Second order differential equations with transcendental coefficients.

*(English)*Zbl 0596.30047Suppose 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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##### 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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