Questions on proving and manipulating inequalities.
5
votes
2answers
35 views
Showing equality in Cauchy-Schwarz inequality
With $\mathbf{u,v}$ being vectors in $\mathbb{R}^n$ euclidean space, the Cauchy–Schwarz inequality is
$$
(\sum_{i=1}^{n} u_i v_i)^2 \leq (\sum_{i=1}^{n} u_i^2)(\sum_{i=1}^{n} v_i^2)
$$
further given ...
2
votes
2answers
47 views
How do I prove the arithmetic-geometric mean inequality?
I am following along with this bare-bones proof of the arithmetic-geometric mean inequality with two real numbers. I'm having difficulty understanding the logic behind this step:
$$
...
5
votes
4answers
93 views
Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)
If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$.
As a first step to prove this ...
1
vote
0answers
18 views
Estimation of a scalar product
I encountered the following, which shouldn't be that hard, but I can't get my head around it.
The problem is the following estimate (part of a bigger equation, but here's just the difficult part):
...
15
votes
1answer
182 views
$x^3-3x-3=0$, prove that $10^x<127$
$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$
I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
1
vote
1answer
51 views
Jensen's inequality and $L^p$ norms
Let $(X,\Sigma,\mu)$ be a probability space; in particular, $\mu(X)=1$. The integral form of Jensen's inequality can be phrased in terms of permuting a convex function $\varphi$ (say, with the ...
0
votes
3answers
31 views
Let $a,b \in \Re$. If 0 < $\epsilon$ < min{|a|, |b|}. Show this inequality
Let $a,b \in \Re$. If 0 < $\epsilon$ < min{|a|, |b|}.
$ {\frac{|a+\epsilon|}{|b+\epsilon|}} \leq {\frac{|a|+\epsilon}{|b|-\epsilon}}$
I tried to use triangular inequality.
But have no idea of ...
1
vote
2answers
28 views
Do inequalities hold under square-root (or exponentiation in general)?
This has been bothering me lately. My proof-skills are rusty (and were never great to begin with). I dimly recall having seen this (or something related to it) in a math course I took a while ago, but ...
0
votes
2answers
42 views
I need help to prove this inequality because I am having problems
I need help solving the following:
$$
\left(\frac{x}{y}\right)^{n+1}<\left(\frac{x}{y}\right)^n, n \geq 1 \quad \mathrm{and} \quad 0<x<y
$$
2
votes
3answers
42 views
How come these two inequalities have different solution sets?
The following inequality has a solution set of $\{x \in \mathbb{R} | \frac{3}{2} < x < 5, x<\frac{-1}{4}\}$:
$$\frac{2x + 1}{2x - 3}>\frac{x + 1}{x - 5}$$
However this inequality has a ...
7
votes
2answers
84 views
Help with an inequality problem
I came to ask this because I am really stuck at this problem. I have tried everything from arithmetic mean, geometric mean and harmonic mean. Also, I have tried playing with the variables and such, ...
2
votes
3answers
29 views
Is it true that if $a>b$ where $a,b\in[0,\infty)$ then $a^x>b^x$ or $a^x<b^x$ for any $x\in\mathbb R-\mathbb Q$ according as $x>0$ or $x<0?$
I know the result that if $a>b$ where $a,b\in[0,\infty)$ then $a^x>b^x$ or $a^x<b^x$ for any $x\in\mathbb Q-\{0\}$ according as $x>0$ or $x<0.$ Does the same hold for irrational number? ...
1
vote
0answers
51 views
a simple doubt about inequality in graph theory
I am working on a problem. Somewhere I am stuck.
I have a formula where 2$e$ $=$ $\sum$ $deg$ $v$, over $n$ vertices, where $v$ $\in$ $V(G)$, where $G$ is a graph. I want to find a polynomial (or ...
2
votes
1answer
16 views
Transforming inequalities over the real numbers
Given integers a and b and the relation a <= b, intuitively I feel I can transform this inequality into a strict inequality like this:
a < b + 1
Conversely, I should be able to transform the ...
1
vote
1answer
58 views
Average of divisors of n.
Let n be a natural number and let $f(n)=\frac{\sigma(n)}{d(n)}$ be the arithmetical average of n's divisors. Either prove or give a counterexample that for all natural numbers like n, which are not ...
