Conditions on the logarithmic derivative of a function implying boundedness

MacGregor, T. H.; Rønning, F.

doi:10.1090/S0002-9947-1995-1277126-9

Conditions on the logarithmic derivative of a function implying boundedness
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by T. H. MacGregor and F. Rønning PDF

Trans. Amer. Math. Soc. 347 (1995), 2245-2254 Request permission

Abstract:

In this paper we investigate functions analytic and nonvanishing in the unit disk, with the property that the logarithmic derivative is contained in some domain $\Omega$. We obtain conditions on $\Omega$ which imply that the functions are bounded and that their first derivatives belong to ${H^p}$ for some $p \geqslant 1$. For certain domains $\Omega$ the sufficient conditions that we give are also, in some sense, necessary. Examples of domains to which the results apply are given.

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