ON BOHR INEQUALITY FOR GENERALIZED SERIES AND DERIVATIVES IN A CERTAIN FAMILY OF ANALYTIC FUNCTIONS

ON BOHR INEQUALITY

Abstract

In this paper, we investigate the Bohr phenomenon for a class of bounded analytic functions
\( f(z) = \sum_{k=0}^{\infty} a_k z^k \) defined in the open unit disk \( \mathbb{D} \),
where the coefficients satisfy \( |a_k| \le 1 - |a_0| \) for all \( k \ge 1 \). Extending earlier results for this class, we establish the Bohr inequality for generalized series of the form \( \sum_{k=0}^{\infty} a_{pk+m} z^{pk+m}\), which leads to the determination of the Bohr radius for subclasses such as odd and lacunary analytic functions. In additon, we derive the Bohr inequality for the derivative of functions belonging to this class and obtain sharp results in each case. These findings refine and generalize several known results, contributing to a broader understanding of the Bohr phenomenon in the theory of bounded analytic functions.