On new Bloch-type spaces

We introduce a new Bloch-type space, so called, the logarithmic Bloch-type space Blogβα(D) on the unit disc DD, as the space of all holomorphic functions f   on DD such thatsupz∈D(1-|z|)αlneβ/α1-|z|β|f′(z)|<∞,where α>0α>0 and β⩾0β⩾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.