Bohr's inequality for analytic functions $\sum_k b_k z^{kp+m}$ and harmonic functions

TL;DR

This paper determines the Bohr radius for specific classes of analytic functions with a particular power series form and extends the concept to harmonic functions, addressing recent conjectures and introducing the p-Bohr radius.

Contribution

The authors establish the Bohr radius for functions with power series of the form $f(z)= extstyle ext{sum of } a_{kp+m} z^{kp+m}$ and introduce the p-Bohr radius for harmonic functions, generalizing classical results.

Findings

Solved the Bohr radius for functions with series $f(z)= extstyle ext{sum of } a_{kp+m} z^{kp+m}$.

Provided a solution to a recent conjecture on the Bohr radius for odd analytic functions.

Introduced the p-Bohr radius concept for harmonic functions, encompassing the classical Bohr radius.

Abstract

We determine the Bohr radius for the class of all functions $f$ of the form $f(z)=\sum_{k=1}^\infty a_{kp+m} z^{kp+m}$ analytic in the unit disk $|z|<1$ and satisfy the condition $|f(z)|\le 1$ for all $|z|<1$. In particular, our result also contains a solution to a recent conjecture of Ali, Barnard and Solynin \cite{AliBarSoly} for the Bohr radius for odd analytic functions, solved by the authors in \cite{KayPon1}. We consider a more flexible approach by introducing the $p$-Bohr radius for harmonic functions which in turn contains the classical Bohr radius as special case. Also, we prove several other new results and discuss $p$-Bohr radius for the class of odd harmonic bounded functions.