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Statement

Notation as on the relations (7)--(11) page: $x'_{nj}$ and $\xi'_{nj}$ are the $j$th positive zeros of $p'_n$ and $q'_n$.

(12) (p. 295). Each such zero lies in the right half of its arch:

$$x'_{nj}>j-\tfrac12\quad(j=1,2,\ldots,n);\qquad \xi'_{nj}>j-\tfrac12\quad(j=2,3,\ldots,n+1).$$

For $j=1$ and $q_n$, (5) gives $\xi'_{n1}=\tfrac12$ exactly.

(13) (p. 296). Consecutive positive zeros of the derivative are more than one unit apart:

$$x'_{n,j+1}-x'_{nj}>1,\qquad \xi'_{n,j+1}-\xi'_{nj}>1,\qquad j=1,2,\ldots,n.$$

The range $j=1,2,\ldots,n$ is printed once for both inequalities. Since $x'_{nj}$ is defined only for $j\le n$, the first inequality has content for $j\le n-1$ (an observation of this page).

The paper attributes both to Bálint, in a different notation: (12) is statement (I), p. 35, and (13) is statement (II), p. 36, of Bálint's 1960 paper in Matematikai Lapok (p. 296). It notes that (13) implies (12), by induction from (5) and (10), and gives a new proof of (13), and hence of (12).

Read depth. Claims checked: (12) and (13) and the attributions were read on the page images of pp. 295--296. The new proof was followed but not checked step by step.

Proof pointer

P. 296. The proof follows that of Statement (I). Differentiating the shift identity $p_n(x+1)=\frac{x+n+1}{x-n}\,p_n(x)$ and evaluating at $x=x'_{nj}$ gives $p'_n(x'_{nj}+1)$ as a negative multiple of $p_n(x'_{nj})$. Since consecutive arches lie on opposite sides of the axis, $p_n$ is still moving away from zero at $x'_{nj}+1$, so its next critical point lies beyond $x'_{nj}+1$. The case of $q_n$ is the same with obvious changes.

Dependencies

Statement (I) (the shift identity), and (5) and (10) on the relations (7)--(11) page for the deduction of (12) from (13).

Bears on

Problem 1114: a related bound only. Inequality (13) bounds each gap between consecutive zeros of the derivative below by the spacing of the zeros; it does not compare consecutive gaps. The monotonicity of the gaps that the problem asks for is Erdős's conjecture, which the paper states (p. 293) and credits to Bálint's proof, written as $\Delta^2x'_{nj}>0$ (p. 297); the paper does not reprove it.