On the frequency of zeros of solutions of second order linear differential equations. (English) Zbl 0635.34007

Bank, Steven B.; Laine, Ilpo; Langley, J. K.

Result. Math. 10, 8-24 (1986).

This paper continues work of the authors [Math. Zeitschr. 183, 355-364 (1983; Zbl 0494.34005); Trans. Am. Math. Soc. 273, 351-363 (1982; Zbl 0505.34026); Comment. Math. Helv. 58, 656-677 (1983; Zbl 0532.34008)] on the zeros of solutions of a differential equation in the complex plane, of the type \(f''+a(z)f=0\). The main result is that, under certain assumptions, for any two linearly independent solutions \(f_ 1(z)\) and \(f_ 2(z)\) we have \(\max (\lambda (f_ 1),\lambda (f_ 2))=+\infty,\) where \(\lambda\) (g) denotes the exponent of convergence of the zeros of g(z). The assumptions are that a(z) is a meromorphic function with some specified growth condition.

Reviewer: E.Roxin

MSC:

34M99

Ordinary differential equations in the complex domain

Keywords:

meromorphic function

Citations:

Zbl 0494.34005; Zbl 0505.34026; Zbl 0532.34008

References:

[1]	S. Bank, G. Frank, I. Laine, Uber die Nullstellen von Lösungen linearer Differentialgleichungen, Math. Zeit 183, 355–364 (1983). · Zbl 0494.34005
[2]	S. Bank, I. Laine, On the Oscillation Theory of f” + Af = 0 where A is entire, Trans. Amer. Math. Soc. 273, 351–363 (1982). · Zbl 0505.34026
[3]	S. Bank, I. Laine, Representation of Solutions of Periodic Second Order Linear Differential Equations, J. Reine Angew Math. 344, 1–21 (1983). · Zbl 0524.34007
[4]	S. Bank, I. Laine, On the Zeros of Meromorphic Solutions of Second Order Linear Differential Equations, Comment. Math. Helv. 58, 656–677 (1983). · Zbl 0532.34008
[5]	R. Bellman, Stability Theory of Differential Equations, McGraw-Hill, New York, 1953. · Zbl 0053.24705
[6]	W. K. Hayman, Slowly Growing Integral and Subharmonic Functions, Comment. Math. Helv. 34, 75–84 (1960). · Zbl 0123.26702
[7]	W. K. Hayman, Meromorphic Functions, Oxford at the Clarendon Press, 1964.
[8]	H. Herold, Ein Vergleichssatz für komplexe lineare Differentialgleichungen, Math. Zeit. 126, 91–94 (1972). · Zbl 0226.34005
[9]	E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, Mass. 1969. · Zbl 0179.40301
[10]	M. Ozawa, On a Solution of w” + ew’ + (az + b)w = 0, Kodai Math. J. 3, 295–309 (1980). · Zbl 0463.34028
[11]	F. Tricomi, Differential Equations, Hafner, New York, 1961. · Zbl 0113.40804
[12]	G. Valiron, Lectures on the General Theory of Integral Functions, Chelsea, New York, 1949.