Companion theorems to G. Szegö’s inequality for pairs of coefficients of bounded polynomials

Abstract

We consider real univariate polynomials \(P_n\) of degree \( \le n \) from class $$\begin{aligned} \mathbf {C}_n = \{P_n:|P_n \left( \cos \displaystyle \frac{(n -i)\pi }{n}\right) |\le 1 \; \text{ for }\; 0\le i\le n \} \end{aligned}$$ which encompasses the unit ball of polynomials with respect to the uniform norm on \([- 1, 1]\). For pairs of consecutive coefficients of \( P_n(x) = \sum \nolimits _{k=0}^{n}a_kx^k\) there holds the inequality $$\begin{aligned} |a_{k-1}|+|a_k|\le |t_{n,k}|, \quad \text{ if }\; k\equiv n\; \text{ mod }\; 2, \end{aligned}$$ (1) where \(T_n(x)=\sum \nolimits _{k=0}^{n} t_{n,k}x^k\) is the \(n\)-th Chebyshev polynomial of the first kind. (1) implies Markov’s classical coefficient inequality of 1892 (Math. Ann. 77:213–258, 1916, p. 248) and goes back to Szego, but was made public by P. Erdos (Bull. Am. Math. Soc. 53:1169–1176, 1947, p. 1176) in 1947. We ask here: will the (nonzero) coefficients of \(T_n\) likewise majorize complementary pairs \(|a_k| + |a_{k+1}|\)? More generally: does there hold $$\begin{aligned} |a_k| + |a_j| \le |t_{n,k}| \quad \text{ for } \text{ all }\; P_n \in \mathbf {C_n}, \end{aligned}$$ (2) \(\text{ where }\; k K\), then (2) holds for all \(k\) with \(k_{*}\le k < j\), where the bound \(k_{*} < K\) is explicitly determined, and is optimal for \(n\le 43\). In Theorem 2.6 we return to G. Szego’s original inequality (1) and constructively prove the non - uniqueness of its extremizer \(\pm T_n\).