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. […]

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?

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 […]

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?

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 […]

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 […]

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 […]

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

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 […]

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 […]

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