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Second order differential equations with transcendental coefficients. (English) Zbl 0596.30047
Suppose that \(w_ 1,w_ 2\) are linearly independent solutions of the differential equation \(w''+Aw=0\), where A is a transcendental entire function. Let \(\lambda\) denote the exponent of convergence of the sequence of zeros of \(w_ 1w_ 2\). S. Bank and I. Laine [Trans. Am. Math. Soc. 273, 351-363 (1982; Zbl 0505.34026)] proved that if the order \(\rho\) (A) of A is \(<\) then \(\lambda =\infty\). Such a result does not hold if \(\rho\) (A) is a positive integer or \(\infty\), but it is a very interesting open question whether it holds for other \(\lambda\). In this paper the author uses the Beurling-Tsuji estimate for harmonic measure, together with a clever formula of Bank-Laine, to prove that \(\lambda =\infty\) when \(\rho (A)=\). He actually proves a more general inequality: If \(\leq \rho (A)<1\) then \(\rho (A)^{-1}+\rho (E)^{-1}\leq 2\). In view of the conjecture mentioned above, it would be quite surprising if this turned out to be sharp. Related results have been obtained by L. C. Shen (preprint, 1985).
Reviewer: A.Baernstein II

MSC:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34M99 Ordinary differential equations in the complex domain
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[2] Steven B. Bank and Ilpo Laine, On the oscillation theory of \?\(^{\prime}\)\(^{\prime}\)+\?\?=0 where \? is entire, Trans. Amer. Math. Soc. 273 (1982), no. 1, 351 – 363. · Zbl 0505.34026
[3] Steven B. Bank and Ilpo Laine, On the zeros of meromorphic solutions and second-order linear differential equations, Comment. Math. Helv. 58 (1983), no. 4, 656 – 677. · Zbl 0532.34008 · doi:10.1007/BF02564659 · doi.org
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