# Tagged Questions

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### Show that $\liminf\limits_{n\rightarrow\infty}a_{n}/a_{n-k}\leq \liminf\limits_{n\rightarrow\infty}a_{n}^{k/n}$

Here is an exercise: Let $\{a_{n}\}$ be a positive increasing sequence, can we prove that: $\liminf_{n\rightarrow\infty}\frac{a_{n}}{a_{n-k}}\leq \liminf_{n\rightarrow\infty}a_{n}^{k/n}$? Could ...
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### Does $xy\geq x+y$?

I just see the GM-AM inequality. But I would like to compare $xy$ with $x+y$ for any $(x, y)\in\mathbb{R}^2$. It looks like $xy>x+y$ since the first one is multiplication and the second one is ...
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### Proving an elementary inequality of real vectors related to the p-Laplacian [duplicate]

How would you prove the following inequality? $$\left|\left|a\right|^{p-2}a-\left|b\right|^{p-2}b\right|\leq C_p\left|a-b\right|^{p-1}$$ where $a,b\in\mathbb{R}^{n}$ and $C_p>0$ is some constant ...
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### Help with Gronwall's Inequality

I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
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### How to show $(s+t)^p\le 2^{p-1}(s^p + t^p)$ for $p\gt1$

How to show $(s+t)^p\le 2^{p-1}(s^p + t^p)$ for $p\gt1$ I know how to prove for $s+t=1$ by $\min\lbrace t^p+(1-t)^p\rbrace=2^{1-p}$. But I do not know to how to generalize for $s+t\gt0$. Could you ...
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### Local equivalency between the Euclidean norm and the CC norm

Let $X_1,...,X_m$ be a family of vector fields defined on an open and connected subset $\Omega$ of $\mathbb{R}^n$ with locally Lipschitz continuous coefficients. Assume that the vector fields are ...
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### Directional derivative in a Sobolev-like inequality

I am trying to do the following problem: Let $\Omega \subset \subset \overline{\mathbb{R}_{+}^{n+1}} = \{ (x, x_{n+1}); \, x_{n+1}\geq 0\}$ (i.e., $\Omega$ is bounded inside the closed upper ...
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### Integral Inequality Question: Effect of Exponents

Is it true that, if $\alpha q < (1-\alpha)q$ then $\int|k(x,y)|^{\alpha q}dy \leq \int|k(x,y)|^{(1-\alpha)q}dy$, where $k(x,y)$ is a positive measurable function?
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### Need help filling in the details of proof of Jensen's Inequality

In a book on PDEs that I'm reading, I am trying to fill in the details of the proof of Jensen's Theorem, and am having a little trouble with the algebra. Here is the statement of Jensen's Theorem in ...
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### Inequality for sum of functions defined on power set

Consider $x_1, \ldots , x_N \in \mathbb{R}^n$, and $p_1, ..., p_N \in \{0,1\}$ such that $\sum_{i=1}^N p_i = 1$. Consider $f : \mathcal{P}\left( \{ x_1, \ldots, x_N \}\right) \rightarrow [0,1]$, ...
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### Complicated Inequality involving Improper Integrals

Let $\ell>k>1$, $r=\frac{\ell}{k}-1$ and $y'$ be positive and $L^k$, then $$\int_{0}^{\infty}\frac{y^\ell}{x^{\ell-r}}dx<K \left( \int_{0}^{\infty} y'^k dx\right)^{\frac{\ell}{k}}$$ ...
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### Problem with an inequality from probability theory (Random matrix theory)

I read the following notes on random matrix theory http://www.umpa.ens-lyon.fr/~aguionne/cours.pdf . While reading Wigner's proof for the semi-cicular law I encoutered the following inequality on page ...
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### Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$ for $|x-y|\le\alpha$.

Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $$|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$$ for $|x-y|\le\alpha$. From the mean value theorem, given any $x,y$ with ...
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### Wirtinger's inequality in higher dimension

Wirtinger's inequality for one-dimensional functions states that if $f(x)$, $f'(x) = \frac{df(x)}{dx}$ $\in$ $\mathcal{L}^2(a,b)$ and either $f(a) = 0$ or $f(b) = 0$ then \int_{a}^{b} ...
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### Is there a means of analytically proving the following identity?

Okay, so before I begin, my background is more in the world of applied, rather than pure, mathematics, so this question is motivated by a physics problem I'm looking at just now. Mathematically, it ...
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### Hölder's Inequality and step functions

Define functions $f(x) = \sum_{k=0}^\infty a_k \chi_{[k,k+1)}(x)$ and $g(x) = \sum_{k=0}^\infty b_k \chi_{[k,k+1)}(x)$ where $\chi_ {[k,k+1)}$ is the indicator function for the given interval. Let ...
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### How to prove that $2\sqrt{a^{ea}b^{eb}}\ge a^{eb}+b^{ea}$ for $a > 0, b > 0$?

Let $a,b\in R^{+}$. Show that $$2a^{\frac{ea}{2}}b^{\frac{eb}{2}}\ge a^{eb}+b^{ea} \>.$$ My attempt I know the following inequality is true: $$a^{ea}+b^{eb}\ge a^{eb}+b^{ea} \>.$$ See this ...
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### Proof of the nonexistence of an identity $\phi$ involving convolution

The Banach space $L^1(\mathbb{R}^n)$ is an algebra with a product (convolution) which is both commutative and associative. But this algebra does not have a multiplicative identity. An attempt to show ...
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### Interesting inequality $\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$ over $L^p$

Consider the function $$F(x)=\int_0^\infty \frac{f(y)}{x+y} \, dy, \quad0<x<\infty$$ Prove that if $1<p<\infty$, $$\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$$ and show that the constant ...
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### Lower bound on a number theoretic function

Let $n$ be a positive odd integer, let $$n_j = \Bigl\{\frac{n}{2^{j+1}}\Bigr\}\,,$$ where $\{x\}$ denotes the fractional part of $x$, and finally let $k = \lceil \log_2 n\rceil$. Consider the ...
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### Monotonicity in $x$ of $\frac{x^\alpha - 1}{x-1}$

I'm trying to show that for fixed $\alpha$, the function $f(x)=\frac{x^\alpha-1}{x-1}$ is monotonic for $x>0$ -- either strictly increasing or strictly decreasing, depending on the choice of ...
Good evening, everyone, Could anyone please tell me how to check if the series $\sum_{k\geq 2}\dfrac{1}{k^4+x}$ is greater than $C\sum_{k\geq 2}\dfrac{1}{k^2+x}$ where $C$ is independent of the ...