Growth and oscillation of differential polynomials in the unit disc. (English) Zbl 1213.34112

El Farissi, Abdallah; Belaidi, Benharrat; Latreuch, Zinelaabidine

Electron. J. Differ. Equ. 2010, Paper No. 87, 7 p. (2010).

The authors consider the linear differential equation
\[ f''+ A_1(z) f'+ A_0(z) f= F,\tag{1} \]
where \(A_1(z)\), \(A_0(z)\not\equiv 0\), \(F\) are analytic functions in the unit disk \(D= \{z:|z|< 1|\). It is well-known that all solutions of equation (1) are analytic functions in \(D\) and that there are exactly two linearly independent solutions of (1).
Let \(\lambda(f)\) \((\lambda_2(f))\) denote the exponent (hyper-exponent) of convergence of a sequence of zeros of the function \(f\); let \(\overline\lambda(f)\) \((\overline\lambda_2(f))\) be the exponent (hyper-exponent) of convergence of a sequence of distinct zeros of \(f\), and use the notation \(\rho_2(f)\) to denote the hyper-order of \(f\). Recall that the order of an analytic function \(f\) in \(D\) is defined by
\[ \rho_M(f):= \limsup_{r\to 1-}{\log^+\log^+ M(r,f)\over -\log(1- r)}. \]
The authors give sufficient conditions for the solutions \(f\) of (1) and the differential polynomials generated by (1) to have the same properties of growth and oscillation. The authors present an application of the above results.
Corollary. Let \(A_1\), \(A_0\), \(d_0\), \(d_1\), \(d_2\) be analytic functions in the unit disk \(D\) such that
\[ \max\{\rho(A_1),\rho(d_j)\quad j= 0,1,2\}< \rho(A_0)= \rho\quad (0< \rho<\infty), \]
\(\tau(A_0)= \tau\), \((0<\tau< \infty)\), and let \(\varphi\neq 0\) be an analytic function in \(D\) with \(\rho(\varphi)<\infty\). If \(f\neq 0\) is a solution of the equation
\[ f''+ A_1(z) f'+ A_0(z) f= 0, \]
then the differential polynomial \(g_f= d_2f''+ d_1 f'+ d_0 f\) satisfies
\[ \overline\lambda(g_f- \varphi)= \lambda(g_f- \varphi)= \rho(g_f)= \rho(f)= \infty, \]
\[ \alpha_M\geq\overline\lambda_2(g_f- \varphi)= \lambda_2(g_f- \varphi)= \rho_2(g_f)= \rho_2(f)\geq \rho(A_0), \]
where \(\alpha_M= \max\{\rho_M(A_j): j= 0,1\}\).

Reviewer: Alexej Timofeev (Syktyvkar)

MSC:

34M10	Oscillation, growth of solutions to ordinary differential equations in the complex domain
34M03	Linear ordinary differential equations and systems in the complex domain
30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Keywords:

linear differential equations; analytic solutions; hyper-order; exponent of convergence; hyper-exponent of convergence