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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
Full Text:
##### References:
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