Bohr radius for locally univalent harmonic mappings

Abstract

We consider the class of all sense‐preserving harmonic mappings f=h+g¯ of the unit disk double-struckD, where h and g are analytic with g(0)=0, and determine the Bohr radius if any one of the following conditions holds: 1.h is bounded in double-struckD. 2.h satisfies the condition Re h(z)≤1 in double-struckD with h(0)>0. 3.both h and g are bounded in double-struckD. 4.h is bounded and g′false(0false)=0. We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in double-struckD is strictly less than 1. In addition, we determine the Bohr radius for the space scriptB of analytic Bloch functions and the space BH of harmonic Bloch functions. The paper concludes with two conjectures.

Bohr Radius for Subordination and K-quasiconformal Harmonic Mappings

“…If we choose ϕ(z) = (α−z)/(1−αz) with |α| < 1, then ϕ(0) = α and dist(ϕ(0), ∂Ω) = 1−|α| and this clearly give the following corollary (see also [16] or Theorem C with K → ∞).…”

“…Obviously k → 1 corresponds to the case K → ∞. Harmonic extension of the classical Bohr theorem was established in [16]. For example, they proved the following results.…”

“…Some of the recent results from [4,[13][14][15] are included in the latest two surveys. More generally, harmonic version of Bohr's inequality was discussed by Kayumov et al in [16] which will also be recalled below. For certain other results on harmonic Bohr's inequality, we refer to [9,16].…”

Bohr Phenomenon for Locally Univalent Functions and Logarithmic Power Series

“…(2) It is interesting to note that Theorem 1.1 and Theorem 1.3 (and hence Corollary 1.4 and Corollary 1.5) from the paper [14] can also be established using the Lemma 1. One should, however, note that the part of Corollary 1.4 which remarks on the cases that a 0 = 0 or |a 0 | being replaced by |a 0 | 2 would produce a better Bohr radius 1/3 instead of 1/5, has to be proved separately, as [14, Theorem 1.2] can not be derived from the Lemma 1.…”

On a powered Bohr inequality

“…Without lost of generality we may assume that ||h|| ∞ = 1. As in [14], the condition |g ′ (z)| ≤ |h ′ (z)| gives that for each r ∈ [0, 1),…”