The classical inequality of Bohr asserts that if a power series converges in the unit disk and its sum has modulus less than or equal to 1, then the sum of absolute values of its terms is less than or equal to 1 for the subdisk $$|z|<1/3$$ | z | < 1 / 3 and 1/3 is the best possible constant. Recently, there has been a number of investigations on this topic. In this article, we present related inequalities using $$\sum _{n=0}^{\infty }|a_n|^2r^{2n}$$ ∑ n = 0 ∞ | a n | 2 r 2 n that generalize for example the well known inequality $$\begin{aligned} \sum _{n=0}^{\infty }|a_n|^2r^{2n} \le 1. \end{aligned}$$ ∑ n = 0 ∞ | a n | 2 r 2 n ≤ 1 .

中文翻译：

---

单模有界函数系数的新不等式

Bohr 的经典不等式断言，如果幂级数在单位圆盘中收敛，并且其和的模数小于或等于 1，则其项的绝对值之和对于子圆盘 $$| 小于或等于 1。 z|<1/3$$ | | | < 1 / 3 和 1/3 是最好的常数。最近，有一些关于这个话题的调查。在本文中，我们使用 $$\sum _{n=0}^{\infty }|a_n|^2r^{2n}$$ ∑ n = 0 ∞ | 来呈现相关的不等式。一个 | 2 r 2 n 概括了例如众所周知的不等式 $$\begin{aligned} \sum _{n=0}^{\infty }|a_n|^2r^{2n} \le 1. \end{aligned} $$ ∑ n = 0 ∞ | 一个 | 2 r 2 n ≤ 1 。