A lower bound for the maximum of a polynomial in the unit disc (Q2300132)

For a polynomial \(P(z) = \sum_{i=0}^d a_iz^i \in \mathbb{C}[z]\) of degree \(d > 0\) with \(P(0) = a_0 \ne 0\), it is known that \[ \Bigl( \sum_{i=0}^d | a_i| \Bigr)^{1/2} \le \| P \| := \max_{| z| \le 1} | P(z)| \le \sum_{i=0}^d | a_i|. \tag{1} \] Note that for \(d = 1\), we have \(\| P \| = | a_0| + | a_1| \); the upper bound follows from the right-hand side equality in (1), and equality \(| P(z _0)| = | a_0 | + | a_1 | \) holds for \(z_0 = e^{i\phi_{0}}\), where \(\phi_0\) is the argument of the nonzero complex number \(a_0/a_1\). Therefore, the paper under review deals with the case of \(d \ge 2\) (and \(a_0 a_d \ne 0\)). Its main results are stated as three theorems which show together that \(\| P\| \ge | a_s| + | a_t| \), for some pairs \(s < t\). Specifically, Theorem 4 states that if \(d \ge 4\) and \(a_{d-1} \ne 0\), then \(\| P\| > | a_0| + | a_{d-1}| \). Theorem 1 states that the inequality \(\| P\| \ge | a_s| + | a_t| \) is satisfied, provided that \(s\) and \(t\) are integers with \(0 \le s < t \le d\) and \(s \le \min ((t - 1)/2, 2t - d - 1)\); this inequality is stronger than the lower bound given in (1), in the case where \(\sum_{j \ne s,t} | a_j|^2 < 2| a_s| | a_t| \). When \((s, t) = (0, d)\), the inequality \(\| P\| \ge | a_s| + | a_t| \) has been obtained by \textit{C. Visser} [Nederl. Akad. Wet., Proc. 48, 276--281 (1945; Zbl 0060.14805)]; another special case also appears in earlier sources, see Problem G47 in [\textit{G. J. Székely} (ed.), Contests in higher mathematics. Miklós Schweitzer competitions 1962--1991. Berlin: Springer-Verlag (1996; Zbl 0867.00004)], \textit{S. V. Konyagin} and \textit{V. F. Lev} [Chebyshevskiĭ Sb. 3, No. 2(4), 165--170 (2002; Zbl 1102.30004)]. The third main result of the paper under review, stated as Theorem 2, shows in a strong form that the latter inequality in the former part of Theorem 1 cannot be removed or even weakened. More precisely, by this theorem, for any integers \(s\), \(t\) and \(d \ge 2\) with \(0 \le s < t \le d\) and \(s > \min((t - 1)/2, 2t - d - 1)\), and any nonzero complex numbers \(a_s\), \(a_t\), there exists a polynomial \(Q(z) = \sum_{i=0}^d b_iz^i \in \mathbb{C}[z]\) satisfying the conditions \(b_0 b_d \ne 0\) and \((b_s, b_t) = (a_s, a_t)\), and such that \(\| Q\| < | a_s| + | a_t| \).