Inequalities between height and deviation of polynomials (Q2053635)

For a polynomial \(f(x)=\sum_{k=0}^da_kx^k\in R[x]\) let \(H(f)=\max_k|a_k|\) be its height and \(L(f)=\sum_{k=0}^d|a_k|\) its length. The author studies the constants \(\xi(d)\) and \(\eta(d)\) satisfying \[H(P)\le \xi(d)H(Q)\tag{1}\] for any \(P,Q\in R[x]\) of degree \(d\) for which one has \[ |P(x)|\le |Q(x)| \] for \(x\ge0\) and \[H(P)\le \eta(d)H(Q)\tag{2}\] for any \(P,Q\in R[x]\) of degree \(d\) for which one has \[ |P(x)|\le |Q(x)| \] for \(x\in R\). In Theorem 1 effective lower bounds for \(\xi(d)\) and \(\eta(d)\) are given, showing that \(\xi\) and \(\eta\) grow exponentially with \(d\) and Theorem 3 shows that \(\xi(d)=3^{3d/2}\sqrt d\) and \(\eta(d)=3^{3d/4}\sqrt{7d}\) satisfy (1), resp. (2). Theorem 2 proves that if \(P,Q\in R[x]\) satisfy \[ \max_{0\le x\le1}|P(x)|\le \max_{0\le x\le1}|Q(x)|, \] then \(L(P)\le \alpha(d)L(Q)\), where \(d=\deg P\) and \[ \alpha(d) =\left((3+2\sqrt2)^d+(3-2\sqrt2)^d\right)/2, \] and if \[ \max_{-1\le x\le1}|P(x)|\le \max_{-1\le x\le1}|Q(x)|, \] then \(L(P)\le \beta(d)L(Q)\) with \[ \beta(d) = \left((1+\sqrt2)^d+(1-\sqrt2)^d\right)/2. \] The proofs utilize the properties of Chebyshev polynomials.