Meštrović Romeo
Maritime Faculty, University of Montenegro, Dobrota 36, 85330 Kotor, Montenegro.
ScientificWorldJournal. 2014 Jan 28;2014:901726. doi: 10.1155/2014/901726. eCollection 2014.
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.
我们考虑复平面上开单位圆盘𝔻上的全纯函数类M(p) （1 < p < ∞）。这些类实际上是Kim（1986）引入的类M的推广。配备由度量ρp定义的拓扑的空间M(p)，其中ρp(f, g) = ||f - g|| p = (∫0(2π) log(p) (1 + M(f - g)(θ))(dθ/2π))(1/p)，f, g ∈ M(p)且Mf(θ) = sup 0 ⩽ r<1 |f(re(iθ))|，成为一个F - 空间。根据Stoll（1977）的一个结果，配备由Stoll度量dp给出的拓扑的Privalov空间N(p) （1 < p < ∞）是一个F - 代数。利用这两个事实，我们证明空间M(p)和N(p)重合且具有相同的拓扑结构。因此，我们描述了M(p)上连续线性泛函的一般形式（相对于度量ρp）。此外，我们给出了M(p)空间中有界子集的一个特征。而且，我们给出了M(p)中不是相对紧的有界子集的例子。
