A note on mixed norm spaces of analytic functions (Q2895750)

The quasi-normed spaces \(H(p,q,\alpha)\) of analytic functions \(f(z)\) in the unit disc \(\mathbb{D}\) with the finite quasi-norm \(\| f\|_{p,q,\alpha}\) are considered. For \(p=q<\infty \) these quasi-normed spaces coincide with the known weighted Bergman spaces. The mixed norm spaces \(h(p,q,\alpha)\) consisting of harmonic functions are recalled, too. The paper gives a new direct and elementary proof of some continuous inclusions of Hardy-Littlewood-Flett type. As an application in mixed norm spaces it is proved (Theorem 2) that the considered inclusions for analytic functions from \(H(p,q,\alpha)\) are strict and best possible in a certain sense.