An extension of Krzyż's conjecture (Q5953947)

Let \(B_\alpha\), \(0\leq\alpha< 1\), denote the class of functions \(f\) holomorphic in the unit disk \(U\) of the form \(f(z)= a_0+ a_1z+ a_1z^2+\cdots\) and satisfying the condition \(\alpha<|f(z)|< 1\) for \(z\in U\). If \(A_n= \sup|a_n|\), \(f\in B_0\), \(n= 1,2,\dots\), then we may state what we call \textit{Z. Krzyż's} conjecture [Coefficient problem for bounded non-vanishing functions, Ann. Pol. Math. 20, 314 (1968, per. bibl.)] states that \(A_n= {2\over e}\), \(n= 1,2,\dots\), and this maximum is attained only for functions \(k_n(z)= \exp[(z^n- 1)/(z^n+ 1)]\), \(n= 1,2,\dots\) and its rotations. In this paper the author considers the problem of determining \(S_n(\alpha)= \sup|a_n|\), \(f\in B_\alpha\) and solves it for \(n= 0,1,2\). The author considers also \(\sup|a_0+ a_1|\), \(f\in B_\alpha\).