A Lower Bound for the Norm of the Minimal Residual Polynomial

Let S be a compact infinite set in the complex plane with 0 ∉ S, and let R_n be the minimal residual polynomial on S, i. e., the minimal polynomial of degree at most n on S with respect to the supremum norm provided that R_n(0)=1. For the norm L_n(S) of the minimal residual polynomial, the limit exists. In addition to the well-known and widely referenced inequality L_n(S) ≥ κ(S)ⁿ, we derive the sharper inequality L_n(S) ≥ 2κ(S)ⁿ/(1+κ(S)²ⁿ) in the case that S is the union of a finite number of real intervals. As a consequence, we obtain a slight refinement of the Bernstein-Walsh lemma.