Bohr's Phenomenon for Some Univalent Harmonic Functions

Abstract

In 1914 Bohr proved that there is an $r_0\in (0,1)$ such that if a power series ${\sum }_{m=0}^{\infty }{c}_m{z}^m$ is convergent in the open unit disc and $|{\sum }_{m=0}^{\infty }{c}_m{z}^m|<1$ then, ${\sum }_{m=0}^{\infty }|c_m{z}^m|<1$ for $|z|<r_0$. The largest value of such $r_0$ is called the Bohr radius. In this article, we find Bohr radius for some univalent harmonic mappings having different dilatations and in addition, also compute Bohr radius for the functions convex in one direction.

13 pages, 3 figures