How does one prove the determinant inequality...
Reposted on MathOverflowLet $\,A,B,C\in M_{n}(\mathbb C)\,$ be Hermitian and positive definite matrices such that $A+B+C=I_{n}$, where $I_{n}$ is the identity matrix. Show that...
View ArticleA logarithmic minimum with equal variables for $n= 6$ and unequal variables...
For an integer $n\ge2$, let $a,b,c>0$ satisfy $a+b+c=1$, and let $s_n=1-a^n-b^n-c^n$. Consider$$F_n(a,b,c)=\frac{\displaystyle\sum_{k=1}^{n-1}\binom nk...
View ArticleAsymptotic expansions involving three Gumbel variables
Let $G_1,G_2,G_3\sim\operatorname{Gumbel}(0,1)$ be independent. For $\eta_1,\eta_2,\eta_3\in\mathbb R$ and $\tau>0$,...
View ArticleDoes every triangle with side $a,b,c$ and inradius $r$ satisfy...
Let $a,b,c$ be the sides of a triangle with inradius $r$ and semi-perimeter $s$. Experimental data show that every triangle satisfies$$ 1728r^2|(a-b)(b-c)(c-a)| \le s^5 $$and that $1728$ is the optimal...
View ArticleInequality including the minimum value of the Gamma function and the...
It's a problem I found unconventional :Problem :Let $0<x$ be a real the define :$$f\left(x\right)=\prod_{n=1}^{292}\left(\frac{n+1}{x!+n}\right)^{\frac{1}{n}}$$Shows that :$$f(x)<1+C$$Where $C$...
View ArticleWhy can't we multiply both sides by $x$ when solving $\frac{1}{x} > 1$?...
Given the inequality $\frac{1}{x} > 1$, why can't we simply multiply both sides by $x$ to solve it? Multiplying both sides by $x$ yields $1 > x$, but I'm told this result is incorrect. Someone...
View ArticleA sharp bound for the logarithmic derivative of a hypergeometric function
Following my earlier question, for $0<x<1$, what is the smallest $C(x)$ such that$$0\le\frac{x}{(1-x)^2}-\left(x\frac{\mathrm d}{\mathrm...
View ArticleOriginal source of a classical problem on leading digits of $2^n$ and $5^n$.
I am looking for the earliest known source (or oldest appearance) of the following problem.Let $n$ be a positive integer such that the decimal representations of $2^n$ and $5^n$ begin with the same...
View ArticleBernoulli inequality application
On some level of math in school we learn about Bernoulli's inequality. Proof of its correctness is very common in textbooks as exercise, when we learn mathematical induction.Is Bernoulli's inequality...
View ArticleDvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions?
I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
View ArticleWhat does $a
What does $a <b<c$ really mean?Somebody says that it means “three distinct numbers”and, for example, they said $1\le a < b < c \le N$ means three numbers between $1$ and $N$.We want to say...
View ArticleA sharp refinement of a fractional-power inequality
For $1<r\le5$ and $0<t<1$, let $\mathbf x=(x_1,\ldots,x_4)$ and $\mathbf y=(y_1,\ldots,y_4)$ satisfy $0\le x_1\le\cdots\le x_4\le r$, $0\le y_1\le\cdots\le y_4\le r$, and...
View ArticleAn inequality for ${}_2F_1$ involving $\operatorname{erfc}$
This question arose from my earlier question, where the optimal coefficient is $C(x)=x(1+x)(1+4x+x^2)/(1-x)^5$. Since $C(x)\to\infty$ as $x\uparrow1$, I am looking for a bound that remains finite...
View ArticleA sharp bound for ${}_2F_1(1,-n;n+1;-x)$
For $0<x<1$, what is the smallest $C(x)$ such that$$0\le{}_2F_1(1,-n;n+1;-x)-\frac1{1-x}+\frac{x(1+x)}{n(1-x)^3}\le\frac{C(x)}{n^2}$$holds for every integer $n\ge1$?Here is the attempt: From the...
View ArticleGiven three positive numbers $a,b,c$, prove that $(abc+a+b+c)^{3} \geqq...
Given three positive numbers $a,b,c$, prove that$$(abc+a+b+c)^{3} \geqq 8\,abc(1+a)(1+b)(1+c).$$My own problem is given a solution, and I'm looking forward to seeing a nicer one(s), thank you...
View ArticleGiven three non-negative numbers $a, b, c$ so that $a+ b+ c= 3,\,a^{2}+...
Problem. Given three non-negative numbers $a, b, c$ so that $a+ b+ c= 3,\,a^{2}+ b^{2}+ c^{2}= 5$. Prove:$$a^{3}b+ b^{3}c+ c^{3}a\leqq 8$$My solution in M&Y : (and I'm looking forward to seeing a...
View ArticleGiven two positive numbers $b,\,c$. Prove $\left ( \frac{3}{b}- 1 \right )(3-...
Given two positive numbers $b,\,c$. Prove $\left ( \dfrac{3}{b}- 1 \right )(3- b)^{2}+ \left ( \dfrac{b}{c}- 1 \right )(b- c)^{2}+ (c- 1)^{3}\geqq 0$ .My problem is given a solution by user...
View ArticleAsymptotics of a binomial sum involving $\ln(2\cosh t)$
For $h>0$ and $t\in\mathbb R$, let $g(t)=\cosh t\ln(2\cosh t)-t\sinh t$ and$$S_n(h,t)=\frac{\sqrt n}{2^n}\sum_{k=1}^{n-1}\binom nkg\left(\left(k-\frac n2\right)h+t\right).$$What is a closed-form...
View ArticleA logarithmic inequality involving hyperbolic functions
For $h>0$, let $S_h(t)=\sum_{j\in\mathbb Z}g(jh+t)$, where $g(t)=\cosh t\ln(2\cosh t)-t\sinh t$.For which $h>0$ does$$\left(\cosh\frac{2t}{7}\right)^{7/2}\sum_{j\in\mathbb...
View ArticleIs there a "greater than about" symbol?
To indicate approximate equality, one can use ≃, ≅, ~, ♎, or ≒.I need to indicate an approximate inequality. Specifically, I know A is greater than a quantity of approximately B.Is there a way to...
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