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 Szegö, but was made public by P. Erdös (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)

$\text{ where }\; k < j \;\text{ and }\; k\equiv n\mod 2\;\text{ but }\; j\not \equiv n\mod 2 ?$ We treat the marginal cases $n < 12$ separately, and for $n \ge 12$ we provide answers to this question with the aid of the explicitly determined optimal bound $K \sim \lceil \frac{n}{\sqrt{2}}\rceil$ which incorporates the height and the length of $\frac{T'_n(x)}{n}$. Theorem 2.1: (2) holds, provided $K \le k < j$; in particular, provided $\frac{n}{\sqrt{2}}<k<j$. As a corollary we reveal new extremal properties of the leading coefficients of $\pm T_n$. Theorem 2.4: (2) does not hold if $k < j < K$. Theorem 2.5: If $k < K < j$, then (2) holds for certain, but not for all, $k$ and $j$. If we keep fixed $j = n-1> K$, 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. Szegö’s original inequality (1) and constructively prove the non - uniqueness of its extremizer $\pm T_n$.