[Paper Review] A note on the Bohr inequality

Bappaditya Bhowmik, Nilanjan Das|arXiv (Cornell University)|Nov 15, 2019

Advanced Banach Space Theory14 references4 citations

TL;DR

This paper establishes improved Bohr radius estimates for the derivatives of analytic functions and for subordinating families of odd analytic functions, using subordination and majorization techniques. It proves that for odd analytic functions $ f $ with $ f(0) = 0 $, the Bohr radius for $ f' $ is at least $ 1/ ho $, and provides sharp radii $ r_s $ and $ r_c $ for spherically convex and convex functions of bounded type, respectively, under geometric constraints on the image domain.

ABSTRACT

This article focuses on the Bohr radius problem for the derivatives of analytic functions, along with a technique of establishing Bohr inequalities in classical and generalized settings.

Motivation & Objective

• To investigate the Bohr phenomenon for the derivatives of analytic self-maps of the unit disk, particularly under the condition $ f(0) = 0 $.
• To extend classical Bohr inequalities to generalized settings involving subordination and majorization for odd analytic functions.
• To derive sharp Bohr radius estimates for spherically convex functions $ ilde{\mathcal{K}}_s(\alpha) $ and convex functions of bounded type $ CV(R_1, R_2) $.
• To establish comparison theorems between $ \mathcal{M}_{g'}(r) $ and $ \mathcal{M}_{f'}(r) $ under subordination or majorization, even when $ f $ is not locally univalent.
• To determine conditions under which the majorant series of $ f' $ remains bounded by the distance from $ f(0) $ to the boundary of $ f(\mathbb{D}) $.

Proposed method

• Utilizes the concept of subordination $ g \prec f $, where $ g = f \circ \phi $ for $ \phi $ analytic with $ \phi(0) = 0 $, $ |\phi(z)| < 1 $.
• Applies Lemma 1 (from [6]) to compare majorant series $ \mathcal{M}_g(r) $ and $ \mathcal{M}_f(r) $, extending it to derivative series $ \mathcal{M}_{g'}(r) $ and $ \mathcal{M}_{f'}(r) $.
• Derives sharp bounds for $ \mathcal{M}_f(r) \leq \delta = d(f(0), \partial f(\mathbb{D})) $ using curvature and geometric constraints on $ f $.
• For spherically convex functions $ f \in \mathcal{K}_s(\alpha) $, uses the representation $ f(z)/\alpha z \prec 1/(1 + z\sqrt{1 - \alpha^2}) $ to bound coefficients.
• For $ CV(R_1, R_2) $ functions, applies the subordination $ f(z) \prec Bz/(1 - Az) $ with $ A = (R_2 - \delta)/R_2 $, $ B = (2R_2 - \delta)\delta/R_2 $.
• Establishes sharpness via direct computation on extremal functions $ k_\alpha(z) = \alpha z / (1 - \sqrt{1 - \alpha^2} z) $ and $ k_a(z) = z / (1 - a z) $.

Experimental results

Research questions

• RQ1 Can a Bohr radius independent of $ f $ be established for $ f' $ when $ f: \mathbb{D} \to \mathbb{D} $ is analytic and $ f(0) = 0 $?
• RQ2 What is the best possible radius $ r $ such that $ \mathcal{M}_{f'}(r) \leq \delta $ for odd analytic functions $ f $ with $ f(0) = 0 $?
• RQ3 How do subordination and majorization relations between analytic functions influence the majorant series of their derivatives?
• RQ4 What are the sharp Bohr radii for spherically convex functions $ \mathcal{K}_s(\alpha) $, and how do they depend on $ \alpha $ and $ \delta $?
• RQ5 Under what conditions on $ R_1, R_2, \delta $ does the Bohr inequality $ \mathcal{M}_f(r) \leq \delta $ hold for functions in $ CV(R_1, R_2) $?

Key findings

• For odd analytic functions $ f(z) = \alpha z + \sum_{n=2}^\infty a_n z^n \in \mathcal{K}_s(\alpha) $ with $ 0 < \alpha \leq \sqrt{3}/2 $, the majorant series satisfies $ \mathcal{M}_f(r) \leq \delta $ for all $ r \leq r_s = 1/(1 + 2\sqrt{1 - \alpha^2}) $, and this radius is sharp.
• The extremal function $ k_\alpha(z) = \alpha z / (1 - \sqrt{1 - \alpha^2} z) $ achieves equality in the bound at $ r = r_s $, confirming sharpness.
• For $ f \in CV(R_1, R_2) $ with $ R_2 \geq 2\delta $, the majorant series satisfies $ \mathcal{M}_f(r) \leq \delta $ for all $ r \leq r_c = R_2 / (3R_2 - 2\delta) $, and this radius is sharp for $ k_a(z) = z / (1 - a z) $ with $ a \in [1/2, 1) $.
• The radius $ r_s $ is at least $ 1/3 $ when $ \alpha \leq \sqrt{3}/2 $, and $ r_c \geq 1/3 $ under the same condition, showing improvement over classical results.
• The results demonstrate that subordination and majorization techniques can yield sharper Bohr radii than classical methods, especially when symmetry (oddness) or geometric convexity is present.
• The condition $ R_2 \geq 2\delta $ ensures $ r_c \leq A $, allowing the use of Lemma 1, and the radius $ r_c $ is independent of $ R_1 $, depending only on $ R_2 $ and $ \delta $.