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

Jung, Il Bong; Kim, Yong Chan; Srivastava, H. M.

J. Math. Anal. Appl. 176, No. 1, 138-147 (1993).

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 \(\alpha\).

Reviewer: D.Drasin (Lafayette)

MSC:

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)

Keywords:

integral transforms