Uniformly locally univalent functions and Hardy spaces

Abstract

We consider the class of uniformly locally univalent functions on the unit disk with prescribed pre-Schwarzian norm. In the present paper, we show that the class is contained in the Hardy space of a specific exponent depending on the norm.

Uniformly locally univalent harmonic mappings

“…Obviously, the assertion (1) in Theorem B remains valid if we replace B(λ) by B H (λ). By [16,Theorem 1], we see that the assertion (3) does not hold for B H (λ). On the other hand, we will show that B H (λ) is contained in some Hardy space.…”

“…It is known that for λ > 1, the class B(λ) and the subclass B(λ) ∩ S are contained in the Hardy space H p with 0 < p < 1/(λ 2 − 1) and 0 < p < 1/(λ − 1), respectively (cf. [14,16]).…”

Connections Of Linear Operators Defined By Analytic Functions With Qp Spaces

“…Now assume ∈ ∞ and is a sequence in such that → 0. We have Kim and Sugawa in [8,9] investigated various properties of the functions belong to the class . The following Lemma is doue to Becker [3].…”