# Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

377 questions
55 views

### Proving a sequence is Cauchy in metric

Consider the sequence, $f_n(x)= \begin{cases} (2x)^n & 0 \leq x\leq \frac{1}{2} \\ 1 & \frac{1}{2} \leq x \leq 1\\ \end{cases}$ Then we need to show that $\{ f_n\}$ is Cauchy ...
29 views

### Show that $f'(0) \geq -\sqrt2$ [duplicate]

Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \geq 0$,$f'(x) \leq 0$ and $f''(x) \leq f(x)$, for all $x \geq 0$. Show ...
11 views

### Notation Question Regarding Integral Kernels

I'm reading a proof of an estimate on an integral kernel. They sponateously introduce notation and I'm having trouble following what's happening. (The offending reference is the bottom of pg 60, Lemma ...
108 views

### Let $P$ be a polynomial with positive real coefficients. Prove that if $P(1/x) \geq 1/P(x)$ holds for $x = 1$, then it holds for every $x > 0$.

Let $P$ be a polynomial with positive real coefficients. Prove that if $$P\left( \frac{1}{x} \right) \geq \frac{1}{P(x)}$$ holds for $x = 1$, then it holds for every $x > 0$. What I did: I was ...
37 views

### Inequalities involving the greatest common divisor

Question What are some inequalities involving the greatest common divisor (GCD) function? Typical Answer $$\gcd(a,bc) \leq \gcd(a,b)\gcd(a,c)$$ My Attempt I tried checking the Wikipedia page, ...
21 views

56 views

### Inequality for convex function $f:(0,\infty)\to\mathbb{R}$ [duplicate]

I have been working through the Exercises at the end of Chapter $1$ of Bollobas' Linear Analysis. Chapter $1$ is on inequalities, and the text is fairly brief. I have found the problems unexpectedly ...
39 views

### Functional Space Inequality for Sobolev Space and Lp Space

Let $X = C_{0}(\Omega) := \{ u \in C(\overline{\Omega})\,|\,u|_{\partial\Omega}=0\}$ and define $F : X \to X$ as Lipschitz continuous function and $F(0) = 0$. Let $\Omega\subset \mathbb{R}^{N}$ be a ...
903 views

28 views

### Alternative form for Liapunov inequality

Let $1<p<q<\infty$, and $r\in [p,q]$ whith $\frac{1}{r}= \frac{\alpha}{p}+ \frac{1-\alpha}{q}$. If $f\in L_p\cap L_q$ then $$\|f\|_r \leq \|f\|_p^\alpha\|f\|_q^{(1-\alpha)}$$ My teacher ...
45 views

### Functional is weak lower semicontinuous but not weak continuous

I want to show that the functional $L(u)=\int_0^1 \sqrt{1+(u'(x))^2} dx$ is lower semicontinuous in terms of weak convergence in $W^{1,p}(0,1), p\in(1,\infty)$ but not continuous. Our definition of ...
16 views

331 views

47 views

41 views

21 views

24 views

### Equivalence of norm in Banach Space

Let $A$ be a densely defined closed linear operator in a Banach space $X$ and $\sigma(A)$ be its spectrum. We define its spectral radius $r_{A} := \sup\limits_{\lambda \in \sigma(A)}|\lambda|$. Now, ...
34 views

### Determine the set of all complex number z satisfying following conditions

I’m having some troubles of calculating complex numbers where I need to deal with absolute values and inequalities. Here is an example I’ve been working on but I get stuck Re(2/z)+Im(4/z)<1 I use ...
133 views

### Why does proving inductively $n < 2^n$ for $n \geq 1$ imply it is true for real values of $n$?

If we prove by induction that $2^n > n$ for $n \geq 1$ where $n \in N^+$, how can one know this inequality holds for real values of n like $2^{2.5} > 2.5$? Maybe a bit silly question but I can'...
66 views

### How is this proposition true?

I have a function of $n$, where $n$ is an integer, defined as follows $$D(n)=\frac{nK}{K+\bigg\lceil\frac{n}{M}\bigg\rceil(n-1)},$$ where $K$ are $M$ are some constant positive integers. For this ...
36 views

I have function $$D(M\gamma)=\frac{M\gamma K}{K+\gamma(M\gamma-1)}\tag{1}$$ where $K$ is some positive constant. In this case, how to show that D(M(\gamma-1))<D(M\gamma),~~\text{for } \gamma<\... 1answer 112 views ### Interpolation inequalities Let \Omega be a regular domain of \mathbb{R}^d, d=2,3. Let \mathcal{T}_h be a triangulation of \Omega of size h>0. Assume we can prove \begin{equation*} \begin{aligned} \|v\|_{L^2(\... 1answer 21 views ### Best way to check to see if conditions are (or can be) satisfied? Sorry to bother you guys, but a project I'm working on necessitates checking to see whether somewhat obscure conditions can all hold, given some bounds on possible values for each variable of interest.... 0answers 49 views ### A generalisation of a Hardy inequality I found the following result in Brezis' book. Let \Omega be a bounded open set of class C^1. Let d(x):=\operatorname{dist}(x,\partial\Omega). Then there exists C>0 such that \|\...
Let $f(x)=x^r, 1<r<\infty, x\in \mathbb{R^+}=[0,\infty) ~and ~n\in \mathbb{N}.$ Define $$\pi(x)= 1 ~~if~~ x\leq n ~and =0 ~if~ x< (n+1).$$ Then for any $x,y\in\mathbb{R},~~$ I want to ...
Define a pair of functions $p, h: \mathbb{Q} \to [0,1]$ over the rational numbers, with the properties \begin{align} 0 \le p(x) \le 1 && 0 \le h(x) \le 1 && (x\in\mathbb{Q}), \\ \end{...