Bohr phenomenon for analytic functions $ f $ where $ f(z)=\sum_{ n=0}^{ \infty}a_nz^n $, first introduced by Harald Bohr in $ 1914 $, deals with finding the largest radius $ r_f $, $ 0<r_f<1 $, such that the inequality $ \sum_{ n=0}^{ \infty}|a_nz^n|<1 $ holds whenever $ |f(z)|<1 $ holds in the unit disk $ \mathbb{ D}=\{ z\in\mathbb{ C} : |z|<1\} $. The Bohr phenomenon for the harmonic functions of the form $ f(z)=h+\overline{ g} $, where $ h(z)=\sum_{ n=0}^{ \infty}a_nz^n $ and $ g(z)=\sum_{ n=1}^{ \infty}b_nz^n $ is to find the largest radius $ r_f $, $ 0<r_f<1 $ such that \begin{ equation*} \sum_{ n=1}^{ \infty}\left(|a_n|+|b_n|\right)|z|^n\leq d(f(0),\partial f(\mathbb{ D})) \end{ equation*} holds for $ |z|\leq r_f $, where $ d(f(0),\partial f(\mathbb{ D})) $ is the Euclidean distance between $ f(0) $ and the boundary of $ f(\mathbb{ D}) $. In this paper, we prove several improved versions of the sharp Bohr radius for the classes of harmonic and univalent functions. Further, we prove several corollaries as a consequence of the main results.
Categories: math.CV