Bohr operator on analytic functions

TL;DR

This paper explores the properties of the Bohr operator on analytic functions within the unit disk, establishing bounds and inequalities related to subordination, sections, and Schwarz functions using normed space approaches.

Contribution

It introduces normed theoretic methods to analyze the Bohr operator, extending classical results on subordination and sections of analytic functions with new inequalities.

Findings

Established bounds for the Bohr operator under subordination.

Derived inequalities for sections of analytic functions.

Proved a von Neumann-type inequality for Schwarz functions.

Abstract

For $f(z) = \sum_{n=0}^{\infty} a_n z^n$ and a fixed $z$ in the unit disk, $|z| = r,$ the Bohr operator $\mathcal{M}_r$ is given by \[\mathcal{M}_r (f) = \sum_{n=0}^{\infty} |a_n| |z^n| = \sum_{n=0}^{\infty} |a_n| r^n.\] This papers develops normed theoretic approaches on $\mathcal{M}_r$. Using earlier results of Bohr and Rogosinski, the following results are readily established: if $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ is subordinate (or quasi-subordinate) to $h(z)=\sum_{n=0}^{\infty} b_{n}z^{n}$ in the unit disk, then \[\mathcal{M}_{r}(f) \leq \mathcal{M}_{r}(h), \quad 0 \leq r \leq 1/3,\] that is, \[\sum_{n=0}^{\infty} \ | a_{n}\ | |z|^{n} \leq \sum_{n=0}^{\infty} \ | b_{n}\ |t |z|^{n}, \quad 0 \leq |z| \leq 1/3. \] Further, each $k$-th section $s_k(f) = a_0 + a_1 z + \cdots + a_kz^k$ satisfies \[\ | s_k(f)\ | \leq \mathcal{M}_r \ ( s_k(h)\ ), \quad 0 \leq r \leq 1/2,\] and \[\mathcal{M}_{r}\ ( s_{k}(f) \ ) \leq \mathcal{M}_{r}(s_{k}(h)), \quad 0 \leq r \leq 1/3.\] A von Neumann-type inequality is also obtained for the class consisting of Schwarz functions in the unit disk.