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A remark on boundedness of Bloch functions

Published online by Cambridge University Press:  17 April 2009

Krzysztof Samotij
Krzysztof Samotij
Affiliation:
Instytut Matematyki, Politechnika Wrocławska, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
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Abstract

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Two consequences of a theorem of Dahlberg are derived. Let f be a holomorphic function in the unit disk D of the complex plane, and let E be an Fσ subset of the unit circle T. Suppose that |f(rw)| ≤ M, ω ∈ T/E, for some constant M.

Then f is bounded in either of the two cases:

(i) if f is in the Bloch space and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log log (2πee/t),

(ii) if f is integrable with respect to the planar Lebesgue measure on D and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log(2πee/t).

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 3 , December 1991 , pp. 527 - 528
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Bonilla, A. and Gonzales, F. Perez, ‘Radial growth and boundedness for Block functions’, Bull. Austral. Math. Soc. 42 (1990), 33–39.Google Scholar
[2]Dahlberg, B.E.I., ‘On the radial boundary values of subharmonic functions’, Math. Scand. 40 (1977), 301–317.CrossRefGoogle Scholar
[3]Goolsby, R.C., ‘Boundedness for Bloch functions’, Rocky Mountain J. Math. 16 (1986), 717–726.CrossRefGoogle Scholar