Articles of inequality

Endpoint-average inequality for a line segment in a normed space

Let $X$ be a normed vector space over $\mathbb R$. What is the smallest universal constant $C>0$ such that the inequality $$\|x\|\le C\int_0^1 \|x+tv\|\,dt\tag{1}$$ holds for all $x,v\in X$? Geometrically, (1) means that the norm of an endpoint of a line segment can be majorized by the average of the norm over said line segment. […]

Inequality problem $(a+b+c)^5\geq27(ab+bc+ca)(ab^2+bc^2+ca^2)$

While solving one inequality, I arrived at a much simpler, but still nontrivial inequality $$(a+b+c)^5\geq27(ab+bc+ca)(ab^2+bc^2+ca^2)$$ where $a,b,c$ are positive real numbers. It apparently holds, but I can’t seem to find a proof. The problem is it’s not symmetric and applying inequalities $(a+b+c)^2\geq3(ab+bc+ca)$ or $a^3+b^3+c^3\geq ab^2+bc^2+ca^2$ won’t work. Any ideas?

Where does the premise of this idea come from?

Let $x$ , $y$ be positive real numbers. Prove the inequality $$x^ y + y^x \ge 1$$ This is the solution provided by my textbook: Where does this first idea (proving that $a^b \ge \frac{a}{a+ b – ab}$) come from? I’m repeatedly frustrated by problems like these because the solution uses some forlorn idea which […]

Sum of real powers: $\sum_{i=1}^{N}{x_i^{\beta}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta}$

Let $\{x_i\}_{i=1}^{N}$ be positive real numbers and $\beta \in \mathbb{R}$. Can we say that: $$ \sum_{i=1}^{N}{x_i^{\beta}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta}$$ I know that this holds if $\beta \in \mathbb{N}$. Does the above inequality have a name in case it’s true?

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in \ell^2(\mathbb{Z}^n)$ where the above sums converge? My ideas so far: $ij \leq i^2 + j^2$ shows that $C=1$ works. Is 1 the sharpest possible bound? (This comes from showing that for $u,f$ periodic and […]

there exist some real $a >0$ such that $\tan{a} = a$

How can i prove that there exist some real $a >0$ such that $\tan{a} = a$ ? I tried compute $$\lim_{x\to\frac{\pi}{2}^{+}}\tan x=\lim_{x\to\frac{\pi}{2}^{+}}\frac{\sin x}{\cos x}$$ We have the situation ” $\frac{1}{0}$ ” which leads us ” $\infty$ ” $$\lim_{x\to\frac{\pi}{2}^{-}}\tan x=\lim_{x\to\frac{\pi}{2}^{-}}\frac{\sin x}{\cos x}$$ We have the situation ” – $\frac{1}{0}$” which tells us ” $- \infty$” This […]

Show that, for all $n > 1: \log \frac{2n + 1}{n} < \frac1n + \frac{1}{n + 1} + \cdots + \frac{1}{2n} < \log \frac{2n}{n – 1}$

I’m learning calculus, specifically limit of sequences and derivatives, and need help with the following exercise: Show that for every $n > 1$, $$\log \frac{2n + 1}{n} < \frac1n + \frac{1}{n + 1} + \cdots + \frac{1}{2n} < \log \frac{2n}{n – 1} \quad \quad (1)$$ Important: this exercise is the continuation of a previous problem […]

Proving $\left(A-1+\frac1B\right)\left(B-1+\frac1C\right)\left(C-1+\frac1A\right)\leq1$

This question already has an answer here: Inequality $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$ 1 answer

If $a+b+c=3$ show $a^2+b^2+c^2 \leq (27-15\sqrt{3})\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

I have become interested in constrained relations among simple cyclic sums involving three positive variables. By simple, I mean so simple that they are also fully symmetric. The “building blocks” of the constraints and relations I have been looking at are: $$ \sum_{\mbox{cyc}} 1 \equiv 3 \\ \sum_{\mbox{cyc}} a \\ \sum_{\mbox{cyc}} ab \\ \sum_{\mbox{cyc}} a^2 […]

Inequalities of expressions completely symmetric in their variables

We often encounter inequalities of symmetric expressions, i.e. the expression doesn’t change if the variables in it are interchanged, with the prior knowledge of a certain relation between those variables. In all such cases that I have encountered thus far, we can find the extremum of the expression by letting the variables equal. Here are […]