Bohr's inequalities for the analytic functions with lacunary series and harmonic functions

Abstract

We determine the Bohr radius for the class of all functions f of the form f(z)=zm∑k=0∞akpzkp analytic in the unit disk |z|<1 and satisfy the condition |f(z)|≤1 for all |z|<1. In particular, our result also contains a solution to a recent conjecture of Ali et al. [9] for the Bohr radius for odd analytic functions, solved by the authors in [17]. 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.