Linear differential equations with exceptional fundamental sets. II

Author:
Norbert Steinmetz

Journal:
Proc. Amer. Math. Soc. **117** (1993), 355-358

MSC:
Primary 34A20; Secondary 30D35, 34C10

DOI:
https://doi.org/10.1090/S0002-9939-1993-1081702-X

MathSciNet review:
1081702

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a sharp order estimate for entire functions of completely regular growth, whose zeros are distributed near finitely many rays in terms of the angles . This result then leads immediately to a proof of a conjecture of Hellerstein and Rossi concerning the distribution of zeros of the solutions of linear differential equations with polynomials coefficients.

**[1]**D. A. Brannan and W. K. Hayman,*Research problems in complex analysis*, Bull. London Math. Soc.**21**(1989), 1-35. MR**967787 (89m:30001)****[2]**F. Brüggemann,*On solutions of linear differential equations with real zeros*;*proof of a conjecture of Hellerstein and Rossi*, Proc. Amer. Math. Soc.**113**(1991), 371-379. MR**1057941 (91m:34007)****[3]**A. Edrei, W. H. J. Fuchs, and S. Hellerstein,*Radial distribution of the values of a meromorphic function*, Pacific J. Math.**11**(1961), 135-151. MR**0122999 (23:A330)****[4]**G. Gundersen,*On the real zeros of solutions of**where**is entire*, Ann. Acad. Sci. Fenn. Ser. A. I Math.**11**(1986), 275-294. MR**853961 (87j:34013)****[5]**S. Hellerstein, L. C. Shen, and J. Williamson,*Real zeros of derivatives of meromorphic functions and solutions of second-order differential equations*, Trans. Amer. Math. Soc.**285**(1984), 759-776. MR**752502 (85j:30065)****[6]**E. Hille,*Lectures on ordinary differential equation*, Wiley, New York, 1976. MR**0499382 (58:17266)****[7]**B. J. Lewin,*Nullstellenverteilung ganzer Funktionen*, Akademie-Verlag, Berlin, 1962. MR**0150301 (27:302)****[8]**N. Steinmetz,*Exceptional values of solutions of linear differential equations*, Math. Z.**201**(1989), 317-326. MR**999730 (90i:30044)****[9]**-,*Linear differential equations with exceptional fundamental sets*, submitted.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
34A20,
30D35,
34C10

Retrieve articles in all journals with MSC: 34A20, 30D35, 34C10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1081702-X

Article copyright:
© Copyright 1993
American Mathematical Society