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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
##### Keywords:
Beurling-Tsuji estimate for harmonic measure
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##### References:
 [1] Albert Baernstein II, Proof of Edrei’s spread conjecture, Proc. London Math. Soc. (3) 26 (1973), 418 – 434. · Zbl 0263.30024 · doi:10.1112/plms/s3-26.3.418 · doi.org [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 [4] A. È. Erëmenko, The growth of entire and subharmonic functions on asymptotic curves, Sibirsk. Mat. Zh. 21 (1980), no. 5, 39 – 51, 189 (Russian). [5] L. C. Shen, On a problem of Bank and Laine concerning the product of two linear independent solutions to $$y'' + Ay = 0$$ (to appear). [6] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959. · Zbl 0087.28401
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