Questions on proving and manipulating inequalities.

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2
votes
1answer
18 views

If $q^k n^2$ is an odd perfect number with Euler factor $q^k$, can $q = 73$ hold?

Call a number $N$ perfect if $\sigma(N)=2N$ where $\sigma$ is the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number, can $q = 73$ hold? Here is my attempt: Since ...
0
votes
0answers
11 views

Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions?

I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
0
votes
0answers
29 views

Proof: $|||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| $?

I am looking for a proof of the following: \begin{equation*} |||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| \end{equation*} Where A, B are positive, hermitian matrices, and $|||⋅|||$ ...
0
votes
3answers
65 views

Prove inequality $ab+bc+ca\ge 3,\ abc=1$

How can I prove \begin{equation*} ab+bc+ca\ge 3,~a,b,c \in\mathbb{R},~ a,b,c>0\ \end{equation*} and the product $abc=1$? I obtained only $(a+b+c)^2-(a^2+b^2+c^2)\ge6\ and \ ...
0
votes
0answers
19 views

Poincarè inequality in probability

I'm looking for a proof of the poincarré inequality in a probabilitic setting. That is to say, let $\mu$ be a probability on $\Bbb R^n$, what are the hypothesis in order to have, for any f smooth ...
1
vote
0answers
18 views

First moment inequality and time-average limits

Suppose $\{A(t)\}_{t \geq 0}$ and $\{B(t)\}_{t \geq 0}$ are two non-negative stochastic processes such that $$ \frac{1}{T} \int_{s=0}^T A(s) \, {\rm d} s \stackrel{\text{a.s.}}{\rightarrow} a \in ...
-2
votes
4answers
78 views

Prove the inequality $a^2 + b^2 +c^2 \ge ab +bc +ac$ [on hold]

How do I prove the inequality \begin{equation*} a^2 + b^2 +c^2 \ge ab +bc +ac \end{equation*} where $ a,b,c\in\mathbb{R} $ and $a,b,c>0$?
1
vote
0answers
13 views

Maximum density linear combination chi squares

I have a positive linear combination of chi square variables \begin{equation*} X=\sum_{i=1}^k \lambda_i \chi^2(r_i), \end{equation*} the degrees of freedom satisfy $r_i>1$. I need an upperbound ...
-4
votes
2answers
25 views

help with alternating series test [on hold]

How do I show that $$\large{\frac{\ln(n)}{n} \geq \frac{\ln(n+1)}{(n+1)}}$$ for $\large{n \geq 1}$?
0
votes
0answers
33 views

Inequality on complex polynomial

For every $a\geq 0$, let $p_a(z)=1-z+az^3$. What is the maximal value of $a$ such that $$ p_a(|z|)\leq |p_a(z)| $$ for all $z\in \mathbb C$? EDIT: I claim that $a=\frac{4}{27}$ is the maximal value. ...
3
votes
6answers
97 views

Visualize $z+\frac{1}{z} \ge 2$

As we know, always $$z+\frac{1}{z} \ge 2,~~~~~~~~~ z\in \mathbb{R}^+$$ However, is there any geometric way to visualize this equation for some one who is not that expert in math? I know this ...
0
votes
1answer
33 views

Prove using mathematical induction that $n^2 > n+1$ for all $n \ge 2$

I have proved for the initial case $P(2)$ that this is true, but I'm stuck at substituting in $n=k+1$, $(k+1)^2 > (k+1)+1$ = $k^2 + 2k + 1 > k+2$, where do I go from here or have I made a ...
-1
votes
2answers
29 views

How many books can fit in a box? [on hold]

Each book is 450kg An empty box is 200kg The total mass of the book(s) and box cannot exceed 6,500kg. How many books can fit in a box?
0
votes
1answer
24 views

Semilinear Poisson PDE - proving a (hopefully) simple inequality

This is from page 557 of PDE Evans, 2nd edition. My question is at the bottom of this post, but for now, here is some context for my question: LEMMA 2 (Boundary estimates). Let $u \in ...
1
vote
1answer
19 views

Two exercises about hypermetric spaces

Take $S$ to be the collection of all subsets of $\{1,\dots,n\}$. If $x, y$ are in $S$, define $d(x,y)$ as the number of elements of the symmetric difference $x\triangle y$. Exercise 2.1. Show ...
-1
votes
1answer
27 views

Doubt with Intervals and Inequalities

This doubt has been bothering me for ages. I would be truly grateful for any help. Problem 1: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals Solution: $x\in(2,4)\cup(4,6)$ ...
0
votes
3answers
38 views

Show that $\frac{ n^{1/3} }{n-1} > \frac{ (n+1)^{1/3} }{n}$

I am trying to demonstrate that: $$\frac{ n^{1/3} }{n-1} > \frac{ (n+1)^{1/3} }{n}$$ for $n>0$ I am really struggling. I can get to the point $-2n^3+2n-1>0$ but I am really unsure of how to ...
0
votes
1answer
13 views

System of inequalities any real numbers

Here is a system of inequalities I've been trying to solve, and nothing so far; $y>x,$ $1>y,$ $1>x,$ $x>y/2, x>(2-y)/3, y>0, x>0, 1>x+y$
2
votes
3answers
27 views

Absolute Value Inequality Problem

Problem: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals My attempt using the Definition of Modulus: $$\dfrac{2-|x-4|}{|x-4|}>0$$ $$CASE A:x-4\ge 0\Rightarrow x\ge4\Rightarrow ...
-3
votes
2answers
39 views

How should I go about this proof? [on hold]

Let $a,b > 0$ be real numbers. Prove that $2ab \leq (a+b)\sqrt{ab}$. I'm new to proofs and would like some help understanding how to approach this proof. Thank You.
0
votes
1answer
20 views
+50

Upperbound on the following logarithmic function with matrix

I am trying to find an upperbound the expression below with a function $f$ that is a function of the identity matrix $$\log(1+\mathbf{h}^* \mathbf{\Sigma} \mathbf{h}) \leq f( {\bf I},{\bf h })$$ ...
9
votes
2answers
145 views

Proving $\sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2}$

I've been going through some of my notes when I found the following inequality for $a,b,c>0$ and $abc=1$: $$ \begin{equation*} \sqrt{2}(a+b+c) \geq \sqrt{1+a^2} + \sqrt{1+b^2} + \sqrt{1+c^2} ...
0
votes
1answer
16 views

How to prove the inequality on relative entropy?

Here is the definition of Relative Entropy Now I am only interested in the simplest condition that the index set is finite and discrete, as the naive probability distribution vectors. Now if the ...
1
vote
0answers
13 views

solving non-linear systems inequalities

I am trying to solve a non-linear systems of 14 inequalities with 12 variables, involving exponential and polynomial functions. I have been searching over the web for leads,but without any success.I ...
3
votes
0answers
45 views

Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$ [on hold]

Find all (real) numbers $a $ and $b$ such that $|a| + |b| \ge 2/\sqrt{3} $ and for any $x$ the inequality $|a\sin x + b \sin 2x | \le 1$ holds. In other words, find the set $Q$ defined as ...
5
votes
1answer
87 views

Seeking a More Elegant Proof to an Expectation Inequality

Let $X$ and $Y$ be i.i.d. random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|].$$ This question is a re-posting of An expectation inequality. I can ...
1
vote
2answers
32 views

Inequality $\left|z+w\right|\geq\left||z|-|w|\right|$ when $|w|\leq A|z|$.

Let $z,w\in\mathbb C$ and $|w|\leq A|z|$ for $A>0$. I want estimate from below $|z+w|$. I proceeded as follows. Since $$\left|z+w\right|\geq\left||z|-|w|\right|$$ and $-|w|\geq -A|z|$, I write ...
-3
votes
1answer
37 views

The integer part of $x+1$ is the integer part of $x$ plus $1$ [on hold]

How do you solve the proof: If $x$ is a real number, then: $[x+1] = [x] + 1$. For my proof, I tried to describe the interior of the argument inside the parentheses, but I was unsuccessful. Please ...
2
votes
0answers
19 views

$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$

Let $c_i\in\mathbb R$, $a_i\geq0$ with $\sum_{i=1}^n a_i=1$, prove $$\sum_{i=1}^n\sum_{j=1}^n|c_i+c_j|a_ia_j\geq\sum_{i=1}^n\sum_{j=1}^n|c_i-c_j|a_ia_j$$ This inequality comes from there, when $X$ is ...
1
vote
2answers
36 views

Solve for real value of $x$: $|x^2 -2x -3| > |x^2 +7x -13|$

Here I have a question: Solve for real value of $x$: $$|x^2 -2x -3| > |x^2 +7x -13|$$ I got the answer as $x = (-\infty, \frac{1}{4}(-5-3\sqrt{17}))$ and ...
2
votes
5answers
107 views

Determine whether $f(x)$ is increasing or decreasing

Let $f(x) = -x + (x^3/3!) + \sin(x)$ How do I determine if $f(x)$ is increasing or decreasing? I have already found the derivative of this function which is: $f'(x) = -1 + (x^2/2) + \cos(x)$ And I ...
0
votes
1answer
34 views

About inequality $\sum_{k=1}^n |a_k|^2 \lessgtr \sum_{k\neq s} |a_k| |a_s|$

Let $a_k$ a sequence of complex number. We have $$\left(\sum_{k=1}^n |a_k|\right)^2 \geq \sum_{k=1}^n |a_k|^2$$ It is a basic fact because $$\left(\sum_{k=1}^n |a_k|\right)^2 = \sum_{k=1}^n |a_k|^2 + ...
2
votes
1answer
33 views

Find an example that the following equality doesn't apply

I need a sequence $(f_n)_{n\in\mathbb{N}}, f_n:X\to\mathbb{R}^\star$ which is measurable, $f_n \ngeq 0$ and doens't accept the following equality: $$\int_X\sum_{n=1}^\infty f_n \, d\mu = ...
5
votes
2answers
73 views

Proving that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$ using derivatives

Let $a,b,c\in\mathbb{R}^+$ and $abc=1$. Prove that $$\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge\frac32$$ This isn't hard problem. I have already solved it in following way: Let ...
1
vote
2answers
41 views

prove using Lagrange multipliers that for $x,y>0,\space n\in \mathbb N,\space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2 $

I have been asked to prove using Lagrange multipliers that for \begin{equation*} \space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2,~x,y>0,~n\in \mathbb {N} \end{equation*} I am familiar with the ...
1
vote
1answer
49 views

Struggling to prove inequality

I've been given to following inequality to prove: (The hint given was not to evaluate the integral) \begin{equation*} \frac{1}{4} \leq \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{sin(x)}{x}dx\leq ...
0
votes
1answer
45 views
+50

Can the following inequality be directly infered?

If we have a condition as follows $$\log(1+\mathbf{h}_2^* \mathbf{\Sigma} \mathbf{h}_2) \leq \log(1+\mathbf{h}_1^* \mathbf{\Sigma} \mathbf{h}_1)$$ where $\Sigma$ is positive semi definite matrix ...
0
votes
3answers
36 views

Proving inequality involving real numbers [on hold]

$x, y$ are real positive numbers. Let $m$ be the smallest number among $x, y + \frac{1}{x}, \frac{1}{y}$. How to prove that $m \le \sqrt{2}$? I really don't know how to start.
0
votes
0answers
20 views

A smart way to bound this function and get rid of covariance matrix

I have the following function which I am trying to bound as follows $$A({\bf h},\Sigma)= \log(1+ {\bf h}^* \Sigma {\bf h} )$$ $$\Sigma = \begin{bmatrix} 1-|\rho_1|^2 & \rho_3 - \rho_1 \rho_2^* ...
1
vote
0answers
21 views

Completing Karlin's proof of variation diminishing transformation theorem

In S Karlin's book total positivity there's a theorem that says if $K(x,y)$ is $TP_r$ (totally positive with degree $r$) and the sign change count of function $h$, $S(h) = n\leq r-1$, then ...
0
votes
1answer
25 views

How to show this is decreasing

I'd like to show $$\sum_{i=1}^n \frac{1}{i((n+1)-i)} $$ is decreasing for n>1, which is Cauchy product of $$\sum_{i=1}^n \frac{1}{i}$$ Numerical computation until n=50 shows it's decreasing but I ...
-1
votes
0answers
32 views

How to solve this inequality using AM-GM? [duplicate]

Let $a,b,c>0$ and $a+b+c=1$. Prove $$\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ca}{b+ca}}\le\frac32$$
0
votes
1answer
30 views

Chebiyshev Inequality

In proving the Chebyshev inequality in Probability theory an important step is to observe that: $P((|x-E(x)|≥a))=P(|x-E(x)|^2≥a^2)$. It is assumed that X has a moment of order 2. Can somebody help ...
1
vote
1answer
46 views

Proving $(n+1)c^{1/(n+1)} - nc^{1/n} \le 1$ from first principles

Is it possible to prove that \begin{align*} (n+1)c^{1/(n+1)} - nc^{1/n} \le 1 \qquad c \in \mathbb{R}_+, n \in \mathbb{N} \end{align*} using only elementary techniques? (No calculus, no appeasement to ...
1
vote
1answer
26 views

Upperbound a logarithmic expression that has a covariance matrix

Let $\Sigma$ be a $2\times 2$ covariance matrix and ${\bf h}$ a vector of complex values entries. $$A= \log(1+ {\bf h}^* \Sigma {\bf h} )$$ $$\Sigma = \begin{bmatrix} 1-|\rho_1|^2 & \rho_3 - ...
0
votes
3answers
48 views

The inequality $k(n-1)<n^2-2n$ for all odd $n$ and $k<n$

How one can prove the following statement: $k(n-1)<n^2-2n$ for all odd $n$ and $k<n$ Tried so far: induction on $n$, graphing, and rewriting $n^2−2n$ as $(n−1)^2−1$.
0
votes
1answer
32 views

Hilbert's inequality for $\left|\sum_{n,m}a_n \bar a_m\right|$.

We know that, an Hilbert's inequality states $$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$ Give $a_n, b_n$ two sequences of complex numbers. Then write an inequality ...
1
vote
1answer
53 views

Convex function inequality for Euclidean norm: $\|(f(x_1),\cdots,f(x_n))\|_2\leq f(\|x\|_2)$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a positive, convex, continuous function such that $f(0)=0$. (If you wish you can also suppose $f$ to be monotone increasing.) I would like to prove or to ...
0
votes
0answers
22 views

Determine the order of $\exp Q(z)$ when $Q$ is a polynomial of degree $q$.

I am looking to determine the order of $f(z) = \exp Q(z)$ when $Q$ is a polynomial of degree $q$. I think the order is $q$, but I am struggling to prove it. The definition of order is: An entire ...
1
vote
1answer
46 views

Inequalities using AM-GM

Use the AM-GM inequality to prove $(5xy + 6y)^3$ ≥ $1215xy^3$ for all real numbers x, y > 0. Not sure if I was on the right track but so for my understanding is: since there is a power of 3 on the ...