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

MathSciNet review:
884470

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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]**Kihachiro Arima,*On maximum modulus of integral functions*, J. Math. Soc. Japan**4**(1952), 62–66. MR**0049320****[2]**Steven B. Bank and Ilpo Laine,*On the oscillation theory of 𝑓′′+𝐴𝑓=0 where 𝐴 is entire*, Trans. Amer. Math. Soc.**273**(1982), no. 1, 351–363. MR**664047**, 10.1090/S0002-9947-1982-0664047-6**[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. MR**728459**, 10.1007/BF02564659**[4]**A. Edrei,*The problem of Bank and Laine*, unpublished manuscript.**[5]**B. Ja. Levin,*Distribution of zeros of entire functions*, Revised edition, Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. MR**589888**

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DOI:
http://dx.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