Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

381 questions
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Showing a certain map is a norm.

Define $$\|x\|=\sqrt{(|x_1|^2+|x_2|^2)^{3/2}+|x_3|^3}$$ for any $x=(x_1,x_2,x_3)\in \Bbb R^3$. Show that $\|\cdot\|$ is a norm on $\Bbb R^3$. I stuck on showing triangle inequality. I don't know ...
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How to solve $| x - a | + | x + a | < b$ where $a \neq 0$?

The options are : A ) no solution if $b\leq 2|a|$ B ) has a solution set ${ ( -b/2 , b/2 ) }$ if $b > 2|a|$ C ) has a solution set ${ ( -b/2 , b/2 ) }$ if $b < 2|a|$ D ) All of ...
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Cauchy functional inequality

Given a function on a closed interval $f\colon I\subset \mathbb{R}\to \mathbb{R}$ with $$f(x+y) \leq f(x) + f(y).$$ Moreover, I know that $f$ is monotonic increasing continuous on all points except ...
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Prove that $\log x<\sqrt{x}$ for $x\geq 1$

Prove that $\log x<\sqrt{x}$ for $x\geq 1$ Let $f(x)=\sqrt{x}- \log x$. So, $f(1)=1>0$. $f'(x)=\frac{1}{2\sqrt{x}}-\frac{1}{x}>0$ only when $x>4$. When I draw the graph of $f$ in ...
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Functional equation $(n-1)^2 < f(n) f(f(n)) < n^2 +n$.

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$(n-1)^2 < f(n) f(f(n)) < n^2 +n$$ for all $n \in \mathbb{N}$.
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A tighter family of Markov-like inequalities

I believe there should exist tighter-than-Markov Inequalities that use the same information as the markov inequality (just expectation). Consider the proof of the markov inequality in the link below: ...
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Extending continuous linear functional of the derivative to continuous linear functional of the function

Suppose given $f,g\in L^2(\mathbb{R})$ and $f', g' \in L^2(\mathbb{R})$, the linear functional defined by $$F(g):= \int_{\mathbb{R}} f'g' dx$$ is continuous with respect to the derivative, that is ...
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Functional inequality $f(y)+xf(x)≤yf(x)+f(f(x))$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function sucht that: $$f(y)+xf(x)≤yf(x)+f(f(x))$$ for all $x,y\in\mathbb{R}$. Show that $$f(x)+yf(x+y)≤0$$ for all $x,y\in\mathbb{R}$. I tried some ...
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$t\in (0,1)$ and $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$. Show that if strict inequality holds for even one $t$, then it holds for all $t$.

This is a part of a solution to a problem in showing that if $f$ is continuous and satisfies the condition $f([x+y]/2)\lt [f(x)+f(y)]/2$, then $f$ is convex. Let $t\in (0,1)$. We have the weak ...
I am reading the book Elliptic Partial Differential Equation by Fanghua Lin and I got stuck at the lemma 1.36 ( the Caccioppoli inequality). The conditions of this lemma are: $u\in C^1(B_1)$ ($B_1$ is ...