On new Bloch-type spaces

Abstract

We introduce a new Bloch-type space, so called, the logarithmic Bloch-type space Blogβα (D) on the unit disc D, as the space of all holomorphic functions f on D such thatunder(sup, z ∈ D) (1 - | z |)α fenced(ln frac(eβ / α, 1 - | z |))β | f′ (z) | < ∞,where α > 0 and β ≥ 0, and present some basic properties of the space. A necessary and a sufficient condition for a function with Hadamard gaps to belong to the logarithmic Bloch-type space are given, as well as some applications of these results to a composition operator.