1
vote
0answers
34 views

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 ...
0
votes
4answers
118 views

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 ...
1
vote
0answers
21 views

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 ...
0
votes
0answers
53 views

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 ...
1
vote
2answers
35 views

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 ...
0
votes
0answers
6 views

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 ...
0
votes
1answer
35 views

Integral inequality in $\Bbb R^n$

I came across this problem : Let $f\colon [a,b]\rightarrow \mathbb{R}^n$ a continuous vector valued function. Then it is true that: $$\left\Vert\int \limits_a ^b f(t) dt\right\Vert \leq \int ...
4
votes
3answers
184 views

Prove that the logarithmic mean is less than the power mean.

Prove that the logarithmic mean is less than the power mean. $$L(a,b)=\frac{a-b}{\ln(a)-\ln(b)} < M_p(a,b) = \left(\frac{a^p+b^p}{2}\right)^{\frac{1}{p}}$$ such that $$p\geq \frac{1}{3}$$ That is ...
0
votes
1answer
34 views

when one can expect $\left \| |x|^{2}x- |y|^{2}y \right \| \leq \frac{1}{2} \left \| (x-y) \right \|$?

Suppose $(X, \left\|\cdot \right \|)$ is Banach algebra with the property that $\left\| (|x|^{2}x) \right\| \leq \left\|x\right \|^{3};$ for every $x\in X.$ Let $y_{0}\neq 0 \in X$ and fix it; and ...
0
votes
1answer
15 views

Is this function biLipschitz?

Let $\Omega = (1,2) \times (1,2) \subseteq \mathbb{R}^2$ and define $f$ on $\Omega$ to $\mathbb{R}^2$ as follows: $$ f(x,y) = (x + y, x + y^2). $$ Does there exist a constant $C>0$ such that for ...
0
votes
0answers
44 views

An inequality with hyperbolic function and exponential function

I was encountering with a conjecture about the following statement: Let $Y>0$ and $X \neq 0$ it follows that , $$\cosh(KX) >e^{K^2Y}$$ for all $0<K<\frac{|X|}{2Y}$ it seems to me I need ...
4
votes
1answer
109 views

Equality in Minkowski's inequality proof(no integrals)

So what I'm looking for is a proof for when does the equality hold in Minkowski's inequality? I'm talking about this form of inequality: $\left( \sum_{K=1}^n |x_k + y_k|^p \right)^{\frac{1}{p}} \leq ...
1
vote
1answer
74 views

the best constant in an inequality?

I learnt how to show the below inequality by C-S inequality: k is from $0$ to $\infty$ If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. next,I tried to show that 3 is the best possible ...
1
vote
1answer
63 views

An application of Cauchy-Schwarz ineq. on infinite series

If $\sum a_{k}^{2}9^{k}\le 5$ then $\sum |a_{k}|2^{k}\le 3$. sums are from $0$ to $\infty$. could you please help with this question.
0
votes
1answer
32 views

Deriving Holder's Inequality

This is classwork: Let $m,n > 1$, where $1/m + 1/n = 1$. Fix $A > 0$ and have a function $f(x) = \frac{x^m}{m} + \frac{A^n}{nx^n}$ for $x > 0$. Show that $f$ has a global minimum at $x = ...
0
votes
1answer
51 views

Validity of an integral inequality

Suppose we have two functions $f(x)$ and $g(x)$ And $f(x)<g(x)$ for all values of $n$ Then for arbitrary $a$ and $b$ (within the range) is it true that $$\int_b^{a}\frac{dx}{f(x)} > ...
2
votes
2answers
69 views

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 ...
2
votes
1answer
19 views

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?
2
votes
0answers
47 views

Unexplained “It Suffices to Show”

For $u \in C^{\infty}(\Omega)$, for $\Omega$ convex, and $p$, $q$ such that $\frac{1}{p}-\frac{1}{q}<n$, then in order to show that $$||u-u_{\Omega}||_{L^{q}} \leq c_{n} \left[ ...
4
votes
0answers
41 views

Soft Question: Inequalities like this

I am studying signed and complex measure and at a point in a proof the following lemma is being used: Lemma. If $z_1,...,z_n$ are complex numbers, then there exists a subset $S\subset\{1,2,...,n\}$ ...
10
votes
3answers
287 views

Prove that $\,\,\displaystyle\inf_{n\in\mathbb N}\sum_{k=0}^{p}\lvert\sin{(n+k)^p}\rvert>0$

For any poistive integer number $p$, show that $$\inf\left\{ {\left\vert\sin{(n^p)}\right\vert+\left\vert\sin{(n+1)^p}\right\vert+\cdots+ \left\vert\,\sin{(n+p)^p}\right\vert\, :\,n\in ...
1
vote
1answer
34 views

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 ...
0
votes
1answer
38 views

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 $f : \mathcal{P}\left( \{ x_1, \ldots, x_N \}\right) \rightarrow [0,1]$, ...
2
votes
0answers
41 views

Complicated Inequality involving Improper Integrals

Let $\ell>k>1$, $r=\frac{\ell}{k}-1$ and $y'$ be positive and $L^k$, then $$\int_{0}^{\infty}\frac{y^\ell}{x^{\ell-r}}dx<K \left( \int_{0}^{\infty} y'^k dx\right)^{\frac{\ell}{k}} $$ ...
0
votes
0answers
21 views

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 ...
0
votes
1answer
127 views

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 ...
1
vote
1answer
59 views

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 \begin{equation} \int_{a}^{b} ...
6
votes
1answer
298 views

The series $\sum\limits_{n=1}^\infty \frac n{\frac1{a_1}+\frac1{a_2}+\dotsb+\frac1{a_n}}$ is convergent

If a series $\sum\limits_{n=1}^\infty a_n$ is convergent, and $a_n\gt0$... Do not refer to Carleman's inequality or Hardy's inequality, show that the series $$\sum_{n=1}^\infty \frac ...
1
vote
0answers
53 views

$L^{\infty}$ is p-concave

I want to show that the $L^{\infty}$ on a Banach lattice $X$ is $p$-concave with $M_{(p)}(L^{\infty})=1$. Where $L^\infty=L^\infty(X,\mathcal{M},\mu)$. Recall that a Banach lattice $X$ is said ...
2
votes
2answers
224 views

Let $f,g$ be differentiable with $f(0)=g(0)$ and $f'(x)<g'(x)$. Prove that $f(x)<g(x)$.

Let $f,g:\mathbb{R} \rightarrow \mathbb{R}$ be differentiable with $f(0)=g(0)$ and $f'(x) < g'(x)$ for all $x$ belonging to the set of real numbers. Prove that $f(x)<g(x)$ for all $x>0$. Any ...
2
votes
0answers
44 views

Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following: $$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$ and $$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$ Note that ...
1
vote
0answers
63 views

Prove that $0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$

Let $f'$ be integrable. Prove that $$0\le\frac1{b-a}\int_a^b|f(x)|dx-\left|\frac1{b-a}\int_a^bf(x)dx\right|\le\frac{b-a}3\sup_{a\le x\le b}|f'(x)|$$ Source: ...
7
votes
1answer
141 views

Bound on $|f(x)|^2 + |f'(x)|^2$

Let $f\in C^2(\mathbb{R})$ be a twice differentiable function satisfying $$|f(x)|^2\le a$$ and $$|f'(x)|^2 + |f''(x)|^2\le b$$ for all real $x$, where $a$ and $b$ are positive constants. Prove that ...
10
votes
6answers
183 views

Proof of the following inequality $ \frac{x - y}{\log x - \log y} > \sqrt{xy} $, $x>y$.

I have seen the following inequality $$ \frac{x - y}{\log x - \log y} > \sqrt{xy} \ , \quad \forall x>y $$ be stated as a near "obvious" fact in another question, on the site. The inequality ...
1
vote
2answers
44 views

prove that for $n \ge 4, {{2n}\choose{n}} \ge n\cdot2^n$

Prove that for $n \ge 4$ $${{2n}\choose{n}} \ge n\times2^n$$ I tried like that: $T_4$: ${{8}\choose{4}} = 70 \ge 4\times2^4$ = 64 so it's ok $T_{n+1}$: $$\frac{(2n+2)!}{(n+1)!)(n+1)!} \ge ...
1
vote
3answers
48 views

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 ...
1
vote
1answer
42 views

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 ...
3
votes
1answer
325 views

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 ...
0
votes
1answer
54 views

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 ...
3
votes
1answer
66 views

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 ...
3
votes
1answer
98 views

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 ...
0
votes
1answer
50 views

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 ...
0
votes
1answer
37 views

about function series

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 ...
3
votes
2answers
65 views

Proof of $|x^{\alpha} - y^{\alpha}| \le \alpha^{\alpha} |x-y|$ for $\alpha \ge 1, x,y\in [0,1]$

I want to prove $$ |x^{\alpha} - y^{\alpha}| \le \alpha^{\alpha} |x-y| $$ for $\alpha \ge 1$ and $x,y \in [0,1]$. For $\alpha \in \mathbb N$ I already got the proof by using the formulae $$ (x^n - ...
6
votes
4answers
94 views

How to prove $\sum_{k=1}^n \frac{2^k}{k}< 3\frac{2^n}{n}$?

How to prove $$\sum_{k=1}^n \frac{2^k}{k}< 3\frac{2^n}{n}$$ and further $$\lim_{n\rightarrow \infty}\frac{n}{2^n}\sum_{k=1}^n \frac{2^{k}}{k} = 2$$? These results are verified by computer, yet I ...
3
votes
2answers
61 views

The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive ...
1
vote
0answers
84 views

Is there such an example?

Is there an example of a sequence of point sets $\left\{ S_{n}\right\} _{n=1}^{\infty}$in which $S_{n}$ is a set of $n$ points inside the unit triangle, such that the minimum altitude of the triangles ...
6
votes
2answers
176 views

How find this$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+…+\frac{1}{{{p}_{n}}}<10$

Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that $$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$$ This problem is from this ...
3
votes
2answers
40 views

How to check whether a given inequality is correct for a large span of integers?

The inequality $\sqrt{n+ 1}−\sqrt n < \frac{1}{\sqrt n}$ is false for all n such that $101 ≤ n ≤ 2000$. Is the statement true?
2
votes
1answer
70 views

Trigonometric inequality for the sum of sin and cos

I need a proof for the following trigonometric inequality $$\frac{|\sin x| + |\cos x|}{\sqrt2} \leq 1- \frac{\cos^2(2x)}{8}$$ Can someone please help me with this?