Integral means spectrum and the modified Bessel function of zero order

Kayumov, I.

doi:10.1090/S1061-0022-06-00914-9

Integral means spectrum and the modified Bessel function of zero order
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by I. R. Kayumov
Translated by: S. V. Kislyakov

St. Petersburg Math. J. 17 (2006), 453-463

DOI: https://doi.org/10.1090/S1061-0022-06-00914-9

Published electronically: March 9, 2006

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Abstract:

A new characteristic $\beta ^*_f(t)$ of a conformal mapping $f$ of the disk $\Bbb D$ onto a simply connected domain is introduced and its relationship with the so-called integral means spectrum $\beta _f(t)$ is studied. The Brennan conjecture (saying that $\beta _f(-2)\le 1$) is confirmed in the case where the Taylor series of $\log f’(z)$ is Hadamard lacunary with sufficiently large lacunarity exponent.

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