A note on Bohr's phenomenon for power series
Let \(f(z)=\sum_{n=1}^{\infty}a_n z^n\) be a power series with radius of convergence 1 and \(\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}\). For \(z\in\mathbb{D}\), the authors use the notation: \[ M_f(z)=\sum_{n=1}^{\infty}|a_n| |z|^n \;\text{ and } A_f(z)=\sum_{n=1}^{\infty}(-1)^n|a_n| |z|^n. \] Let \(\mathcal{S}=\{f\in H(\mathbb{D}): f(\mathbb{D})\subset \overline{\mathbb{D}}\}\). It is known [\textit{H. Bohr}, Proc. Lond. Math. Soc. (2) 13, 1--5 (1914; JFM 44.0289.01)] that \[ \sup\big\{r>0: M_f(r)\leq 1 \;\forall f\in\mathcal{S} \big\}=\frac{1}{3}. \] Using the above fact and basic properties of power series, the authors prove that \[ \sup\big\{r>0: M_f(r)\leq 1 \;\forall f\in\mathcal{S}, \;f \text{ even}\big\}=\frac{1}{\sqrt{3}} \] \[ \sup\big\{r>0: |A_f(r)|\leq 1 \;\forall f\in\mathcal{S}\big\}=\frac{1}{\sqrt{3}}, \] and they give a lower and an upper bound for \[ \sup\big\{r>0: M_f(r)\leq 1 \;\forall f\in\mathcal{S}, \;f \text{ odd}\big\}. \] In the second part of the article, they prove that, for \(\alpha\in[1,2]\), \[ \sup\big\{r>0: M_f(z)\leq \mathrm{dist}(f(0), \partial W_{\alpha}), \;\forall f\in \mathcal{S_{\alpha}}\big\}=\frac{2^{\frac{1}{\alpha}}-1}{2^{\frac{1}{\alpha}}+1}, \] where \(W_{\alpha}=\{w\in\mathbb{C}: |\mathrm{arg }w|<\frac{\pi \alpha}{2}\}\) and \(\mathcal{S_{\alpha}}\) contains all analytic functions \(f\) that map \(\mathbb{D}\) into \(W_{\alpha}\).
scientific article; zbMATH DE number 6675583
scientific article