The Hardy space of analytic functions associated with certain one-parameter families of integral operators. (English) Zbl 0774.30008

The authors consider classes of analytic functions \(f(z)\) \((z\in U=\{| z|<1\})\) with \(\operatorname{Re}\;f'(z)>0\) \((z\in U)\). They show (Theorem 1 and (2.16)) that the image of \(f\) under a variety of integral transforms is continuous on the closed disk, and in particular belongs to all Hardy spaces \(H_ p\). (Note that \(f\) itself need not even be bounded!). One such transform is \[ F(z)=c\int_ 0^ z(\log(z/t)^{\alpha-1}f(t)\,dt \] for a specific \(c\) and any \(\alpha>1)\). Such transforms make sense for a wider range of \(\alpha\) and the authors ask if their conclusions are valid for such \(\alpha\).


30D55 \(H^p\)-classes (MSC2000)
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
46E20 Hilbert spaces of continuous, differentiable or analytic functions
47B38 Linear operators on function spaces (general)
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