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On the complex oscillation theory of \(f^{\prime \prime } + A (z)f = 0\) where \(A (z)\) is analytic in the unit disc. (English) Zbl 1193.34174
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 \(\sigma(A)\leq\overline\lambda(f)\).

34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Full Text: DOI
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