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On the complex oscillation of meromorphic solutions of second order linear differential equations in the unit disc. (English) Zbl 1218.34103
This paper is devoted to considering the oscillation theory of
\[ f''+A(z)f=0 \]
in the unit disc \(D\), provided that \(A(z)\) is meromorphic and of finite iterated order
\[ \sigma_{n}(A):=\limsup_{r\rightarrow 1-}\frac{\log_{n}T(r,A)}{-\log (1-r)} \]
in \(D\). Here, \(\log_{n}\) means the usual iterated logarithm starting from \(\log_{0}t=t\). The iterated exponent of convergence for zeros is defined similarly by using \(N(r,1/A)\) instead of \(T(r,A)\). The case when \(A(z)\) is analytic in \(D\), meaning that all solutions are analytic as well, has been treated by T.-B. Cao and H.-X. Yi [Math. Nachr. 282, No. 6, 820–831 (2009; Zbl 1193.34174)]. To obtain the present results in the meromorphic case, ideas and reasoning from [S. B. Bank and I. Laine, Comment. Math. Helv. 58, 656–677 (1983; Zbl 0532.34008)] are being effectively applied. A key notion to describe the results in this paper is the finiteness degree \(i(f)\) of \(f\), see [L. Kinnunen, Southeast Asian Bull. Math. 22, No. 4, 385–405 (1998; Zbl 0934.34076)], where this notion had been introduced in the whole plane case. In the unit disc setting, \(i(f)=0\), if \(D(f):=\limsup_{r\rightarrow 1-}\frac{T(r,f)}{\log 1/(1-r)}<\infty\), while if \(D(f)=\infty\) whenever \(\sigma_{n}(f)=\infty\) for all \(n\), and \(i(f)=\min\{ n | \sigma_{n}(f)<\infty\}\) otherwise. Examples obtained in the present paper are: (1) if \(0<i(A)=n<\infty\) and \(\overline{\lambda}_{n}(A)<\sigma_{n}(A)\), then \(\sigma_{n}(A)\leq\max (\overline{\lambda}_{n}(f),\overline{\lambda}_{n}(1/f))\) for all non-trivial solutions \(f\) of \(f''+A(z)f=0\); (2) if, in addition, \(\delta (\infty ,A)>0\), and \(f_{1},f_{2}\) are linearly independent solutions, then \(\max (\overline{\lambda}_{n+1}(f_{1}),\overline{\lambda}_{n}(f_{2}))=\overline{\lambda}_{n+1}(E)\leq\sigma_{n+1}(f_{1})=\sigma_{n+1}(f_{2})\), here \(E=f_{1}f_{2}\).
MSC:
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
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