Generic covering properties for spaces of analytic functions. II. (English) Zbl 0602.30055

Stegenga, David A.; Stephenson, Kenneth

Pac. J. Math. 126, No. 1, 153-163 (1987).

[For part I see the authors in Pac. J. Math. 119, 227-243 (1985; Zbl 0534.30040).]
It is known that for \(0<p<\infty\) the Hardy space \(H^ p\) contains a residual set

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of functions, each of which has range equal to the whole plane at every boundary point

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of the unit disc. With quite new general techniques, we are able to show that this result holds for numerous other spaces. The space BMOA of analytic functions of bounded mean oscillation, the Bloch spaces, the Nevanlinna space and the Dirichlet spaces \(D_ a\) for \(0\leq a\leq\) are examples. Our methods involve hyperbolic geometry

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, cluster set analysis and the ”depth” function which we have used previously for determining geometric properties of the image surfaces of functions.

MSC:

30H05

Spaces of bounded analytic functions of one complex variable

Keywords:

analytic functions of bounded mean oscillation; Bloch spaces; Nevanlinna space; Dirichlet spaces; hyperbolic geometry; cluster set

Citations:

Zbl 0564.30040; Zbl 0534.30040