Proof of a conjecture of Bank and Laine regarding the product of two linearly independent solutions of

Author:
Li-Chien Shen

Journal:
Proc. Amer. Math. Soc. **100** (1987), 301-308

MSC:
Primary 34A20; Secondary 30D35

DOI:
https://doi.org/10.1090/S0002-9939-1987-0884470-4

MathSciNet review:
884470

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a transcendental entire function of order . If and are two linearly independent solutions of the differential equation , then at least one of has the property that the exponent of convergence of its zeros is .

**[1]**K. Arima,*On maximum modulus of integral functions*, J. Math. Soc. Japan**4**(1952), 62-66. MR**0049320 (14:155d)****[2]**S. Bank and I. Laine,*On the oscillation theory of**where**is entire*, Trans. Amer. Math. Soc.**273**(1982), 351-363. MR**664047 (83k:34009)****[3]**-,*On the zeros of meromorphic solutions of second order linear differential equations*, Comment. Math. Helv.**58**(1983). MR**728459 (86a:34008)****[4]**A. Edrei,*The problem of Bank and Laine*, unpublished manuscript.**[5]**B. Ja. Levin,*Distribution of zeros of entire functions*, Transl. Math. Monos., vol. 5, Amer. Math. Soc., Providence, R. I., 1963. MR**589888 (81k:30011)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1987-0884470-4

Keywords:
Linearly independent solutions,
entire functions,
Cartan's lemma,
Carleman's differential inequality

Article copyright:
© Copyright 1987
American Mathematical Society