# Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

115 questions
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$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
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### How to prove this polynomial inequality?

How can we prove the following? If $\frac{dP_{n}}{dz}|_{z=z_{0}}=0$ then $|P_{n}(z_{0})|<2$ for all $n>1$, where $P_{n}(z)\equiv P_{n-1}^{2}+z$ and $P_{1}\equiv z$ $z$ is in the complex plane. ...
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### Question about the proof of General Sobolev Inequality in P.D.E. by Evan

I have been reading the chapter of Sobolev Space in Partial Differential Equations by Lawrence C. Evan, and I came across the General Sobolev Inequality stated as follows: Theorem (General Sobolev ...
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### Functional inequality with a strong RHS

Consider a continuous function $f:[0,1]\to\mathbb{R}^{+}$. Show that $$\int_0^1 f(x)dx-\exp\left(\int_0^1 \log(f(x)) dx\right)\le \max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2$$ I ...
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### Evaluating constants of an inverse estimate

In some analysis on a domain $\Gamma$, I want to employ a type of inverse estimate $$\|F\|_X \le \frac{k}{\Delta{x}}\|F\|_{L^2(\Gamma)}$$ where $F$ belongs to a finite-dimensional subspace, $\|X\|$ is ...
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### Jensens function on the real line exactly and conditional equations f(xy) + f(xy−1) = 2f(x) and f(xy) + f(y−1x) = 2f(x)

What is meant by Jensens function on the real line exactly F(x+y)+F(x-y)=2F(x); jensens equation on a R rather than just a continuousinterval(a closed and bounded real valued (continuum interval such ...
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### Strictly Monotonic, Continuous Sub-linear functions with F(1)=1; Any real difference between these and linear functions?

Is there a great deal of difference between sub (or super-linear functions) Functions F, and linear functions, ie, when a sub-linear function $F$, is also strictly monotonic increasing and continuous ...
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### Inequalities and limits?

Given a function $f$ such that $\lim \limits_{x \to \infty} f(x) =0$, and want to see if $f(x) >0$ or $f(x)<0$, but its so hard to tell(its very complicated function). My approach to solve ...
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### Best minimum constant for a functional inequality

Let $f$ be a twice differentiable function from $[0,1]$ to $\mathbb{R}$ with $f"$ continuous on $[0,1]$ and $\int_a^b f(x)dx=0$ where $0<a<b<1$. What is the minimum constant $C$, function ...
### A question on $\arctan x$
Let's suppose that I need to know if the angle $2^i \arctan x$ are in the first, second, third or fourth quadrant, for $i = 1, 2, 3... n$ and some real number $x$. Is it possible to know in which ...