[Paper Review] Bohr inequality for odd analytic functions

I. R. Kayumov, Saminathan Ponnusamy|arXiv (Cornell University)|Jan 14, 2017

Advanced Banach Space Theory11 references4 citations

TL;DR

This paper resolves the Bohr inequality for odd analytic functions by determining the sharp Bohr radius at $ r_2 = 0.789991\ldots $, confirming a conjecture by Ali, Barnard, and Solynin. It generalizes the result to $ p $-symmetric functions and establishes a new Bohr radius for subordinations to odd univalent functions using coefficient estimates and Schwarz lemma techniques.

ABSTRACT

We determine the Bohr radius for the class of odd functions $f$ satisfying $|f(z)|\le 1$ for all $|z|<1$, settling the recent conjecture of Ali, Barnard and Solynin \cite{AliBarSoly}. In fact, we solve this problem in a more general setting. Then we discuss Bohr's radius for the class of analytic functions $g$, when $g$ is subordinate to a member of the class of odd univalent functions.

Motivation & Objective

• Resolve the conjecture by Ali, Barnard, and Solynin on the Bohr radius for odd analytic functions bounded by 1 in the unit disk.
• Generalize the Bohr inequality to $ p $-symmetric functions, where $ f(z) = z\sum_{k=0}^\infty a_{pk+1}z^{pk} $, and derive the sharp radius $ r_p $.
• Establish a new Bohr radius for functions subordinate to odd univalent functions, where coefficient estimates are insufficient due to lack of sharp bounds.
• Provide a sharp bound for the sum of absolute values of coefficients in odd univalent functions using Rogosinski’s theorem and $ \ell^2 $-norm estimates.

Proposed method

• Use the $ p $-symmetric structure of $ f(z) $ to derive a characteristic equation: $ -6r^{p-1} + r^{2(p-1)} + 8r^{2p} + 1 = 0 $, whose maximal positive root in $ (0,1) $ gives the Bohr radius $ r_p $.
• Construct extremal functions of the form $ z(z^p - a)/(1 - a z^p) $, with $ a $ derived from $ r_p $, to verify sharpness.
• Apply subordination theory: if $ g \prec f $, then $ g(z) = f(w(z)) $ with $ w(0) = 0 $, $ |w(z)| < 1 $, and use the Schwarz lemma to bound $ |g'(0)| $.
• Use the $ \ell^2 $-norm bound $ \sum_{k=1}^n |a_{2k-1}|^2 \leq n $ for odd univalent functions to derive $ \sum |a_{2k-1}| r^{2k-1} \leq \frac{r}{1 - r^2} $.
• Apply the Cauchy-Schwarz inequality to $ \sum |b_k| r^k $, leading to $ \sum |b_k| r^k \leq \frac{r}{(1 - r)\sqrt{1 + r}} $, which is $ \leq 1 $ when $ r^3 - 2r^2 - r + 1 \geq 0 $.
• Improve the bound via constrained optimization of $ \psi(x,y) $, leading to a refined radius of $ r \approx 0.564 $, and show the upper bound $ (\sqrt{5} - 1)/2 \approx 0.618 $ is sharp for $ f(z) = z/(1 - z^2) $.

Experimental results

Research questions

• RQ1 What is the sharp Bohr radius for the class of odd analytic functions $ f $ with $ |f(z)| \leq 1 $ in the unit disk?
• RQ2 Can the Bohr inequality be generalized to $ p $-symmetric functions, and what is the corresponding Bohr radius $ r_p $?
• RQ3 What is the Bohr radius for functions $ g $ subordinate to an odd univalent function $ f $, given the lack of sharp coefficient bounds for odd univalent functions?
• RQ4 Can the Bohr radius be improved beyond $ r = 0.564 $ using constrained optimization of coefficient norms?
• RQ5 Is the radius $ (\sqrt{5} - 1)/2 \approx 0.618 $ sharp for the class of odd univalent functions with $ a_1 = 1 $?

Key findings

• The Bohr radius for odd analytic functions satisfying $ |f(z)| \leq 1 $ is exactly $ r_2 = 0.789991\ldots $, confirming the conjecture of Ali, Barnard, and Solynin.
• For $ p $-symmetric functions, the Bohr radius $ r_p $ is the maximal positive root of $ -6r^{p-1} + r^{2(p-1)} + 8r^{2p} + 1 = 0 $, with extremal function $ z(z^p - a)/(1 - a z^p) $.
• The Bohr radius for functions $ g \prec f $, where $ f $ is odd univalent, is at least $ r \approx 0.564 $, improving upon the initial bound from $ r^3 - 2r^2 - r + 1 \geq 0 $.
• The upper bound $ (\sqrt{5} - 1)/2 \approx 0.618 $ is sharp for the class of odd univalent functions with $ a_1 = 1 $, achieved by $ f(z) = z/(1 - z^2) $.
• For odd univalent functions with $ |a_1| = \alpha \leq 1 $, the Bohr radius is $ r_\alpha = \frac{-\alpha + \sqrt{4 + \alpha^2}}{2} $, with extremal function $ \alpha z / (1 - z^2) $.
• The bound $ \sum |a_{2k-1}| r^{2k-1} \leq \frac{r}{1 - r^2} $ holds for odd univalent functions, derived from $ \ell^2 $-norm constraints and Rogosinski’s theorem.