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On Connections Between Growth and Distribution of Zeros of Integral Functions

  • Q. I. Rahman (a1)

Extract

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).

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References

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1. Boas, R.P. Jr., Entire functions (New York, 1954).
2. Boas, R.P. Jr., Integral functions with negative zeros, Can. J. Math., 5 (1953), 179-84.
3. Clunie, J., On a theorem of Noble, J. Lond. Math. Soc. 32 (1956), 138-44.
4. Paley, R. E. A. C. and Wiener, N., Fourier transforms in the complex domain (New York, 1934).
5. Pfluger, A., Ueber gewisse ganze Funktionen vom Exponentialtypus, Comm. Math. Helvet. 16 (1944), 118.
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On Connections Between Growth and Distribution of Zeros of Integral Functions

  • Q. I. Rahman (a1)

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