Bohr--Rogosinski inequalities for bounded analytic functions

Abstract

In this paper we first consider another version of the Rogosinski inequality for analytic functions $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ in the unit disk $|z|<1$, in which we replace the coefficients $a_n$ $(n=0,1,\dots ,N)$ of the power series by the derivatives $f^{(n)}(z)/n!$ $(n=0,1,\dots ,N)$. Secondly, we obtain improved versions of the classical Bohr inequality and Bohr's inequality for the harmonic mappings of the form $f=h+\overline{g}$, where the analytic part $h$ is bounded by $1$ and that $|g^{\prime }(z)|\le k|h^{\prime }(z)|$ in $|z|<1$ and for some $k\in [0,1]$.

11 pages; To appear in Lobachevskii Journal of Mathematics