Abstract
Let f be a regular function, r > 0 and ω a complex number. Denote by n(ω) the total number of roots of f − ω that lie in the disc ¦z¦ < r. An expression for the (two-dimensional) Fourier transform of n is derived from the Hardy-Stein-Spencer identities, and some consequences of it are explored. In particular, taking account of another identity due to Stein, we obtain a triple identity involving the real part of f, when this is positive, which doesn’t appear to have been noticed before.
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Dedicated to Walter Hayman to mark his 80th birthday
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Holland, F. Some Relatives of the Hardy-Stein-Spencer Identities. Comput. Methods Funct. Theory 8, 363–372 (2008). https://doi.org/10.1007/BF03321693
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DOI: https://doi.org/10.1007/BF03321693