The Landau problem for nonvanishing functions with real coefficients (Q5957972)

Let \({\mathcal H}\) denote the class of all holomorphic functions in the unit disc \(\Delta =\{z:|z|<1\}\) and \(B_0=\{f(z)=\sum_{k=0}^{\infty} f_kz^k\in {\mathcal H}: 0<|f(z)|<1\), \(z\in \Delta\}\). \textit{Z. Lewandowski} and \textit{J. Szynal} [J. Comput. Appl. Math. 105, 367-369 (1999; Zbl 0945.30011)] showed that NEWLINE\[NEWLINE\max_{f\in B_0}|f_0+f_1|\leq 2e^{-1/2}\approx 1.21 \cdots \quad\text{and}\quad \max_{f\in B_0}|f_0+f_1+f_2|\approx 1.33 \cdots NEWLINE\]NEWLINE and, in addition, they have conjectured that for each \(f\in B_0\), one has the property that NEWLINE\[NEWLINE\max_{f \in B_0, f_n \in {\mathbb R}}\left|\sum_{k=0}^{n}f_k \right |\leq LNEWLINE\]NEWLINE for some \(L>1\) and for \(n\in {\mathbb N}=\{1,2,\dots \}\). \textit{A. Ganczar, M. Michalska} and \textit{J. Szynal} [Ann. Univ. Mariae Curie-Skłodowska Sect. A 52, No. 2, 37-46 (1998; Zbl 1009.30008)] showed that \(\max_{f\in B_0, f_n\in {\mathbb R}}|\sum_{k=0}^{3}f_k |=1.40315 \cdots .\) In the present article the authors have proved that for each \(f\in B_0\) one has NEWLINE\[NEWLINE\max \left(\sum_{k=0}^{4}f_k \right)=\max \left|\sum_{k=0}^{4}f_k\right |=1.46109 \cdots, \quad\text{and}\quad \min \left(\sum_{k=0}^{4}f_k\right)= -0.713082 \cdots .NEWLINE\]NEWLINE The proof of these assertions are established using Carathéodory-Toeplitz conditions and expressing them in such a way that each Taylor coefficient can be converted separately to a polynomial of several variables.