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)