Articles of inequality

inequalities with fraction problem x

$\frac{1}{x} > \frac{2x} {x^2 +2}$ solving this inequalities: My long solution (wrong) : multiplying $(x^2 + 2)^2 (x)^2 \dots$ (multiplying square of each denominator, getting rid of the > or < 0) $x (x^2 +2)^2 > 2x(x^2) ( x^2 +2)$ $x(x^4+4x^2 +4)>2x^2(x^2+2)$ $x^5+4x^3+4x>2x^4+4x^2$ $x^5+4x^3+4x-2x^4-4x^2>0$ $x ( x^4 + 4x^2 + 4 +2x^3 -4x) >0 $ […]

Is $\frac{\lfloor{x}\rfloor+1}{2} \le \lfloor\frac{x}{2}\rfloor + 1$

The answer seems to be yes. Here’s my reasoning: For $x < 1$, $\frac{1}{2} < 1$ For $1 < x < 2$, $1 = 1$ For $2 \mid x$, $\frac{x}{2} + \frac{1}{2} < \frac{x}{2} + 1$ For $2 \mid x-1$, $\frac{x+1}{2} = \frac{x-1}{2} + 1$ For all other values, the value is equal to one […]

Small inequality on unit open disc

For $|u|,|z|<1$, $u,z$ complex numbers, how to show the inequality: $|\frac{u-z}{1-\bar uz}|<1$?

A contradiction or a wrong calculation? $3\lt\lim_{n\to\infty}log_n(p)\lt3$, $\forall n$ (sufficiently large) where $n^3\lt p\in\Bbb P\lt(n+1)^3$?

This is probably a very trivial observation but at the same time somehow interesting to me. For every sufficiently large $n$, it has been already proved that: $$\exists p \in \Bbb P: n^3 \lt p \lt (n+1)^3$$ Then applying $log_n$ two all the terms: $$log_n n^3 \lt log_np \lt log_n(n+1)^3$$ By the properties of logarithms: […]

Solving inequalities comparing $f(x)$ to $0$ where $f$ is an elementary function

Any inequality comparing elementary functions can be rearranged to compare some elementary function $f$ to $0$. What is the best way to approach, in general, solving such inequalities at the precalculus level?

What are the limitations of non-metric distances?

If the triangle inequality does not hold for a distance function (i.e. it is not metric), will this limit its application in some area?

To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$

To prove there exist $M>0$ and $a_0>0$ such that for $|z|>a_0$, $$\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$$ where $p_n$ and $q_m$ are the polynomials of degree $n$ and $m$ respectively with $n<m$. I encountered this question in this post. According to the hint, it is enough to show that for large enough $R$ and for every $|z|>R$, we have […]

How to go upon proving $\frac{x+y}2 \ge \sqrt{xy}$?

This question already has an answer here: Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$ 5 answers How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$? [duplicate] 3 answers

notion of the minimum of a function over a polytope

Let $P$ be a polytope with $M$ vertices. (The polytope $P$ is the intersection of the hypercube $0≤x _j ≤1$ with the hyperplane $\sum_{j=1}^nx_j=t$, $0\leq t\leq n$). Suppose that the volume of $P$ is $Vol(P)=A$. We subdivide $P$ into $M$ pieces each of the volume $A/M$. Let $f$ be a function, such that the integral […]

On a criterion for almost perfect numbers using the abundancy index

Let $\sigma(x)$ denote the sum of the divisors of $x$, and let $I(x) = \sigma(x)/x$ be the abundancy index of $x$. A number $y$ is said to be almost perfect if $\sigma(y) = 2y – 1$. In a preprint titled A Criterion For Almost Perfect Numbers Using The Abundancy Index, Dagal and Dris show that […]