bibkey: xuyang2026millsratio
authors: Hui-Zuo Xu and Zhen-Hang Yang
year: 2026
title: Higher order monotonicity involving the Mills ratio with applications
doi: 10.1186/s13660-026-03505-9
url: https://doi.org/10.1186/s13660-026-03505-9
claim: "Conjecture 7.1 proposes complete monotonicity of the signed asymptotic remainders of Y at a = -1."
strata_touched:
  - D5/S3/Analytic/XuYangMillsRemainder/Result
license: citation-only
triage: anchor

Mills-ratio asymptotic remainders

Verified locator

https://doi.org/10.1186/s13660-026-03505-9

Journal of Inequalities and Applications (2026) 2026:98. The locators below are printed article pages.

Source statements

Mills-ratio definition, printed p. 2:

The Mills ratio [3] is defined by

$$R\equiv R(x)=\frac{1-\Phi (x)}{\varphi (x)}>=e^{x^2/2}{\int }_x^{\infty }{e}^{-t^2/2}dt,x\ge 0,$$

which satisfies the differential equation of first order:

$$R^{\prime }=xR-1.(1.1)$$

Here $\varphi (x)=e^{-x^2/2}/\sqrt{2\pi }$ and $\Phi (x)={\int }_{-\infty }^x\varphi (t)dt$ are the standard Gaussian density and cumulative distribution function used in the preceding source sentences.

The bound family in (1.5), printed p. 2, is

$${\phi }_{p,q,r}(x)=\frac{p}{\sqrt{x^2+q}+rx},p,q>0,r\ge 0.$$

Equation (1.8), printed p. 3, with its complete surrounding statement:

Note that all bounds of the form ${\phi }_{p,q,r}(x)$ can be derived from the inequality

$$Y_{a,b}(x):=(x^2+a)R^2(x)+bxR(x)<(>)c(1.8)$$

for all $x>0$, $a,b\in \mathbb{R}$, where

$$c=\underset{x\to 0}{\lim }{Y}_{a,b}(x)=\frac{a\pi }{2}>\text{or}c=\underset{x\to \infty }{\lim }{Y}_{a,b}(x)=b+1.$$

Equation (3.7), printed p. 6, gives the coefficients following the asymptotic expansion in Theorem 3.1:

where

$$u_n\equiv u_n(a,b)=(b+1)\frac{2^n\Gamma (n+1/2)}{\Gamma (1/2)}>+(n-a)(n-1)!\sum _{k=0}^{n-1}>\frac{2^k\Gamma (k+1/2)}{\Gamma (1/2)k!}.(3.7)$$

In particular, $u_1=b+2-a$, $u_2=3b+7-2a$, $u_3=15b+36-7a$, and $u_4=105b-36a+249$.

Conjecture 7.1, printed p. 27, complete statement:

Moreover, inspired by Theorem 4.2, we propose the following conjecture.

Conjecture 7.1 Let

$$D_n(x,a)=Y_{a,a-2}(x)-\left(a-1+\sum _{k=1}^n>\frac{(-1{)}^k{u}_k}{x^{2k}}\right),$$

where $u_k$ is defined by (3.7). Then $x\mapsto (-1{)}^n{D}_n(x,-1)$ is CM on $(0,\infty )$.

Mathematical correspondence

millsRatio uses the displayed Gaussian-tail normalization of $R$. Y is the defining expression in (1.8), and u retains the Gamma factors and finite sum of (3.7). D is the remainder in Conjecture 7.1. CompletelyMonotoneOn means $C^{\infty }$ on $(0,\infty )$ together with $(-1{)}^m{f}^{(m)}(x)\ge 0$ for every nonnegative integer $m$ and every $x>0$. claim quantifies over all integers $n\ge 2$.

At $(a,b)=(-1,-3)$, the source coefficients satisfy $u_1=u_2=0$. Theorem 4.2, printed p. 14, treats $n=2,3,4$; the all-index assertion extends those cases. Since $D_1=D_2$, the signed remainder at $n=1$ is $-D_2$ and is excluded. The question for general $a$ is separate.