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 ρp (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 ρp). 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.