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

We consider the classes M (p) (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 M (p) equipped with the topology given by the metric ρ (p) defined by ρ (p)(f, g) = ||f − g||(p) = (∫ (0) (2π)log(p)(1 + M(f − g)(θ))(dθ/2π))(1/p), with f, g∈M (p) and Mf(θ) = sup(0⩽r<1) ⁡|f(re (iθ))|, becomes an F-space. By a result of Stoll (1977), the Privalov space N (p) (1 < p < ∞) with the topology given by the Stoll metric d (p) is an F-algebra. By using these two facts, we prove that the spaces M (p) and N (p) coincide and have the same topological structure. Consequently, we describe a general form of continuous linear functionals on M (p) (with respect to the metric ρ (p)). Furthermore, we give a characterization of bounded subsets of the spaces M (p). Moreover, we give the examples of bounded subsets of M (p) that are not relatively compact.