), \dots ,a_{k-1}(z)\) are analytic in \(\mathbb{D}\) such that \(\rho _{M,p}(a_{j})\leq \rho _{M,p}(a_{0})\) for all \(j=1,\dots ,k-1\) and \begin{equation*} \max \left\{ \tau _{M,p}(a_{j}):\rho _{M,p}(a_{j})=\rho _{M,p}(a_{0})\right\} < \tau _{M,p}(a_{0}), \end{equation*} then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\rho _{M,p+1}(f)=\rho _{M,p}(a_{0})\).

Our proofs depend mainly upon the following lemmas. Before starting these lemmas, we recall the concept of logarithmic measure. The logarithmic measure of a set \(S\subset (0,1)\) is given by \begin{equation*} lm(S):=\int_{S}\frac{dt}{1-t}. \end{equation*} The set \(F\subset \lbrack 0,1)\) in all this paper is not necessarily the same at each occurrence, but it is always of finite logarithmic measure, that is \(lm(F)< +\infty \). To avoid some problems of the exceptional sets, we need the following lemma.

Lemma 1. ([2, 14) Let \(g:[0,1)\mapsto \mathbb{R}\) and \(h:[0,1)\mapsto \mathbb{R}\) be monotone non-decreasing functions such that \(g(r)\leq h(r)\) holds outside of an exceptional set \(F\subset \lbrack 0,1)\) of finite logarithmic measure. Then there exists a \(d\in (0,1)\) such that if \(s(r)=1-d(1-r)\), then \( g(r)\leq h(s(r))\) for all \(r\in \lbrack 0,1)\).

Lemma 2.[12, Theorem 3.1] Let \(k\) and \(j\) be integers satisfying \(k>j\geq 0\), and let \( \varepsilon >0\) and \(d\in (0,1)\). If \(f\) is a meromorphic in \(\mathbb{D}\) such that \(f^{(j)}\not\equiv 0\), then \begin{equation*} \left\vert \frac{f^{(k)}(z)}{f^{(j)}(z)}\right\vert \leq \left[ \left( \frac{ 1}{1-|z|}\right) ^{2+\varepsilon }\max \left\{ \log \frac{1}{1-|z|} ;T(s(|z|),f)\right\} \right] ^{k-j} \end{equation*} for \(|z|\not\in F,\) where \(F\subset \lbrack 0,1)\) is a set of finite logarithmic measure, and where \(s(|z|)=1-d(1-r)\).

Lemma 3. ([10]) \label{lem2}Let \(f\) be a meromorphic function in the unit disc with \(\rho _{p}(f):=\rho < +\infty \) for some \(p\in \mathbb{N}\), and let \(\varepsilon >0\) be a given constant. Then, there exists a set \(F\subset (0,1)\) of finite logarithmic measure such that for all \(z\) with \(|z|=r\not\in F\) and for all integer \(j\geq 1\), we have:

• (i) If \(p=1\), then
  \begin{equation} \left\vert \frac{f^{(j)}(z)}{f(z)}\right\vert \leq \frac{1}{(1-r)^{j(\rho +2+\varepsilon )}}. \label{conlem3} \end{equation}

  (2)
• (ii) If \(p\geq 2\), then
  \begin{equation} \left\vert \frac{f^{(j)}(z)}{f(z)}\right\vert \leq \exp _{p-1}\left\{ \frac{1 }{(1-r)^{\rho +\varepsilon }}\right\} . \label{conlem4} \end{equation}

  (3)

Lemma 4. ([10]) Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions in the unit disc \(\mathbb{D}\). Then, every solution \(f\not\equiv 0\) of the Equation \((1)\) satisfies \begin{equation*} \rho _{p+1}(f)=\rho _{M,p+1}(f)\leq \max \left\{ \rho _{M,p}(a_{j}):j=0,\dots ,k-1\right\} . \end{equation*}

Remark 4. If \(p\geq 2\), then by Remark 2 and Lemma 4, we obtain that very solution \(f\not\equiv 0\) of the Equation \((1)\) satisfies \begin{equation*} \rho _{p+1}(f)\leq \max \left\{ \rho _{p}(a_{j}):j=0,\dots ,k-1\right\} . \end{equation*}

Lemma 5. ([2, 3, 7]) Let \(f\) be a meromorphic function in the unit disc \(\mathbb{D}\) and let \(k\in \mathbb{N}\). Then \begin{equation*} m\left( r,\frac{f^{(k)}}{f}\right) =S(r,f), \end{equation*} where \(S(r,f)=O\left( \log ^{+}T(r,f)+\log \left( \frac{1}{1-r}\right) \right) \), possibly outside a set \(F\subset \lbrack 0,1)\) with finite logarithmic measure.

Lemma 6. ([15]) Let \(f\) be a meromorphic function in the unit disc \(\mathbb{D}\) for which \(i\left( f\right) =p\geq 1\) and \(\rho _{p}\left( f\right) =\rho < +\infty \), and let \(k\geq 1\) be an integer. Then for any \(\varepsilon >0,\) \begin{equation*} m\left( r,\frac{f^{\left( k\right) }}{f}\right) =O\left( \exp _{p-2}\left\{ \frac{1}{1-r}\right\} ^{\rho +\varepsilon }\right) \end{equation*} holds for all \(r\) outside a set \(F\) \(\subset \lbrack 0,1)\) with \(\int_{F}\frac{dr}{1-r}< +\infty .\)

Lemma 7. For an integer \(p\geq 2\), let \(f\) be a meromorphic function in \( \mathbb{D}\) such that \(0< \rho _{p}(f)=\rho < +\infty \) (see Definition 7), \(0< \tau _{p}(f)=\tau < +\infty \) and \(0< \tau _{p}^{\ast }(f)=\tau ^{\ast }< +\infty \). Then for any given \(\eta < \tau ^{\ast }\), there exists a subset \(E\subset \lbrack 0,1)\) that has an infinite logarithmic measure \(\int_{E}\frac{dr}{1-r}=+\infty \) such that for all \(r\in E,\) we have \begin{equation*} \log _{p-2}T(r,f)>\eta \exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }} \right\} . \end{equation*}

Proof. By the definition of \(\tau _{p}^{\ast }(f)\), there exists an increasing sequence \(\{r_{m}\}_{m=1}^{+\infty }\subset \lbrack 0,1)\) satisfying \(\frac{1 }{m}+\left( 1-\frac{1}{m}\right) r_{m}< r_{m+1},\;(r_{m}\longrightarrow 1^{-}, \) \(m\longrightarrow +\infty )\) and \begin{equation*} \lim_{m\rightarrow +\infty }\frac{\log _{p-2}T(r_{m},f)}{\exp \left\{ \frac{ \tau }{\left( 1-r_{m}\right) ^{\rho }}\right\} }=\tau ^{\ast }. \end{equation*} Then, for any given \(0< \varepsilon < \tau ^{\ast }\), there exists a positive integer \(m_{0}\) such that for all \(m\geq m_{0}\), we have

For \(r\in \left[ r_{m},\frac{1}{m}+\left( 1-\frac{1}{m}\right) r_{m}\right] \), we get \begin{equation*} \lim_{m\rightarrow +\infty }\frac{\exp \left\{ \tau \left[ \left( 1-\frac{1}{ m}\right) \left( \frac{1}{1-r}\right) \right] ^{\rho }\right\} }{\exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}\right\} }=1. \end{equation*} Then for any given \(0< \eta < \tau ^{\ast }-\varepsilon \), there exists a positive integer \(m_{1}\) such that for all \(m\geq m_{1}\), and for all \(r\in \left[ r_{m},\frac{1}{m}+\left( 1-\frac{1}{m}\right) r_{m}\right] \), we have By (4) and (5), for all \(m\geq m_{2}=\max \left\{ { m_{0};m_{1}}\right\} \) and for all \(r\in \left[ r_{m},\frac{1}{m}+\left( 1- \frac{1}{m}\right) r_{m}\right] \), we have \begin{eqnarray*} \log _{p-2}T(r,f)&\geq& \log _{p-2}^{+}T(r_{m},f)>(\tau ^{\ast }-\varepsilon )\exp \left\{ \frac{\tau }{\left( 1-r_{m}\right) ^{\rho }}\right\}\\ &\geq& (\tau ^{\ast }-\varepsilon )\exp \left\{ \tau \left[ \left( 1-\frac{1}{m }\right) \left( \frac{1}{1-r}\right) \right] ^{\rho }\right\} >\eta \exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}\right\} . \end{eqnarray*} Set \begin{equation*} E=\bigcup_{m=m_{2}}^{+\infty }\left[ r_{m},\frac{1}{m}+\left( 1-\frac{1}{m} \right) r_{m}\right] . \end{equation*} Then \begin{equation*} \int_{E}\frac{dt}{1-t}=\sum_{m=m_{2}}^{+\infty }\int_{r_{m}}^{\frac{1}{m} +\left( 1-\frac{1}{m}\right) r_{m}}\frac{dt}{1-t}=\sum_{m=m_{2}}^{+\infty }\log \frac{m}{m-1}=+\infty . \end{equation*}

By using similar reasoning as in the proof of Lemma 7, we easily obtain the following lemma.

Lemma 8. For an integer \(p\geq 2\), let \(f\) be a meromorphic function in \(\mathbb{D}\) such that \(0< \rho _{M,p}(f)=\rho < +\infty \), \(0< \tau _{M,p}(f)=\tau < +\infty \) and \(0< \tau _{M,p}^{\ast }(f)=\tau ^{\ast }< +\infty \). Then for any given \(\eta < \tau ^{\ast }\), there exists a subset \(E\subset \lbrack 0,1)\) that has an infinite logarithmic measure \(\int_{E} \frac{dr}{1-r}=+\infty \) such that for all \(r\in E,\) we have \begin{equation*} \log _{p-1}M(r,f)>\eta \exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }} \right\} . \end{equation*}

Lemma 9. ([16]) Let \(f\) be a solution of Equation \(\left( 1\right) ,\)where the coefficients \(a_{j}\left( z\right) \) \(\left( j=0,...,k-1\right) \) are analytic functions in the disc \(\Delta _{R}=\left\{ z\in\mathbb{C}:\left\vert z\right\vert < R\right\} ,\) \(0< R\leq \infty .\) Let \(n_{c}\in \left\{ 1,...,k\right\} \) be the number of nonzero coefficients \(a_{j}\left( z\right) \) \(\left( j=0,...,k-1\right) ,\) and let \(\theta \in \left[ 0,2\pi \right] \) and \( \varepsilon >0.\) If \(z_{\theta }=\nu e^{i\theta }\in \Delta _{R}\) is such that \(a_{j}\left( z_{\theta }\right) \neq 0\) for some \(j=0,...,k-1,\) then for all \(\nu < r< R,\) \begin{equation*} \left\vert f\left( re^{i\theta }\right) \right\vert \leq C\exp \left( n_{c} \overset{r}{\underset{\nu }{\int }}\underset{j=0,...,k-1}{\max }\left\vert a_{j}\left( te^{i\theta }\right) \right\vert ^{\frac{1}{k-j}}dt\right) , \end{equation*} where \(C>0\) is a constant satisfying \begin{equation*} C\leq \left( 1+\varepsilon \right) \underset{j=0,...,k-1}{\max }\left( \frac{ \left\vert f^{\left( j\right) }\left( z_{\theta }\right) \right\vert }{ \left( n_{c}\right) ^{j}\underset{n=0,...,k-1}{\max }\left\vert a_{n}\left( z_{\theta }\right) \right\vert ^{\frac{j}{k-n}}}\right) . \end{equation*}

Lemma 10. Let \(\{a_{j}(z)\}_{0\leq j\leq k-1}\) be analytic functions in the disc \(\mathbb{D}\) such that \(0< p< \infty \) and \(0< \max \{\rho _{M,p}(a_{j}):j=1,\dots ,k-1\}\leq \rho _{M,p}\left( a_{0}\right) =\rho < \infty \) and \(\max \{\tau _{M,p}(a_{j}):j=1,\dots ,k-1\}\leq \tau _{M,p}\left( a_{0}\right) =\tau < \infty \). Then, every solution \(f\not\equiv 0\) \textit{\ }of the Equation \(\left( 1\right) \) with \(\rho _{p+1}(f)=\rho \) satisfies \(\tau _{p+1}(f)\leq \tau \).

Proof. Let \(f\not\equiv 0\) be a solution of \(\left( 1\right) \) with \(\rho _{p+1}(f)=\rho .\) Let \(\theta _{0}\in \left[ 0,2\pi \right) \) be such that \( \left\vert f\left( re^{i\theta _{0}}\right) \right\vert =M\left( r,f\right) . \) By Lemma 9, we have

We have \(\max \left\{ \rho _{M,p}(a_{j}):j=0,1,...,k-1\right\} =\rho _{M,p}\left( a_{0}\right) =\rho \). By the definition of \(\tau _{M,p}(a_{j})\) , for any given \(\varepsilon >0\) and \(r\rightarrow 1^{-},\) we obtain By (6) and (7), we have for \(r\rightarrow 1^{-}\) Then, it follows from (8), arbitrariness of \(\varepsilon >0\) that \(\tau _{p+1}(f)=\tau _{M,p+1}(f)\leq \tau .\)

3. Main results

In this article, we aim to answer the following questions:

1. What can be said about the growth of solutions of the Equation \((1)\) in the case when \(\rho _{M,p}(a_{j})\leq \rho _{M,p}(a_{0})\) for all \(j=1,\dots ,k-1\) and \begin{equation*} \max \left\{ \tau _{M,p}(a_{j}):\rho _{M,p}(a_{j})=\rho _{M,p}(a_{0})\right\} \leq \tau _{M,p}(a_{0})? \end{equation*}
2. What happened when we replace \(\rho _{M,p}\) and \(\tau _{M,p}\) by \(\rho _{p}\) and \(\tau _{p}\)?

As the first result, we give an improvement to Theorems 1 and 2 of [13].

Theorem 4. Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be meromorphic functions in the unit disc \(\mathbb{D}\), and \(a_{0}(z)\not\equiv 0\). Suppose that there exist a point \(\omega \in \partial \mathbb{D}\), a curve \(\gamma \) tending to \(\omega \) and a set \(F_{1}\subset (0,1)\) of finite logarithmic measure such that for \(z\in \gamma \) and \(|z|=r\not\in F_{1},\) we have for the largest integer \(p\geq 1\)

for all \(\lambda >0\) and \(\alpha >0\). Then, every nontrivial meromorphic solution \(f\) of the Equation \((1)\) satisfies \(\rho _{p+1}(f)=+\infty \).

Proof. Suppose that \(f\not\equiv 0\) is a solution of the Equation (1) with \(\rho _{p+1}(f)=\rho < +\infty \). By (1), \(f\) satisfies

For \(p\geq 1\), by (3), for all \(z\) satisfying \(|z|=r\not\in F\) (\( F\) has finite logarithmic measure), we obtain where \(\alpha >0\) is a constant which depends on \(\rho \), \(\varepsilon \) and \(j=1,\dots ,k\). By substituting (10) into (11), it yields By (9), for \(z\in \gamma \) such that \(|z|=r\not\in F_{1}\) (\( F_{1} \) has finite logarithmic measure), we know that as \(z\rightarrow \omega \) Thus, for all \(z\in \gamma \) with \(|z|=r\not\in F_{1}\cup F\), by (12) and (13), we get a contradiction. Hence, every meromorphic solution \( f\not\equiv 0\) of (1) has an infinite \((p+1)\)-order.

Remark 5. In [13], under the same hypotheses of Theorem 4, Hamouda obtained that \(\rho _{p+1}(f)\geq \alpha .\)

In all the next, we consider \(p\in \mathbb{N}\backslash \{1\}\). In trying to give an answer on the above questions, we prove the following results.

Theorem 5. Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions in the unit disc \(\mathbb{D}\) satisfying \(\rho _{M,p}(a_{j})\leq \rho _{M,p}(a_{0})=\rho \) \((0< \rho < +\infty )\) and \(\tau _{M,p}(a_{j})\leq \tau _{M,p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\). Suppose that there exist two positive real numbers \(\alpha \) and \(\beta \) with \( 0\leq \beta < \alpha \), such that

and as \(|z|=r\rightarrow 1^{-}\) for \(r\in E_{1}\) (\(E_{1}\) is of infinite logarithmic measure). Then, every solution \(f\not\equiv 0\) of the Equation \( \left( 1\right) \) satisfies \(\rho _{p}\left( f\right) =+\infty ,\) \( \rho _{p+1}\left( f\right) =\rho \) and \(d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).

Proof. Let \(f\not\equiv 0\) be a solution of the Equation (1). By (1), \(f\) satisfies

By hypotheses of Theorem 5 and Lemma 4, we know that \( \rho _{p+1}(f)\leq \rho \). Suppose that \(\rho _{p+1}(f)=\rho _{1}< \rho \). Then, by (3) for all \(0< \varepsilon < \rho -\rho _{1}\), we have where \(|z|=r\not\in F\). By substituting (14), (15) and (17) into (16) we obtain for all \(r\in E_{1}\backslash F\). Hence, we get for some constant \(C_{1}>0,\) which is a contradiction as \(|z|=r\rightarrow 1^{-}\), \(r\in E_{1}\backslash F\), since \(\alpha >\beta \geq 0\) and \(\rho >\rho _{1}+\varepsilon \). Thus, \(\rho _{p+1}(f)=\rho \). Now, by Lemma 2, we have for \(j=1,\dots ,k\) for all \(r\not\in F_{2}\cup \lbrack 0,1]\). From, (14), (15), (16) and (20), we obtain for all \(r\in E_{1}\backslash (F_{2}\cup \lbrack 0,1])\). Hence for some constant \(C_{2}>0\) and for all \(r\in E_{1}\backslash (F_{2}\cup \lbrack 0,1])\). Setting \(R=s(r)=1-d(1-r)\), \(d\in (0,1)\). We have \(1-r=\frac{ 1-R}{d},\) \(d\in (0,1)\). Then by Lemma 1, we obtain we obtain for \( R\longrightarrow 1^{-}\) Since \(0< \rho _{p+1}(f)=\rho < \infty ,\) from (23) we deduce that \begin{equation*} \tau _{p+1}(f)=\underset{R\rightarrow 1^{-}}{\lim \sup }\frac{\log _{p}^{+}T(R,f)}{\frac{1}{\left( 1-R\right) ^{\rho }}}\geq d^{\rho }\tau . \end{equation*} By Lemma 10, we conclude that \(d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau .\)

Theorem 6. Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions in the unit disc \(\mathbb{D}\) satisfying \(\rho _{p}(a_{j})\leq \rho _{p}(a_{0})=\rho \) \((0< \rho < +\infty )\) and \(\tau _{p}(a_{j})\leq \tau _{p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\). Suppose that there exist two positive real numbers \(\alpha \) and \(\beta \) with \( 0\leq \beta < \alpha \), such that

and as \(|z|=r\rightarrow 1^{-}\) for \(r\in E_{2}\) (\(E_{2}\) is of infinite logarithmic measure). Then, every solution \(f\not\equiv 0\) of the Equation \( \left( 1\right) \) satisfies \(\ \rho _{p}\left( f\right) =+\infty ,\) \(\rho _{p+1}\left( f\right) =\rho \) and \(d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).

Proof. Let \(f\not\equiv 0\) be a solution of the Equation (1). By (1) we can write

By hypotheses of Theorem 6 and Lemma 4, we know that \( \rho _{p+1}(f)\leq \rho \). Suppose that \(\rho _{p+1}(f)=\rho _{1}< \rho \). Then by Lemma 6, for all \(0< \varepsilon < \rho -\rho _{1}\) and for all \(|z|=r\notin F\), we have for \(j=1,\dots ,k\) Now, let \(p\geq 2\), it follows by (24), (25), (26) and (27) that holds for all \(z\) satisfying \(|z|=r\in E_{2}\backslash F\) as \(r\rightarrow +\infty ,\) and \(M>0\) is some constant. Hence, from (\ref{24}) we obtain \begin{equation*} (\alpha -\beta )\exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }} \right\} \leq \exp \left\{ \frac{1}{(1-r)^{\rho _{1}+\varepsilon }}\right\} +C_{3} \end{equation*} for some constant \(C_{3}>0,\) which is a contradiction as \(|z|=r\rightarrow 1^{-}\), \(r\in E_{2}\backslash F\), since \(\alpha >\beta \geq 0\) and \(\rho >\rho _{1}+\varepsilon \). Thus, \(\rho _{p+1}(f)=\rho \). Now, it follows by (24), (25), (26) and Lemma 5 that for all sufficiently large \(|z|=r\in E_{2}\backslash F\). Then, for all sufficiently large \(|z|=r\in E_{2}\backslash F\) for some constant \(C_{4}>0.\) Then by Lemma 1, we obtain for \( |z|=r\in E_{2},\) \(s\left( r\right) \rightarrow 1^{-}\) where \(s(r)=1-d(1-r)\), \(d\in (0,1)\). Hence, by (31) we obtain \begin{equation*} \tau _{p+1}(f)=\underset{s\left( r\right) \rightarrow 1^{-}}{\lim \sup } \frac{\log _{p}^{+}T(s\left( r\right) ,f)}{\frac{1}{\left( 1-s\left( r\right) \right) ^{\rho }}}\geq d^{\rho }\tau . \end{equation*} By Lemma 10, we conclude that \(d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau .\)

Hamouda in [17], to study the growth of meromorphic solutions of differential equations with finite iterated \(p\)-order in complex plane, introduced new type of growth (see[17, p. 46]). According to the definition of this new type of growth, we introduce a new definition of type of growth that we note \(\tau _{p}^{\ast }(f)\) related to iterated \(p\)-order of meromorphic function \(f\) in the unit disc, as follows.

Definition 7. For \(p\geq 2,\) let \(f\) be a meromorphic function of finite iterated \(p\) -order in \(\mathbb{D}\) such that \(0< \rho _{p}\left( f\right) =\rho < +\infty \) and \(0< \tau _{p}\left( f\right) =\tau < +\infty \), we define \(\tau _{p}^{\ast }(f)\) by \begin{equation*} \tau _{p}^{\ast }(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log _{p-2}^{+}T(r,f) }{\exp \left\{ \frac{\tau }{\left( 1-r\right) ^{\rho }}\right\} }. \end{equation*} If \(f\)\ is an analytic function in \(\mathbb{D}\) with \(0< \tau _{M,p}\left( f\right) =\tau _{M}< +\infty \), we also define \begin{equation*} \tau _{M,p}^{\ast }(f)=\limsup_{r\rightarrow 1^{-}}\frac{\log _{p-1}^{+}M(r,f)}{\exp \left\{ \frac{\tau _{M}}{\left( 1-r\right) ^{\rho }} \right\} }. \end{equation*}

The following theorems improves and extends Theorems 2 and 3.

Theorem 8. Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions in the unit disc \(\mathbb{D}\) satisfying \(\rho _{M,p}(a_{j})\leq \rho _{M,p}(a_{0})=\rho \) \((0< \rho < +\infty )\) and \(\tau _{M,p}(a_{j})\leq \tau _{M,p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\) and \ \begin{equation*} \max \left\{ \tau _{M,p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \tau _{M,p}^{\ast }(a_{0}). \end{equation*} Then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\ \rho _{p}\left( f\right) =+\infty ,\) \(\rho _{p+1}\left( f\right) =\rho \) and \( d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).

Proof. Suppose that all coefficients of the Equation (1) satisfy the hypotheses of Theorem 8. Now, let \(\alpha \) and \(\beta \) be two real numbers such that \begin{equation*} \max \left\{ \tau _{M,p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \beta < \alpha < \tau _{M,p}^{\ast }(a_{0}). \end{equation*} Because all coefficients are analytic, then for \(r\longrightarrow 1^{-}\)

and by Lemma 8, we have for all \(r\in E\) (\(E\) is a set of infinite logarithmic measure). \ From (32) and (33), and by Theorem 5, we obtain the result.

Theorem 9. Let \(a_{0}(z),a_{1}(z),\dots ,a_{k-1}(z)\) be analytic functions in the unit disc \(\mathbb{D}\) satisfying \(\rho _{p}(a_{j})\leq \rho _{p}(a_{0})=\rho \) \((0< \rho < +\infty )\), \(\tau _{p}(a_{j})\leq \tau _{p}(a_{0})=\tau \) \((0< \tau < +\infty )\) for all \(j=1,\dots ,k-1\) and \ \begin{equation*} \max \left\{ \tau _{p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \tau _{p}^{\ast }(a_{0}). \end{equation*} Then all solutions \(f\not\equiv 0\) of \((1)\) satisfy \(\ \rho _{p}\left( f\right) =+\infty ,\) \(\rho _{p+1}\left( f\right) =\rho \) and \( d^{\rho }\tau \leq \tau _{p+1}(f)\leq \tau ,\) \(d\in \left( 0,1\right) \).

Proof. Suppose that all coefficients of the Equation (1) satisfy the hypotheses of Theorem 9. Now, let \(\alpha \) and \(\beta \) be two real numbers such that \begin{equation*} \max \left\{ \tau _{p}^{\ast }(a_{j}):j=1,\dots ,k-1\right\} < \beta < \alpha < \tau _{p}^{\ast }(a_{0}). \end{equation*} Since all coefficients are analytic, then for \(r\longrightarrow 1^{-}\)

and by Lemma 7, we have for all \(r\in E\) (\(E\) is a set of infinite logarithmic measure). From (34) and (35), and by Theorem 6, we obtain the result.

For some related results in the whole complex plane, see [18].

Acknowledgments

The authors would like to thank the referee for his/her valuables remarks, which led to an improvement of the presentation of this paper. This paper is supported by University of Mostaganem (UMAB) (PRFU Project Code C00L03UN270120180005).

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The authors do not have any competing interests in the manuscript.