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Several generalized Bohr-type inequalities with two parameters

Wanqing Hou, Qihan Wang, Boyong Long

TL;DR

This work generalizes Bohr-type inequalities for bounded analytic functions to two-parameter families, introducing radii that depend on $(p,\lambda)$ and, further, on an additional parameter $t$. The authors derive explicit sharp radii $R_{\lambda,p}$ and $R_{t,p}$ by leveraging Schwarz-Pick estimates, coefficient bounds, and convexity arguments, with extremal functions $f(z)=\frac{a-z}{1-az}$ demonstrating sharpness. The results unify and extend known one-parameter Bohr-type inequalities, and specific choices recover classical Bohr phenomena, thereby enriching the theory of Bohr radii for bounded analytic functions.

Abstract

In this paper, several Bohr-type inequalities are generalized to the form with two parameters for the bounded analytic function. Most of the results are sharp.

Several generalized Bohr-type inequalities with two parameters

TL;DR

This work generalizes Bohr-type inequalities for bounded analytic functions to two-parameter families, introducing radii that depend on and, further, on an additional parameter . The authors derive explicit sharp radii and by leveraging Schwarz-Pick estimates, coefficient bounds, and convexity arguments, with extremal functions demonstrating sharpness. The results unify and extend known one-parameter Bohr-type inequalities, and specific choices recover classical Bohr phenomena, thereby enriching the theory of Bohr radii for bounded analytic functions.

Abstract

In this paper, several Bohr-type inequalities are generalized to the form with two parameters for the bounded analytic function. Most of the results are sharp.

Paper Structure

This paper contains 3 sections, 13 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Some Lemmas
  3. Main results

Key Result

Theorem 1.1

kayumov2017bohrrogosinski Suppose that $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ is analytic in $\mathbb{D}$ and $|f(z)|<1$. Then and the radius $\sqrt{5}-2$ is the best possible. Moreover, and the radius $\frac{1}{3}$ is the best possible.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 12 more