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The radial oscillation of solutions to ode's in the complex domain

Published online by Cambridge University Press:  20 January 2009

John Rossi  and
Shupei Wang
John Rossi
Affiliation:
Department of MathematicsVirginia Polytechnic Institute and State UniversityBlacksburg, Virginia 24061–0123, USA
Shupei Wang
Affiliation:
Department of MathematicsUniversity of JoensuuSF-80101 Joensuu, Finland
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Abstract

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We prove three results concerning the oscillation near a ray of solutions to (*)w″ + Aw = 0, where A is an entire function. The first result assumes that A is a polynomial and gives an upper bound on the number of its real zeros if (*) admits a solution with only real zeros and infinitely many. The second result proves that for A of finite order a solution w to (*) has “few” zeros “near” a ray if and only if the same is true for w′. The third result involves the density of the zeros of a solution to (*) “away” from a finite set of rays.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 3 , October 1996 , pp. 473 - 483
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

REFERENCES

1. Barth, K., Brannan, D. and Hayman, W. K., Research problems in complex analysis, Bull. London Math. Soc. 21 (1989), 1–35.Google Scholar
2. Gundersen, G., On the real zeros of solutions of f″ + A(z)f = 0 where A(z) is entire, Ann. Acad. Sci. Fenn. Ser. A I Math. 11 (1986), 275–294.CrossRefGoogle Scholar
3. Hayman, W. K., The local growth of power series – A survey of the Wiman-Valiron method, Canadian Math. Bull. 17 (1974), 317–358.CrossRefGoogle Scholar
4. Hellerstein, S. and Rossi, J., Zeros of meromorphic solutions of second order ordinary differential equations, Math. Z. 192 (1986), 603–612.CrossRefGoogle Scholar
5. Hellerstein, S., Shen, L. C. and Williamson, J., Real zeros of derivatives of meromorphic functions and solutions of second-order differential equations. Trans. Amer. Math. Soc. 285 (1984), 759–776.CrossRefGoogle Scholar
6. Hille, E., Lectures on ordinary differential equations (Addison-Wesley, London, 1969).Google Scholar
7. Hille, E., Ordinary differential equations in the complex domain (Wiley, New York, 1976).Google Scholar
8. Laine, I., Nevanlinna theory and complex differential equations (W. de Gruyter, Berlin, 1993).CrossRefGoogle Scholar
9. Nevanlinna, R., Über die Eigenschaften meromorpher Funcktionen in einem Winkleraum, Acta Soc. Sci. Fenn. 50 (1925), 1–45.Google Scholar
10. Rossi, J., The Tsuji characteristic and real zeros of solutions of second order ordinary differential equation, J. London Math. Soc. 36 (1987), 490–500.CrossRefGoogle Scholar
11. Sheil-Small, T., On the zeros of the derivatives of real entire functions amd Wiman's conjecture, Ann. Math. 129 (1989), 179–193.CrossRefGoogle Scholar
12. Tsuji, M., On Borel's directions of meromorphic functions of finite order I, Tôhoku Math. J. 2 (1950), 97–112.CrossRefGoogle Scholar
13. Wang, S., On the sectorial oscillation theory f″ + A(z)f = 0 (Ann. Acad. Sci. Fenn. A I Math., Dissertationes 92, Helsinki 1994.Google Scholar