Improved version of Bohr's inequality
Let \(f\) be a bounded analytic function on the unit disk \(\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}\) with Taylor expansion \(\displaystyle \sum_{k=0}^{\infty} a_kz^k\). The classical Bohr inequality states that: \[M_f(r)=\displaystyle\sum_{n=0}^{\infty}|a_n|r^n\leq||f||_{\infty}, \ \text{for} \ 0\leq r\leq\frac{1}{3}\] and the constant \(\frac{1}{3}\) is sharp. In this interesting article, the authors give several different improved versions of the aforementioned inequality. Let us state one of them: Let \(f\) be a bounded analytic function on the unit disk \(\mathbb{D}\) with \(||f||_{\infty}\leq 1\). Then \[|a_0|+\sum_{k=1}^{\infty}\biggl( |a_k|+\frac{1}{2}|a_k|^2\biggl)r^k\leq 1, \ \text{for} \ 0\leq r\leq\frac{1}{3}\] and the numbers \(\frac{1}{3}\) and \(\frac{1}{2}\) cannot be improved.
scientific article; zbMATH DE number 6849184