Lorch 1976 monotonicity properties polynomials equally spaced zeros

../

conjecture_21: Lorch's conjecture, from numerical evidence, that the higher differences in j of the zeros of p'_n, q'_n, p''_n and q''_n alternate in sign, whose second-difference case for the first derivative is the Erdős-Bálint result.

equation_13: Bálint's inequalities, reproved by Lorch, that consecutive positive zeros of p'_n and of q'n are more than 1 apart and that each, apart from xi'{n1} = 1/2, lies beyond the midpoint of its arch.

equation_14: Lorch's limit theorem that x'{nj} (j >= 1) and xi'{nj} (j >= 2) decrease to j - 1/2 as n tends to infinity, so consecutive gaps of the derivative's zeros tend to 1, with the monotonicity of these limits in n left open.

equations_7_11: Lorch's inequalities x'{nj} > x'{n+1,j} > xi'{nj} > xi'{n+1,j} for j = 2,...,n, with xi'_{n1} = 1/2, ordering the positive zeros of p'_n and q'_n across consecutive degrees.

statement_i: Lorch's statement that |p_n(x+1)| > |p_n(x)| and |q_n(x+1)| > |q_n(x)| for non-integral 0 < x < n, so the areas and maxima of the successive arches of |p_n| and |q_n| increase for x > 0.

statement_ii: Lorch's statement that |p_{n+1}(x)| > (n+1)|q_n(x)| > (n+1)|p_n(x)| for non-integral 0 < x < n, so in the normalized system each fixed arch gains area and maximum as the degree increases.

statement_iii: Lorch's evaluation |p'_n(j)| = (n+j)!(n-j)! and |q'_n(j)| = (n+j)!(n+1-j)! of the slopes at the zeros, and his statement that these sequences are absolutely monotonic in j.

---

Lorch, L., Some monotonicity properties of polynomials with equally spaced zeros. Acta Math. Acad. Sci. Hungar. 27 (3--4) (1976), 293--300. The copy read for this card is the repository's whole-volume scan (https://real-j.mtak.hu/7424/) with the Akadémiai Kiadó imprint and no copyright notice; the publisher's page for the journal's backfile, read for a 1978 article in the same journal, shows "© Akadémiai Kiadó" under "Reprints and permissions" with subscription access (https://link.springer.com/article/10.1007/BF01902213, read 2026-10-02), and this article's own page (DOI 10.1007/BF01902106) was not consulted; every other right reserved.

For the normalized polynomials $p_n(x)=x\prod_{k=1}^{n}(x^2-k^2)$ (degree $2n+1$) and $q_0(x)=x(x-1)$, $q_n(x)=(x-n-1)p_n(x)$ (degree $2n+2$), Lorch proves statement (I) (p. 294): $|p_n(x+1)|>|p_n(x)|$ and $|q_n(x+1)|>|q_n(x)|$ for non-integral $0<x<n$, so the areas and maxima of the successive arches of $|p_n|$ and $|q_n|$ increase for $x>0$; and statement (II) (p. 294): $|p_{n+1}(x)|>(n+1)|q_n(x)|>(n+1)|p_n(x)|$ for non-integral $0<x<n$, so raising the degree enlarges each fixed arch. Section 3 orders the positive zeros $x'_{nj}$ of $p'_n$ and $\xi'_{nj}$ of $q'_n$ across degrees: $\xi'_{n1}=\tfrac12$ (5), and $x'_{nj}>x'_{n+1,j}>\xi'_{nj}>\xi'_{n+1,j}$ for $j=2,\ldots,n$ (11, p. 295). It reproves Bálint's inequality (13), that consecutive positive zeros of $p'_n$, and of $q'_n$, are more than 1 apart, which implies Bálint's (12) (pp. 295--296), and shows in (14) that $x'_{nj}$ ($j\ge1$) and $\xi'_{nj}$ ($j\ge2$) decrease to $j-\tfrac12$ as $n\to\infty$, the extremum points of $\sin\pi x$, by the uniform convergence of the partial products $P_n(x)$ of $\sin\pi x$ (pp. 296--297). Section 4 evaluates the slopes at the zeros, $|p'_n(j)|=(n+j)!(n-j)!$ (16) and $|q'_n(j)|=(n+j)!(n+1-j)!$ (18), and states in (III) (p. 298) that both sequences are absolutely monotonic in $j$. The paper states Erdős's conjecture that the gaps between consecutive zeros of the derivative increase outward (p. 293), and credits its proof to Bálint without reproving it, writing Bálint's result as $\Delta^2x'_{nj}>0$ (p. 297). It leaves open whether the convergences in (15), and that of these second differences to 0, are monotonic in $n$ (p. 297), and in Section 5 (pp. 299--300) poses conjecture (21), from numerical calculations, that the higher differences in $j$ of the zeros of the first and second derivatives alternate in sign.

Read status: claims checked for the result pages below, read on the page images of the print, pp. 293--300; the proofs were followed but not checked step by step.

Source: https://real-j.mtak.hu/7424/.

Result pages: Statement (I) (p. 294), Statement (II) (p. 294), Relations (5) and (7)--(11) (p. 295), Inequalities (12) and (13) (pp. 295--296), Relations (14) and (15) (pp. 296--297), Statement (III), with (16)--(20) (pp. 297--299), Conjecture (21) (pp. 299--300).

Bears on.

• #1114: background and a proposed generalization. The paper states Erdős's conjecture for a polynomial with simple, equally spaced, real zeros (p. 293) and credits its proof to Bálint, without reproving it. It reproves Bálint's inequality (13), that each gap exceeds the spacing of the zeros, which does not compare consecutive gaps, and its conjecture (21) would extend the gap monotonicity, which the paper says is the case $m=2$, to higher differences.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.