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

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
Full Text:

### 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.
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