On F-Algebras Mp  (1<p<∞) of Holomorphic Functions

Abstract

We consider the classes Mp (1<p<∞) of holomorphic functions on the open unit disk 𝔻 in the complex plane. These classes are in fact generalizations of the class M introduced by Kim (1986). The space Mp equipped with the topology given by the metric ρp defined by ρp(f,g)=f-gp=∫02π‍logp1+Mf-gθdθ/2π1/p, with f,g∈Mp and Mfθ=sup0⩽r<1⁡f(reiθ), becomes an F-space. By a result of Stoll (1977), the Privalov space Np (1<p<∞) with the topology given by the Stoll metric dp is an F-algebra. By using these two facts, we prove that the spaces Mp and Np coincide and have the same topological structure. Consequently, we describe a general form of continuous linear functionals on Mp (with respect to the metric ρp). Furthermore, we give a characterization of bounded subsets of the spaces Mp. Moreover, we give the examples of bounded subsets of Mp that are not relatively compact.