an:05570425
Zbl 1193.34174
Cao, Ting-Bin; Yi, Hong-Xun
On the complex oscillation theory of \(f^{\prime \prime } + A (z)f = 0\) where \(A (z)\) is analytic in the unit disc
EN
Math. Nachr. 282, No. 6, 820-831 (2009).
00250198
2009
j
34M10 30D35
linear differential equations; analytic function; complex oscillation theory; exponent of convergence of zeros; exponent of convergence of distinct zeros
The authors study the complex oscillation theory of the second linear differential equation
\[
f''+ A(z) f= 0,\tag{1}
\]
where \(A(z)\) is analytic in the unit disk \(D= \{z:|z|< 1\}\). The order of meromorphic function \(f\) in \(D\) can be defined either as
\[
\sigma(f):= \limsup_{r\to 1-} {\log^+T(r, f)\over-\log(1- r)},
\]
where \(T(r,f)\) is the Nevanlinna characteristic of \(f\). Let \(\lambda(f)\) denote the exponents of convergence of the sequence of zeros of the function \(f\), \(\overline\lambda(f)\) to denote the respectively the exponents of convergence of the sequence of distinct zeros of \(f\), and use the notation \(\sigma_2(f)\) to denote the hyper-order of \(f(z)\). For analytic function \(f\) in \(D\), we also define
\[
\sigma_M(f):= \limsup_{r\to 1-} {\log^+\log^+ M(r,f)\over -\log(1- r)}.
\]
The authors prove the following theorems:
Theorem 1. Let \(A(z)\) be an admissible analytic function in the unit disc \(D\). Then all nonzero solutions \(f\) of equation (1) are of infinite order and satisfy \(\sigma(A)\leq\sigma_2(f)=\sigma_M(A)\).
Theorem 2. Let \(A(z)\) be an admissible analytic function in the unit disc \(D\). If \(\overline\lambda(A)<\sigma(A)\), then every nonzero solution \(f\) of equation (1) satisfies \(\sigma(A)\leq\overline\lambda(f)\).
Alexej Timofeev (Syktyvkar)