TY - JOUR

T1 - A note on the Bohr's phenomenon for power series

AU - Ali, Rosihan M.

AU - Barnard, Roger

AU - Solynin, Alexander

PY - 2017/5/1

Y1 - 2017/5/1

N2 - Bohr's phenomenon, first introduced by Harald Bohr in 1914, deals
with the largest radius $r$, $0<r<1$, such that the inequality
$\sum_{k=0}^\infty |a_k|r^k\le 1$ holds whenever the inequality
$|\sum_{k=0}^\infty a_kz^k|\le 1$ holds for all $|z|<1$. The exact
value of this largest radius known as the \emph{Bohr's radius},
which is $r_b=1/3$, was discovered long ago. In this paper, we
first discuss Bohr's phenomenon for the classes of even and odd
analytic functions and for alternating series. Then we discuss
Bohr's phenomenon for the class of analytic functions from the
unit disk into the wedge domain $W_\alpha =\{w:\,|\arg
w|<\pi\alpha/2\}$, $1\le \alpha \le 2$. In particular, we find
the Bohr's radius for this class.

AB - Bohr's phenomenon, first introduced by Harald Bohr in 1914, deals
with the largest radius $r$, $0<r<1$, such that the inequality
$\sum_{k=0}^\infty |a_k|r^k\le 1$ holds whenever the inequality
$|\sum_{k=0}^\infty a_kz^k|\le 1$ holds for all $|z|<1$. The exact
value of this largest radius known as the \emph{Bohr's radius},
which is $r_b=1/3$, was discovered long ago. In this paper, we
first discuss Bohr's phenomenon for the classes of even and odd
analytic functions and for alternating series. Then we discuss
Bohr's phenomenon for the class of analytic functions from the
unit disk into the wedge domain $W_\alpha =\{w:\,|\arg
w|<\pi\alpha/2\}$, $1\le \alpha \le 2$. In particular, we find
the Bohr's radius for this class.

M3 - Article

SP - 154

EP - 167

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

ER -