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Statement

Notation as on the Statement (I) page: $p_n$ and $q_n$ are the normalized polynomials (1) and (2) of degrees $2n+1$ and $2n+2$ with zeros at consecutive integers.

Statement (II) (p. 294). For $0<x<n$ with $x$ not an integer,

$$|p_{n+1}(x)|>(n+1)\,|q_n(x)|>(n+1)\,|p_n(x)|. \tag{4}$$

Hence, in the system (1)--(2), increasing the degree increases both the area and the maximum of the absolute value of the polynomial on each fixed arch.

Unlike Statement (I), this monotonicity depends on the coordinate scale fixed by (1) and (2), as the paper notes (p. 294).

Read depth. Claims checked: the statement was read clause by clause on the page image of p. 294. The short proof was followed.

Proof pointer

P. 294. The factorization $p_{n+1}(x)=(x+n+1)\,q_n(x)=(x+n+1)(x-n-1)\,p_n(x)$ and, for $0<x<n$, the bounds $|x+n+1|>n+1$ and $|x-n-1|=n+1-x>1$.

Dependencies

None beyond the definitions (1) and (2).

Bears on

None recorded. The paper presents the analogue of (II) for the zeros of the derivative as relations (7)--(11).