Abstract
One says that an entire function f of finite exponential type belongs to the Cartwright class C, if
Let N+(r)(N−(r)) denote the number of zeros of the function f in the disk ¦z¦⩾R, such that Re z⩾0 (Re z<0, respectively). We give a simple derivation of the folfollowing result of importance in the theory of entire functions, from a weak type Kolmogorov inequality.
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Literature cited
B. J. Levin, Distribution of Zeros of Entire Functions, Amer. Math. Soc. (1972).
R. P. Boas, Entire Functions, Academic Press, New York (1954).
Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York (1968).
E. Titchmarsh, Theory of Functions, Oxford Univ. Press (1939).
Additional information
I am happy to note that this paper was written in connection with the 50th birthday of my friend V. P. Khavin.
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 76–86, 1984.
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Koosis, P. Derivation of the Cartwright-Levinson theorem from the Kolmogorov theorem. J Math Sci 31, 2687–2693 (1985). https://doi.org/10.1007/BF02107252
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DOI: https://doi.org/10.1007/BF02107252