Questions on proving and manipulating inequalities.
2
votes
2answers
219 views
If $ a+b+c = \frac{9}{2}$ and $a,b,c>0$, then what is the minimum value of $\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$
If $a+b+c = \dfrac{9}{2}$ and $a,b,c>0$, then what is the minimum value of $$\dfrac{a}{b^3+54}+\dfrac{b}{c^3+54}+\dfrac{c}{a^3+54} \qquad ?$$
My try:
$$\begin{align*}
...
2
votes
3answers
45 views
Intuition behind $(\{a, b, p, q\} \subset \mathbb{R}^{+} \;\wedge\;\; 1/p +1/q = 1) \Rightarrow a^p/p + b^q/q \geq ab$
If $p$ and $q$ are positive real numbers with1
$$ \frac{1}{p} + \frac{1}{q} = 1,$$
then, for any non-negative real numbers $a, b$,
$$ \frac{a^p}{p} + \frac{b^q}{q} \geq ab$$
My textbook offers a ...
1
vote
3answers
75 views
Infinite Series
How can you show that
$$\left(1-\frac{2}{n^2}\right)^{n^2/2} \le \frac{1}{e}\:\: \qquad\forall n \ge 2$$
Any ideas? Infinite series have never really been my thing. Thanks
2
votes
1answer
28 views
Inequality in inner product space
Given $V$ an inner product space with norm $(‖v‖_V)^2$=$∫_Ω(v^2 (x)+|∇v|^2 )dx$. Prove that
$$(∫_Ω(|v||w|+|∇v||∇w|)dx)^2 ≤ ∫_Ω(|v|^2+|∇v|^2 )dx ∫_Ω(|w|^2+|∇w|^2 )dx=(‖v‖_V)^2(‖w‖_V)^2.$$
Any ...
17
votes
26answers
3k views
2
votes
1answer
52 views
Find the probability that equation has two solutions of different signs
I have 3 random variables $\xi_1, \xi_2,\xi_3$ which are independent and uniformly distributed on segments $[-\sqrt{2}, \sqrt{2}], [-\sqrt{3}, \sqrt{3}], [-\sqrt{\pi }, \sqrt{\pi}]$ respectively.
I ...
1
vote
1answer
33 views
Harmonic mean: show $\max\{ax,by\} \ge \frac{1}{a+b}(x+y)$, $a,b>1$, $x,y\ge 0$
Let $z=x+y$ with $x,y\ge0$ and $a,b>1$.
Show that
$$
\max\{ax,by\} \ge \frac{1}{a+b}z. \tag{1}
$$
This requires either the use of:
(a) the convex function $f(x)=\frac{1}{x}$,
(b) the ...
4
votes
4answers
91 views
Find minimum in a constrained two-variable inequation
I would appreciate if somebody could help me with the following problem:
Q: find minimum
$$9a^2+9b^2+c^2$$
where $a^2+b^2\leq 9, c=\sqrt{9-a^2}\sqrt{9-b^2}-2ab$
2
votes
1answer
52 views
Two inequalities related to norm
We have some difficulties in the following problem:
Let $H$ be a real Hilbert space.
Find $\alpha>0$ such that
$$
\langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq
...
0
votes
1answer
33 views
Normal distribution bound
Let $X$ be a random variable which follows normal distribution.
Is True that
$Pr[|X|\leq \epsilon] \leq \epsilon$ for all $\epsilon \geq 0$.
4
votes
1answer
66 views
A matrix eigenvalue question
If $A, B, C$ are positive definite matrices of size $n$, is it true that $\lambda_j(A(B+C)^2A)\ge \lambda_j(AB^2A)$, $j=1, \dots, n$? $\lambda_j$ means the $j$-th largest eigenvalue.
5
votes
4answers
107 views
Proving $\binom{2n}{n}\le 4^n$ for all $n$ by smallest counterexample
Prove $$\binom{2n}{n}\le 4^n$$ for all natural numbers $n$ by smallest (minimal) counterexample.
My attempt:
First, $$\binom{2n}n = \frac{(2n)!}{(n!)^2} \le 4^n\;.$$ We know that $x\ne 0$ because ...
1
vote
0answers
31 views
Proof of Frechet-Hoeffding Copula bounds
How is the lower Frechet-Hoeffding copula bound proved?
In the bivariate case, it follows from $C(u_1,u_2)-C(u_1,v_2)-C(v_1,u_2)+C(v_1,v_2)\geq0$ by setting $(v_1,v_2)=(1,1)$.
I'm struggling to ...
1
vote
1answer
32 views
Sufficient conditions for an inequality with a log
I need to find sufficient conditions so that $x \geq \frac{1}{a-\ln{x}}$ for $a>1$ and $x > 0$.
Is there a way to get a tight solution to the problem?
1
vote
1answer
32 views
For real $a,b$ show using axioms of ordered field that $a < b$ implie $ a^p < b^p$ whenever $ 0 < a < b$ and $p > 0$.
For real $a,b$ show using axioms of ordered field that $a < b$ implie $ a^p < b^p$ whenever $ 0 < a < b$ and $p > 0$.
I am trying to refresh my memory of maths I learnt 25 years ago. ...
1
vote
1answer
36 views
Special inequality
How to prove the following inequality: $$(a x-b y-c z-d t)^2\geq (a^2-b^2-c^2-d^2) (x^2-y^2-z^2-t^2),$$ if we know $$a^2\geq (b^2+c^2+d^2),$$ $$x^2\geq (y^2+z^2+t^2).$$ Thanks in advance.
3
votes
2answers
55 views
Prove the inequality $x^\alpha \le y^\alpha + z^\alpha$.
Given a triplet of non-negative numbers $x$, $y$, $z$ for which holds $x \le y + z$
one needs to prove that the inequality $x^\alpha \le y^\alpha + z^\alpha$ is also correct for all $\alpha \in ...
1
vote
0answers
39 views
Find the small value of the following functions
Choose $1<x_1<x_2<\cdots<x_M<2$ , such that
$$\left|\sum\limits_{i=1}^{M}x_{i}^{2013}\dfrac{1}{\prod\limits_{1\leq p\leq2013\,,\,,p\neq i}(x_{i}-x_{p})}\right|\leq2$$
where $M=100$
1
vote
0answers
52 views
Azuma's inequality with high probabilistic bounds
Let $(X_n)_{n \geq 0}$ be a super-martingale, that is $\mathbb{E}[X_{n+1} \mid X_1, \dots, X_n] \leq X_n$. Let's further assume that $\Pr[|X_n - X_{n-1}| < c_n] \geq 1-\delta$. Does there exist any ...
0
votes
1answer
29 views
Normal distribution in equality
Let $p(x)=a_1x_1+a_2x_2+. . . .a_n x_n$ be a polynomial such that
$\sum_ia_i^2=1,$ each $x_i \sim N(0,1)$. then we know that $p(x) \sim N(0,1)$.
How can we bound $\Pr_{x\in ...
3
votes
3answers
36 views
How to read this expression?
How can I read this expression :
$$\frac{1}{4} \le a \lt b \le 1$$
Means $a,b$ lies between $\displaystyle \frac{1}{4}$ and $1$?
Or is $a$ less the $b$ also less than equal to $1$?
So $a+b$ ...
1
vote
2answers
56 views
In general $\overline{U_\epsilon(x)}=K_\epsilon(x)$ false
First of all let $(M,d)$ be a metric space. We know that the set $K_\epsilon(x)=\{y\in M \mid d(x,y)\le\epsilon\}$ for arbitrary $x\in M$ and $\epsilon>0$ is closed and ...
1
vote
1answer
33 views
Order of infinite dimension norms
I know that
$$\|{f}\|_{L^1(0,L)}\leq\|{f}\|_{L^2(0,L)}\leq\|{f}\|_{\mathscr{C}^1(0,L)}\leq\|{f}\|_{\mathscr{C}^2(0,L)}\leq\|{f}\|_{\mathscr{C}^{\infty}(0,L)}$$
But I don't know where to put in this ...
3
votes
2answers
40 views
Inequality between norms in $\mathbb{R}^n$
I am trying to prove that given $p>1$ there exists a constant $C=C(p,n)$ such that $\big||x|^px-|y|^py\big|\leq C\big(|x|^p+|y|^p\big)|x-y|$ for all $x,y\in\mathbb{R}^n$. It seems useful to ...
8
votes
3answers
200 views
Polynomial always positive
Is there an elegant way to demonstrate that (for example) $x^{2016}-1008x^2+1007\ge 0$ $\forall x\in \mathbb{R}$ ? I tried to write it as sum of squares, but I didn't succeed.
1
vote
1answer
33 views
Lower bounds for inner product $x^\top y$
Cauchy-Schwartz provides an upper bound for the inner product $x^\top y$. Are there theorems that talk about lower bounds for this quantity? Assume $x\ge 0$ and $y\ge 0$ wlog.
4
votes
1answer
62 views
Poincaré inequality and Rellich Theorem in one dimensional weighted Sobolev space
Consider the weighted Sobolev space $W^{1,2}\big((0,R),r^{N-1}\big)$, $N=2,3,\ldots$ and its subspace $W_0^{1,2}\big((0,R),r^{N-1}\big)$. Anyone knows if the Poincaré inequality is true in this case?
...
1
vote
2answers
86 views
How to evaluate the inequality $|x+1|<-1$?
Okay perhaps the title isn't specific enough, I didn't know how to word it exactly. I'm finding the interval of convergence for a power series and i know the answer to be (-2,0]
I end up with the ...
1
vote
3answers
58 views
If $\frac{1}{2}<a_j<1$ for $j=1,2,\ldots,n$, show that $(1-a_1)(1-a_2)\cdots (1-a_n)>1-\left(a_1+\frac{a_2}{2}+\cdots+\frac{a_n}{2^{n-1}}\right)$
Let $n>1$ be a positive integer and $\frac{1}{2}<a_{j}<1$ for $j=1,2,\ldots,n$. Show that $$(1-a_{1})(1-a_{2})\cdots ...
5
votes
0answers
68 views
Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space
This question is perhaps a little vague; part of what I want to know is what question I should ask.
First, recall the following form of the Cauchy-Schwarz inequality: let $V$ be a real vector space, ...
1
vote
1answer
73 views
Prove two inequalities about limit inferior and limit superior
I wish to prove the following two inequalities:
Suppose $X$ is a subset in $\Bbb R$, and functions $f$ and $g$: $X\to \Bbb R$, and $x_{0}\in X$ is a limit point. Then: $$\lim\sup_{x\to ...
2
votes
1answer
64 views
Show that $\sum_{1\le i<j \le n}x_ix_j \le 2+\sum_{1\le i<j<k \le n}x_ix_jx_k$
Let $x_1,x_2,...,x_n$ be real numbers in $[0;1]$. Prove that
$$\sum_{1\le i<j \le n}x_ix_j \le 2+\sum_{1\le i<j<k \le n}x_ix_jx_k$$
I thinks it can be solved by induction and function, ...
1
vote
1answer
50 views
Inequalities related to infimum and supremum
Let $f,g: A \rightarrow \mathbb{R}$ be integrable functions on a closed rectangle $A \subset \mathbb{R}^n$.
Let $P$ be a partition of $A$ and $S \in P$ a sub-rectangle. Show that:
$m_S(f+g) \geq ...
2
votes
1answer
45 views
Calculate or bound infimum
Let $a_1, \ldots, a_n \in\mathbb R$ and nonnegative let $b\geq1$ and $c\in [0,1]$.
Calculate or bound from above
$$
\inf \left\{d>0: \sum_{i=1}^n \ln ...
3
votes
0answers
243 views
Trigonometric inequality proof
Can anyone help me in proving that $$\cos\theta > \frac{\left(x^a\cos\theta-(x-1\right)^a\cos\frac{\ln x\theta}{\ln(x-1)})\cos(\theta+\gamma)}{\cos\gamma},$$ where $a<1$, $x\in \mathbb{N}$, and ...
1
vote
2answers
137 views
Cauchy-Schwarz Inequality for Integrals for any two functions clarification
I'm trying to work through a homework set, and it states that for any two functions, $f$ and $g$, that the following inequality holds:
$$
\int{fg} \le ||f|| \cdot ||g|| \le \frac{c}{2}||f||^2 + ...
0
votes
3answers
86 views
Proving $\frac{a}{a^2+1}+\frac{b}{b^2+1} ≤ \frac{1}{2}$
How to prove that :
$$\frac{a}{a^2+1}+\frac{b}{b^2+1} ≤ \frac{1}{2}$$
$a,b$ are real positive numbers
4
votes
4answers
94 views
Prove that for every positive integer $n$, $1/1^2+1/2^2+1/3^2+\cdots+1/n^2\le2-1/n$
Base case: n=1. $1/1\le 2-1/1$. So the base case holds.
Let $n=k\ge1$ and assume
$$1/1^2+1/2^2+1/3^2+\cdots+1/k^2\le 2-1/k$$
We want to prove this for $k+1$, i.e.
...
2
votes
1answer
32 views
Regularity and the Varitational Inequality
Let $K = \left\{ v \in H_0^1(\Omega) \, : \, v \geq 0 \right\}$, further suppose $\Omega$ has the regularity property that $||v||_{H^2} \leq C(\Omega)||\Delta v||_{L^2}$, for all $v \in ...
1
vote
0answers
40 views
Inequalities involving regularized incomplete Gamma functions
I am new to the world of the Gamma functions and am wondering if there exist positive functions $f_1(x)>0$ and $g_1(x)>0$, and non-negative functions $f_2(x)\geq0$ and $g_2(x)\geq0$ such that ...
7
votes
1answer
176 views
Prove $ |\vec{a_1}-\vec{b}|+ \cdots +|\vec{a_n}-\vec{b}| > n $
There are $ \vec{a_1},\vec{a_2},\vec{a_3}, \ldots ,\vec{a_n},\vec{b}\; $ such that $ |\vec{a_i}|>1 $, $ |\vec{b}|<1 $, $ \vec{a_1}+\cdots+\vec{a_n}=0 $
.Prove : $ |\vec{a_1}-\vec{b}|+\cdots+ ...
2
votes
1answer
38 views
Need help showing the supremum of a function exists.
I was wondering if anyone knows a technique for proving that this function has a supremum less than infinity for $x \in \mathbb{R}$ ,$x \in [-1,1]$ (I am very certain that it does).
The function is, ...
1
vote
0answers
50 views
Quadratic variation process of $G$–Brownian motion
I would like to prove the inequality
$$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$
where $\langle B ...
0
votes
2answers
44 views
Markov's inequality question about random variable.
How to do this problem, I am really confused. Also, what is the definition of Markov's inequality?
1
vote
1answer
57 views
estimation of a moment for the sum with Bernoulli random variables
Let $x\in R_+^n$ and let $b_i, i=1, \ldots, n$ be $(0,1)$ Bernoulli random variables with $P(b_i=1)=p$. Denote $S=\sum_{i=1}^n x_ib_i$. For $q\geq 2$ estimate from above
$$
E\left|S\right|^q
$$
8
votes
0answers
215 views
Prove $\frac{1}{2a+2bc+1} + \frac{1}{2b+2ca+1} + \frac{1}{2c+2ab+1} \ge 1$
If $a,b$ and $c \ge 0$ and $ab + bc + ca = 1$, prove that the following inequality holds:
$$\frac{1}{2a+2bc+1} + \frac{1}{2b+2ca+1} + \frac{1}{2c+2ab+1} \ge 1$$
I've tried two aproaches, but it ...
0
votes
1answer
19 views
Range of values for optimization?
Example 1: A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 meters of framing materials what must the dimensions of the window be to let in the ...
3
votes
4answers
58 views
Hyperbolic cosine
I have an A level exam question I'm not too sure how to approach:
a) Show $1+\frac{1}{2}x^2>x, \forall x \in \mathbb{R}$
b) Deduce $ \cosh x > x$
c) Find the point P such that it lies on ...
0
votes
0answers
28 views
Condition on inequality of linear combination of powers of positive numbers
Given positive constants $a_1$, $a_2$, .. $a_m$; $b_1$, $b_2$, .. $b_n$, and $0 \leq \lambda_1$, $\lambda_2$, .. $\lambda_m < 1$ and $0 \leq \mu_1$, $\mu_2$, .. $\mu_n <1$, I need to find a ...
0
votes
1answer
43 views
Show that $|\sin(z)|≥1$ at all points on the square with vertices $±(N+1/2)π±(N+1/2)πi $, for any positive integer $ N $.
Show that $|\sin(z)|≥1$ at all points on the square with vertices $±(N+1/2)π±(N+1/2)πi$, for any positive integer $N$.
One of the confusing things is 'at all points on the square with ...'.
I tried ...
