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Computational Methods and Function Theory 8 (2008), No. 2, 363--372
Copyright Heldermann Verlag 2008
Some Relatives of the Hardy-Stein-Spencer Identities
Finbarr Holland
f.holland@ucc.ie
, University College, Mathematics Department, Cork, Ireland.
[Abstract-pdf]
[Abstract-ps]
Let f be a regular function, r>0 and w a complex number. Denote by
n(w) the total number of roots of f-w that lie in the disc
{|z|. 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.
Keywords:
Regular function, integral means, Hardy-Stein-Spencer identities, Fourier transform.
MSC 2000:
Primary 30E20.
[FullText-pdf (236 K)]
[FullText-ps (396 K)]
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