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On Connections Between Growth and Distribution of Zeros of Integral Functions

Published online by Cambridge University Press:  20 November 2018

Q. I. Rahman
Q. I. Rahman*
Affiliation:
Northwestern University and Muslim University, Aligarh
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The following theorem was proved by Paley and Wiener (4, p. 70; 1, p. 136).

Theorem 1. If f(z) is a canonical product of order 1 with real zeros, and f(0) = 1, the conditions

and

are equivalent. n(r) denotes the number of zeros of absolute value not exceeding r.

Instead of assuming the zeros to be all real Pfluger assumed that the zeros are close to the real axis and proved the following theorem (5 or 1, p. 143).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 40 - 44
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Boas, R.P. Jr., Entire functions (New York, 1954).Google Scholar
2. Boas, R.P. Jr., Integral functions with negative zeros, Can. J. Math., 5 (1953), 179-84.Google Scholar
3. Clunie, J., On a theorem of Noble, J. Lond. Math. Soc. 32 (1956), 138-44.Google Scholar
4. Paley, R. E. A. C. and Wiener, N., Fourier transforms in the complex domain (New York, 1934).Google Scholar
5. Pfluger, A., Ueber gewisse ganze Funktionen vom Exponentialtypus, Comm. Math. Helvet. 16 (1944), 1–18.Google Scholar