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Area functional and majorant series estimates in the class of bounded functions in the disk

Ramis Sh. Khasianov

TL;DR

The paper addresses sharp inequalities for weighted coefficient sums and area-type functionals of bounded analytic functions on the disk, framed as Bohr-type questions with $f(z)=\sum a_n z^n$. It develops a Reich-based majorization framework for subordinate functions, applies it to lacunary series and Hadamard convolution operators, and obtains explicit $G(r)$ bounds and extremal functions. Key contributions include lacunary area bounds $S_r f\le \pi (ms+p) r^{2(ms+p)} \|f\|_\infty^2$, universal Bohr-type corollaries, and improved lower bounds for Bohr radii of differentiation operators, with comparisons to prior bounds and numerical data. Overall, the work strengthens Bohr-type inequalities in one-variable function theory and links them to operator theory through Hadamard convolutions and majorant-series techniques.

Abstract

In this article, the new inequalities for the weighted sums of coefficients in the class of bounded functions in the disk are obtained. We develop the methods of I.R.~Kayumov and S.~Ponnusamy, using E.~Reich's theorem on the majorization of subordinate functions. The sharp estimates for the area of the image of the disk of radius $r$ under the action of the function which is expanded into a lacunary series of standard form are obtained. Under significantly lower than in \cite{Khas} restrictions on the initial coefficient, the estimates for the Bohr--Bombieri function of the Hadamard convolution operator are proved. Using the example of the differentiation operator, it is shown that in some cases the new method for calculating the lower bound for the Bohr radius of the Hadamard operator with a fixed initial coefficient is more effective than the known one.

Area functional and majorant series estimates in the class of bounded functions in the disk

TL;DR

The paper addresses sharp inequalities for weighted coefficient sums and area-type functionals of bounded analytic functions on the disk, framed as Bohr-type questions with . It develops a Reich-based majorization framework for subordinate functions, applies it to lacunary series and Hadamard convolution operators, and obtains explicit bounds and extremal functions. Key contributions include lacunary area bounds , universal Bohr-type corollaries, and improved lower bounds for Bohr radii of differentiation operators, with comparisons to prior bounds and numerical data. Overall, the work strengthens Bohr-type inequalities in one-variable function theory and links them to operator theory through Hadamard convolutions and majorant-series techniques.

Abstract

In this article, the new inequalities for the weighted sums of coefficients in the class of bounded functions in the disk are obtained. We develop the methods of I.R.~Kayumov and S.~Ponnusamy, using E.~Reich's theorem on the majorization of subordinate functions. The sharp estimates for the area of the image of the disk of radius under the action of the function which is expanded into a lacunary series of standard form are obtained. Under significantly lower than in \cite{Khas} restrictions on the initial coefficient, the estimates for the Bohr--Bombieri function of the Hadamard convolution operator are proved. Using the example of the differentiation operator, it is shown that in some cases the new method for calculating the lower bound for the Bohr radius of the Hadamard operator with a fixed initial coefficient is more effective than the known one.

Paper Structure

This paper contains 4 sections, 6 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Majorization of subordinate functions
  3. Area functional estimates
  4. Weighted majorant series estimates

Key Result

Theorem 1

Let $r\ge 0,\:$$s\ge 2, \: c_n>0 \:(n\ge 1),$$c_1=1$ and Then The supremum of $G(r)$ is attained at the following functions (i) If $r^2<\inf_{n\ge 1}\dfrac{c_n}{c_{n+1}},$ then $\omega(z)=e^{i\phi} z$; (ii) If $r^2=\inf_{n\ge 1}\dfrac{c_n}{c_{n+1}}$, then or (iii) If $\inf_{n\ge s-1}\dfrac{c_n}{c_{n+1}}<r^2<\inf_{n\ge s}\dfrac{c_n}{c_{n+1}}$ and $s \geq 2$, then (iv) If $r^2=\inf_{n\ge s}\dfr

Theorems & Definitions (18)

  • Definition 1
  • Theorem 1
  • proof : Proof of Theorem \ref{['G(r)']}
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • proof : Proof of Corollary \ref{['cons3']}
  • proof : Proof of Theorem \ref{['Area']}
  • Definition 2
  • Definition 3
  • ...and 8 more