Asymptotic estimates of linear functionals for bounded nonvanishing functions (Q1397588)

Let \(\mathcal B\) denote the class of all functions \(f\) of the form \(f(z)= a_0+a_1z+ \cdots+ a_nz^n+\cdots\), holomorphic in the unit disk \(D=\{z: | z|< 1\}\) and such that \(0<| f(z)| <1\) for \(z\in D\). If \(A_n =\max\{| a_n|\), \(f\in B\}\), \(n=1,2,\dots\), then we may state what we call Krzyż's conjecture states that \(A_n=\frac {e}{n}\), \(n=1,2, \dots\) and this maximum, is attained only for functions \(k_n(z)= \exp\frac {z^n-1} {z^n+1}\), \(n=1,2,\dots\) and its rotations [\textit{J. Krzyż}, Ann. Polon. Math. 20, 314 (1968)]. The problem in the general case is very hard and it is solved in some special cases [see for example, \textit{J. A. Hummel}, \textit{S. Scheinberg}, \textit{L. Zalcman}, J. Anal. Math. 31, 169--190 (1977; Zbl 0346.30013); \textit{W. Szapiel}, Ann. Univ. Mariae Curie-Skłodowska, Sect A 48, 169--192 (1994; Zbl 0868.30025); \textit{Z. Lewandowski}, \textit{J. Szynal}, J. Comput. Appl. Math. 105, 367--369 (1999; Zbl 0945.30011)]. In 1992 \textit{R. Peretz} [Complex Variables, Theory Appl. 17, 213--222 (1992; Zbl 0738.30025)] obtained asymptotic estimates of \(| a_n|\) for \(| a_0|\) close, to 0 or 1. In this paper the author obtains asymptotic estimates for the linear functional \(L(f)=\text{Re} \{a_n+ \alpha_{n-1}a_{n-1} +\cdots+\alpha_1a_1\}\), \(\alpha_1,\dots, \alpha_{n-1} \in\mathbb{C}\), \(f\in\mathcal B\). In the proof an equation of the Loewner's type was applied.