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Integral Functions With Negative Zeros

Published online by Cambridge University Press:  20 November 2018

R. P. Boas Jr.
R. P. Boas Jr.*
Affiliation:
Northwestern University
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If f(z) is an integral function of non-integral order with only real negative zeros, there is a close connection between the rates of growth of the function and of n (r), the number of zeros of absolute value not exceeding r. The best known theorem is that of Valiron [12], which may be stated as follows.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 5 , 1953 , pp. 179 - 184
Copyright
Copyright © Canadian Mathematical Society 1953

References

1. Bowen, N. A., A function-theory proof of Tauberian theorems on integral functions, Quart. J. Math., 19 (1948), 90–100.Google Scholar
2. Bowen, N. A. and Macintyre, A. J., Some theorems on integral functions with negative zeros, Trans. Amer. Math. Soc., 70 (1951), 114–126.Google Scholar
3. Bowen, N. A., An oscillation theorem of Tauberian type, Quart. J. Math. (2), 1 (1950), 243–247.Google Scholar
4. Delange, H., Sur les suites de polynomes ou de fonctions entiéres à zéros réels, Ann. sci. Ec. norm. sup. Paris. (3), 62 (1945), 115–183.Google Scholar
4a. Delange, H., Un théoràme sur les fonctions entiéres à zéros réels et négatifs, J. Math, pures appl. (9), 81 (1952), 55–78.Google Scholar
5. Heins, M., Entire functions with bounded minimum modulus; subharmonic function analogues, Ann. Math. (2), 49 (1948), 200–213.Google Scholar
6. Levinson, N., Gap and density theorems (New York, 1940).Google Scholar
7. Noble, M. E., Extensions and applications of a Tauberian theorem due to Valiron, Proc. Cambridge Phil. Soc, 47 (1951), 22–37.Google Scholar
8. Paley, R. E. A. C. and Wiener, N., Fourier transforms in the complex domain (New York, 1934).Google Scholar
9. Pfluger, A., Die Wertverteilung und das Verhalten von Betrag und Argument einer speziellen Klasse analytischer Funktionen, I, II, Comment. Math. Helv., 11, (1938), 180-214; 12 (1939), 25–65.Google Scholar
10. Titchmarsh, E. C., On integral functions with real negative zeros, Proc. London Math. Soc. (2), 26 (1927), 185–200.Google Scholar
11. Valiron, G., Sur les fonctions entiéres d'ordre fini et d'ordre nul, et en particulier les fonctions à correspondance réguliére, Ann. Fac. Sci. Univ. Toulouse (3), 5 (1914), 117–257.Google Scholar
12. Valiron, G., Sur un théorème de M. Wiman, Opuscula Mathematica A. Wiman Dedicata (Lund, 1930), 1–12.Google Scholar