The Krzyż conjecture revisited (Q2253808)

Denote by \(\mathcal B\) the class of all analytic functions \(f(z)=\sum_{j=0}^{\infty}a_jz^j\) in the unit disk \(\mathbb D\) such that \(0<|f(z)|\leq1\) in \(\mathbb D\). A function \(f\) is extremal if the supremum \(M_n=\sup\{|a_n|:f\in\mathcal B\}\), \(n\geq1\), is attained for \(f\). The Krzyż conjecture states that \(M_n=\frac{2}{e}\) for all \(n\geq1\). For an extremal function \(f\), define \(W_nf(z)=\sum_{j=0}^{\infty}a_{nj}z^{nj}=H(z^n)\) and \(P(z)=a_n+2a_{n-1}z+\dots+2a_0z^n\). There exists a Blaschke product \(B\) and a function \(G\) which is analytic and non-vanishing in \(\mathbb D\) such that \(H=BG\), \(\|G\|_{\infty}=\|H\|_{\infty}\leq1\). Without loss of generality assume that \(B_0=B(0)>0\). The main result can be viewed as a reformulation of the Krzyż conjecture in many ways. Let \(n\geq2\) and consider an arbitrary but fixed extremal function \(f(z)=\sum_{j=0}^{\infty}a_jz^j\) for the Krzyż conjecture: \(M_n=\text{Re}\;a_n\), \(a_0>0\). Write \(f=e^g\) where \[ g(z)=\sum_{j=0}^{\infty}b_jz^j=\sum_{k=1}^Nr_k\frac{\alpha_kz+1}{\alpha_kz-1},\quad r_k>0,\quad |\alpha_k|=1,\quad 1\leq k\leq N\leq n. \] The authors prove the equivalence of 17 different conditions for \(f\), \(g\) and \(P\). In particular, the following statements are equivalent: \newline(1) \(a_n=2a_0\); \newline(2) \(a_k=0\) when \(1\leq k\leq n\);\newline(3) \(f(z)=\exp((z^n-1)/(z^n+1))\); \newline(4) \(g(z)=(z^kH(z)-1)/(z^kH(z)+1)=\) for some analytic function \(H\) in \(\mathbb D\) such that \(|H(z)|\leq1\) for all \(z\in\mathbb D\) and \(k\in\mathbb N\), \(k\geq\frac{n}{2}\);\newline(5) the zeros \(\lambda_1,\dots,\lambda_n\) of the polynomial \(P\) satisfy \((-1)^n\Pi_{k=1}^n\lambda_k=1\);\newline(6) the zeros of \(P\) all lie on the unit circle;\newline (7) the zeros of \(P\) are actually the \(n\)-th roots of (-1);\newline(8) \(a_nb_0+a_0\text{Re}\{b_n\}=0\);\newline(9) \(\text{Re}\{a_1b_{n-1}+\dots+a_{n-1}b_1\}=0\);\newline(10) \(a_n=a_0\text{Re}\{b_n\}\);\newline(11) \(\text{Re}\{b_n\}=2|b_0|\), \(N=n\), and \(r_1=\dots=r_n\);\newline(12) \(B_0=1\);\newline(13) \(W_nf\) does not vanish in \(\mathbb D\);\newline(14) \(W_nf\equiv f\);\newline(15) \(g=W_ng\). Besides, there is a unique extremal function for the Krzyż conjecture if and only if every extremal function for the Krzyż conjecture satisfies one of the conditions (1)--(15), and therefore all of them.