Localisation of linear differential equations in the unit disc by a conformal map (Q2800096)

Let us define inductively, for \(r\in \left( 0,+\infty \right) ,\) \(\exp _{1}r=e^{r}\) and \(\exp _{n+1}r=\exp \left( \exp _{n}r\right) ,\) \(n\in \mathbb{N}\). Let \(g\) be an analytic function in the unit disc \(\Delta =\left\{ z\in\mathbb{C}:\left| z\right| <1\right\} ,\) denoted \(g\in \mathcal{H}\left( \Delta \right) \) for short. Then, the iterated \(n\)th-order of \(g\) in \(\Delta \) is defined by NEWLINE\[NEWLINE \rho _{M,n}\left( g\right) =\underset{r\rightarrow 1^{-}}{\lim \sup }\frac{ \log _{n+1}^{+}M\left( r,g\right) }{\log \frac{1}{1-r}}\text{ }\left( n\geq 1 \text{ is an integer}\right) , NEWLINE\]NEWLINE where \(\log _{1}^{+}x=\log ^{+}x=\max \left\{ \log x,0\right\} ,\) \(\log _{n+1}^{+}x=\log ^{+}\left( \log _{n}^{+}x\right) \) and \(M\left( r,f\right) = \underset{\left| z\right| =r}{\max }\left| f\left( z\right) \right| \) is the maximum modulus function.NEWLINENEWLINEThe main purpose of this paper is to use a localisation method via conformal maps to study the growth of solutions of the differential equation NEWLINE\[NEWLINEf^{\left( k\right) }+\sum\limits_{j=0}^{k-1}a_{j}\left( z\right) f^{\left( j\right) }=0, \tag{1}NEWLINE\]NEWLINE where \(a_{j}\left( z\right) =A_{j}\left( z\right) \exp _{n_{j}}\left\{ \frac{ b_{j}}{\left( 1-z\right) ^{q_{j}}}\right\} ,\) \(k,n_{j}\in\mathbb{N},A_{j}\in \mathcal{H}\left( \Delta \cup \left\{ 1\right\} \right) ,\) \( b_{j},q_{j}\in\mathbb{C}\) for \(j=0,1,\cdots ,k-1\). In the first part, the author considers the case when only \(a_{0}\) in \(\left( 1\right) \) is unbounded near a boundary point of the unit disc and obtains the following result.NEWLINENEWLINETheorem 1. Consider the differential equation NEWLINE\[NEWLINEf^{\left( k\right) }+A_{k-1}\left( z\right) f^{\left( k-1\right) }+\cdots +A_{1}\left( z\right) f^{\prime }+A_{0}\left( z\right) \exp _{n}\left\{ \frac{b}{\left( 1-z\right) ^{q}}\right\} f=0, \tag{2} NEWLINE\]NEWLINE where \(k,n\in\mathbb{N},A_{j}\in \mathcal{H}\left( \Delta \cup \left\{ 1\right\} \right) \) for \( j=0,1,\cdots ,k-1,\) \(A_{0}\not\equiv 0,b,q\in\mathbb{C}\backslash \left\{ 0\right\} \) and \(\mathrm{Re}\left( q\right) >0\). Suppose that Im\( \left( q\right) \neq 0\) or \(\left| \arg \left( b\right) \right| < \frac{1}{2}\pi \left( \mathrm{Re}\left( q\right) +1\right) \). Then all nontrivial solutions \(f\) of \(\left( 2\right) \) satisfy \(\rho _{M,n+1}\left( f\right) \geq \mathrm{Re}\left( q\right) \).NEWLINENEWLINEIn the second part of the paper, the author considers a second-order equation with both coefficients possibly unbounded near the point \(z=1,\) namely NEWLINE\[NEWLINEf^{\prime \prime }+A_{1}\left( z\right) \exp \left\{ \frac{b_{1}}{\left( 1-z\right) ^{q_{1}}}\right\} f^{\prime }+A_{0}\left( z\right) \exp \left\{ \frac{b_{0}}{\left( 1-z\right) ^{q_{0}}}\right\} f=0, \tag{3}NEWLINE\]NEWLINE where \(A_{j}\in \mathcal{H}\left( \Delta \cup \left\{ 1\right\} \right) ,\) \( A_{0}\not\equiv 0,b_{j},q_{j}\in\mathbb{C}\backslash \left\{ 0\right\} \) for \(j=0,1\) and \(\mathrm{Re}\left( q_{0}\right) >0\).NEWLINENEWLINETheorem 2. Let \(q_{1}=q_{0}=q\in \left( 2,+\infty \right) \) and \( \arg \left( b_{1}\right) \neq \arg \left( b_{0}\right) \) in \(\left( 3\right) \). Then all nontrivial solutions \(f\) of \(\left( 3\right) \) satisfy \(\rho _{M,2}\left( f\right) \geq q.\)NEWLINENEWLINETheorem 3. Let \(q_{1}=q_{0}=q,\) Im\(\left( q\right) \neq 0,\) \(\mathrm{Re}\left( q\right) >0\) and \(\left| b_{1}\right| <\left| b_{0}\right| \) in \(\left( 3\right) \). Then all nontrivial solutions \(f\) of \(\left( 3\right) \) satisfy \(\rho _{M,2}\left( f\right) \geq \mathrm{Re}\left( q\right) .\)NEWLINENEWLINETheorem 4. Let \(q_{1}\neq q_{0}\) in \(\left( 3\right) \). Assume that either \(q_{0},q_{1}\in \left( 0,+\infty \right) \) and NEWLINE\[NEWLINE\mathrm{Re}\left( \frac{b_{1}}{e^{i\gamma q_{1}}}\right) <0<\mathrm{Re}\left( \frac{b_{0}}{ e^{i\gamma q_{0}}}\right)NEWLINE\]NEWLINE for some \(\gamma \in \left( -\frac{\pi }{2},\frac{\pi }{2}\right) \) or \(\mathrm{Im} \left( q_{0}\right) \neq 0\) and \(\mathrm{Re}\left( q_{1}\right) <\mathrm{Re}\left( q_{0}\right) .\) Then all nontrivial solutions \(f\) of \(\left( 3\right) \) satisfy \(\rho _{M,2}\left( f\right) \geq \mathrm{Re}\left( q_{0}\right) .\)NEWLINENEWLINEThe final part of the paper is devoted to proving some results for analytic solutions of the homogeneous linear differential NEWLINE\[NEWLINE f^{\left( k\right) }+\sum\limits_{j=0}^{k-1}A_{j}\left( z\right) \exp \left\{ \frac{b_{j}}{\left( 1-z\right) ^{q}}\right\} f^{\left( j\right) }=0,NEWLINE\]NEWLINE where \(k\in\mathbb{N},A_{j}\in \mathcal{H}\left( \Delta \cup \left\{ 1\right\} \right) ,\) \(q\in \left( 0,+\infty \right) \) and \(b_{j}\in\mathbb{C}\) for \(j=0,1,\cdots ,k-1\), and the nonhomogeneous linear differential equation NEWLINE\[NEWLINEf^{\left( k\right) }+\sum\limits_{j=0}^{k-1}A_{j}\left( z\right) \exp _{n_{j}}\left\{ \frac{b_{j}}{\left( 1-z\right) ^{q}}\right\} f^{\left( j\right) }=A_{k}\left( z\right) \exp _{n_{k}}\left\{ \frac{b_{k}}{\left( 1-z\right) ^{q_{k}}}\right\} ,NEWLINE\]NEWLINE where \(k\in\mathbb{N}n_{j}\in\mathbb{N},A_{j}\in \mathcal{H}\left( \Delta \cup \left\{ 1\right\} \right) ,\) \( q,q_{k}\in\mathbb{C}\backslash \left\{ 0\right\} \) and \(b_{j}\in\mathbb{C}\) for \(j=0,1,\cdots ,k\). The results of this paper improve the results of \textit{S. Hamouda} [Electron. J. Differ. Equ. 2012, Paper No. 177, 8 p. (2012; Zbl 1254.34121)].