A space of analytic functions with Montel property. (Q1860690)

Let \(\omega\) be a non-constant continuous, strictly increasing and subadditive function on \([0,\infty)\) such that \(\omega(e^t)\) is a convex function. The Bergman-Orlitz type space \(A(\omega)\) is defined as a set of all analytic functions \(f\) in the unit disk \(D\) such that \(\omega(| f(z)| )\) is integrable over the normalized Lebesgue measure on \(D\). The space \(A(\omega)\) is an \(F\)-space. In the present paper the authors show that \(A(\omega)\) has the Montel property namely that the topologically compact subsets are relatively compact.