Questions on proving, manipulating and applying inequalities.

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0
votes
2answers
35 views

Clarification on inductive proof of Bernoulli's inequality

Prove that if $h > -1$, then $1 + nh ≤ (1+h^n)$ for all nonnegative integers $n$. I've read several solutions and I'm still totally lost on how to go about this. I have the inductive hypothesis:...
0
votes
1answer
28 views

if $\sqrt{\log_4(\log_3(\log_2(x^2-2x+a)))}$ is defined for all x={set of real numbers} then find the valid interval for “a”.

if $\sqrt{\log_4(\log_3(\log_2(x^2-2x+a)))}$ the question asks to find the interval in which the valid values for "a" lies I tried by defining the ${\log_4(\log_3(\log_2(x^2-2x+a)))}$$\ge$0
2
votes
0answers
24 views

On some iterated inequalities and $x \geq 5$

Let $x \in \mathbb{N}$. Suppose that I have a function $f:\mathbb{N}\rightarrow\mathbb{Q}$, with initial bounds $$2 - \frac{2}{x_0} < f(x_0) = \frac{2{x_0}}{x_0 + 1} \leq 2 - \frac{5}{3x_0}.$$ ...
2
votes
1answer
57 views

How to prove that if $E[X^2]$ is finite then $n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0$?

Let $X$ be a random variable with $E[X^2]<\infty$. I want to prove that $$ n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0 \text. $$ I tried to apply Chebyshev's inequality, ...
0
votes
1answer
66 views

On the inequality $|z_1-z_2|^2 \lt (1+c)|z_1|^2+(1+\frac{1}{c})|z_2|^2$

Now, I know this question has been asked here but my question doesn't deal with finding a solution, my question deals with checking the validity of the question. Question:- If $z_1, z_2$ are ...
1
vote
1answer
34 views

Existence of a functional inequality

Does there exist $f=f(x)$ satisfying $f(x)\ge0$ for $x\in\mathbb{R}$, $f(x)=f(-x)$ for $x\in\mathbb{R}$ (i.e. $f$ is even), $\int_{\mathbb{R}} f(x)\,dx<\infty$, and $\int_{\mathbb{R}} x^2\,f(x)\,dx&...
4
votes
1answer
57 views

Proof of logarithm inequality without continuity

Showing that the logarithm function is continuous in its domain boiled down to proving $$\frac{x}{1+x}\le \ln(1+x)\le x \ \ \text{for all}\ x >-1.$$ There are quite a few proofs already online. ...
3
votes
1answer
32 views

How to solve the inequality $\log_2(4^x-2(2^x)+17)>5$?

Find the number of positive integers not satisfying the inequality $$\log_2(4^x-2(2^x)+17)>5$$ My approach: let $2^x=t$ then inequality is rewritten in form $$\log_2(t^2-2t+17)>5$$ then I ...
-3
votes
2answers
41 views

Determine whether $x^3 > 2^{x/2}$ [closed]

Determine whether there exists $c\in\mathbb{R}$ such that $x^3 > 2^{x/2}$ for $x \in [c, \infty)$.
3
votes
0answers
47 views

Linear separability / Number of positive solutions of a random linear system

This one is on linear separability of cyclic patterns. The shorter geometric version: Take a ring of length $p$ of randomly assigned mean-free binary values $x_i = \pm 1$, $i = 1 \cdots p$. ...
4
votes
0answers
75 views

Is the error I noticed a harmless typo?

Here http://arxiv.org/PS_cache/arxiv/pdf/1002/1002.0442v1.pdf , at page $2$ at the bottom, it is stated that the number of primes not exceeding $x$, denoted by $\pi(x)$, satisfies the double-...
2
votes
0answers
19 views

Upper estimate of function such that $\int |x|^2 f(x) dx<\infty$

Let $f$ be a non-negative function on $\mathbb R^d$ satisfying the following: (1) There exists a non-increasing function $g$ on $(0,\infty)$ such that (1-i) $ C_1^{-1} g(|x|) \le f(x) \le C_1 g(|x|)...
-6
votes
0answers
26 views

Proof for $|z_1+z_2| \le |z_1|+|z_2|$ and $|z_1-z_2|\ge |z_1|-|z_2|$ [closed]

I need proof for $$|z_1+z_2| \le |z_1|+|z_2|$$ and $$|z_1-z_2|\ge |z_1|-|z_2|$$
4
votes
1answer
56 views

Solving $\frac{1}{2} < \cos \theta < \frac{\sqrt{3}}{2}$

Find the values of theta which satisfy the given condition on a unit circle $$\frac{1}{2} < \cos \theta < \frac{\sqrt{3}}{2}$$ I'm able to plot the points and answer according to me ...
1
vote
1answer
32 views

is the function $\rho$ a pseudometric?

Let $\Im=\left\{\Im_{n}\right\}_{n\in \mathbb{N}}$ be a sequence of open covers of a topological space $(X,\tau)$. We define a function $\delta:X\times X \rightarrow \mathbb{R}$ as follows If $(x,y)\...
0
votes
2answers
28 views

How is $\left| \exp(iaRe^{i\theta}) \right|\le e^{-aR\sin\theta}$?

In one book on complex variables, in the proof of Jordan's Lemma, For any constant $a>0$, and any radius $R>0$, it is stated that $\left| \exp(iaRe^{i\theta}) \right|\le e^{-aR\sin\theta}$. I ...
2
votes
1answer
66 views

Inequality on a sequence of $n$ reals whose sum is $0$

Consider $n\geq3$ real numbers $a_1,a_2,\dots ,a_n$ satisfying $a_1+a_2+\cdots+a_n=0$ and $$2a_k \leq a_{k-1}+a_{k+1}$$ for all $2\leq k\leq n-1$. Prove that $$|a_k|\leq\frac{n+1}{n-1}\,\max\big\{|a_{...
1
vote
0answers
32 views

Checking feasibility of a system of inequalities with scipy

I have a set of pairwise constraints, like this: a > b, b > c, c > a and need to check if they are satisfiable (in the example above, they are not). ...
0
votes
3answers
44 views

Minimum value of algebraic expression.

If $0\leq x_{i}\leq 1\;\forall i\in \left\{1,2,3,4,5,6,7,8,9,10\right\},$ and $\displaystyle \sum^{10}_{i=1} x^2_{i}=9$ Then $\max$ and $\min$ value of $\displaystyle \sum^{10}_{i=1} x_{i}$ $\...
3
votes
8answers
89 views

How to prove the inequalities between $20^{70^2},30^{60^2},40^{50^2}$

Let $$M=\{ 20^{70^2}, 30^{60^2},40^{50^2}\}$$. What number is the greatest and which is the smallest? I thought about beginning by assuming certain inequalities and trying to prove them, for example: ...
1
vote
0answers
18 views

Inclusion about solution of inequality

Lets consider the following inequality form this question: $$|y^2-y-2|\geq 4 + |y^2+y-2|+|y+4|+|y|$$ User Did said that: $$ |y^2-y-2|=|4+(y^2+y-2)-(y+4)-y|.$$ Using the triangular inequality ...
1
vote
10answers
117 views

A clean proof of $x^2 \geq x$, for any integer $x$

I am trying to prove that $x^2 \geq x$ for any integer $x$. Since we know that for any number $n$, $n^2 \geq 0$ we conclude that if $x \leq 0$ the proposition will hold. Next we must prove that the ...
0
votes
2answers
84 views

Is there any clever way to solve inequalities like $|y^2-y-2|\geq 4 + |y^2+y-2|+|y+4|+|y|$?

I have to solve different types of inequalites of this type: $$|y^2-y-2|\geq 4 + |y^2+y-2|+|y+4|+|y|$$ I know the standard method for solving these inequalities, by finding the all zeros of the ...
1
vote
2answers
33 views

How to solve this inequality problem?

Given that $a^2 + b^2 = 1$, $c^2 + d^2 = 1$, $p^2 + q^2 = 1$, where $a$, $b$, $c$, $d$, $p$, $q$ are all real numbers, prove that $ab + cd + pq\le \frac{3}{2}$.
0
votes
0answers
18 views

Why are generalized inequalities defined over proper cone?

Why generalized inequality is defined over a proper cone? What property does not hold if we define it over non-convex cone? Same with `pointed'. For example, generalized inequality makes sense in a ...
0
votes
1answer
58 views

Brute force way to show that $\rho(x,y) = \min\{1, d(x,y)\}$ is a metric

Following a hint in Short proof that $\rho^\prime(x,y) = \min\{1,\rho(x,y)\}$ is a metric I would like to use the brute force method to show that the standard bounded metric is a metric $$\rho(x,y) ...
2
votes
2answers
69 views

Follow up to a question, why does proof $\rho(x,y) = \dfrac{d(x,y)}{1+d(x,y)}$ work

This is a follow up to a well known question Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric A general proof is as follows: Let $x,y,z \in (X, \rho)$ \begin{align} \rho(x,z) &= \dfrac{...
0
votes
0answers
28 views

Combining a working hypothesis for odd perfect numbers with an inequality for logarithms

Euler's theorem for odd perfect numbers states that if there exists and odd perfect number, that is an odd positive integer $n$ satisfying $\sigma(n)=2n$, where $\sigma(m)=\sum_{d\mid m}d$ denotes the ...
2
votes
1answer
70 views

Why must $|z|\gt 1$ be the necessary condition

Question:- If $\left|z+\dfrac{1}{z} \right|=a$ where $z$ is a complex number and $a\gt 0$, find the greatest value of $|z|$. My solution:- From triangle inequality we have $$|z|-\left|\dfrac{1}{...
-4
votes
0answers
27 views

Why do we set both factors x>0 when solving an inequality?

Here's a page from my math textbook: (http://s31.postimg.org/u2ow52xgb/Whats_App_Image_20160715.jpg) (unfortunately I can't post pictures due to low reputation) Why do we consider both factors as ...
0
votes
1answer
33 views

Largest positive integer $n$ that satisfies a given inequality [duplicate]

Largest positive integer value of $n$ in $$n \cdot \left(\frac{abc}{a+b+c}\right)\leq (a+b)^2+(a+b+4c)^2$$ where $a, b, c \in \mathbb{R^{+}}$. $\bf{My\; Try::}$ We can write $(a+b)^2+(a+b+4c)...
2
votes
4answers
126 views

What is the minimum value for $(\frac{1}{a}-1)(\frac{1}{b}-1)(\frac{1}{c}-1)$ if $a+b+c=1$ and $a,b,c\in\mathbb{R}^+$?

The primary question was: What is the minimum value for $(1-\frac{1}{a})(1-\frac{1}{b})(1-\frac{1}{c})$ if $a+b+c=1$ and $a,b,c\in\mathbb{R}^+$? $\color{red}{\text{But sorry guys! I messed it up! my ...
1
vote
1answer
75 views

Square root inequality revisited

This is a follow-up question of this one: Proof of the square root inequality I am interested in the following generalizations of the square root inequality. Let $\varepsilon,\delta>0.$ Then $$\...
1
vote
1answer
22 views

Query about an algebraic inequality involving $(a-b)^p$

I want to know if there exists any inequality of the type $(a-b)^p \geq C(a^p -b^p)$ or $(a-b)^p \leq C(a^p -b^p),$ where $a>0,\, b>0,$ $C>0$ is a constant and $0<p<1.$ I am aware of ...
4
votes
1answer
95 views

An inequality involving two complex numbers

Let $z_1, z_2 \in \mathbb C$ and $a,b \in \mathbb{R} \setminus \{0\}$. Prove that $$|z_1|^2+|z_2|^2-|z_1^2+z_2^2|\le 2\dfrac{|az_1+bz_2|^2}{a^2+b^2}\le |z_1|^2+|z_2|^2+|z_1^2+z_2^2|$$ ...
3
votes
2answers
63 views

Show that $x^2 \sin{x} + x \cos{x} + x^2 + \frac{1}{2} > 0$

Show that for any real number $x$: $$x^2 \sin{x} + x \cos{x} + x^2 + \frac{1}{2} > 0.$$ $\bf{My\; Try::}$ Using $a\sin x+b\cos x\geq -\sqrt{a^2+b^2}$ So $$x^2\sin x+x\cos x\geq -\sqrt{x^4+x^2}...
7
votes
6answers
1k views

Proof of the square root inequality [duplicate]

I stumbled on the following inequality: For all $n\geq 1,$ $$2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}.$$ However I cannot find the proof of this anywhere. Any ideas how to ...
0
votes
2answers
36 views

Probability in a inequality

I've been practicing probability for a while and now I encountered a math problem which I don't know how to approach. One integer is selected randomly from the set $[1,50]$ . What is the ...
-2
votes
1answer
38 views

Find $x$ where $|(x-3)^2-1|\geq |(x+2)^3+5|$ [closed]

Find all real $x$ such that the following is true: $|(x-3)^2-1|\geq |(x+2)^3+5|$
7
votes
2answers
189 views

Is this inequality provable? $e^{\left(\pi^{e^{\pi^{.^{.^{.^{e^\pi}}}}}}\right)}\ge \pi^{\left(e^{\pi^{e^{.^{.^{.^{\pi^e}}}}}}\right)}$

I am interested in proving the following inequalities: $e^\pi\ge\pi^e$, $\quad \pi^{(e^\pi)}\ge e^{(\pi^e)}$, and $\quad e^{(\pi^{(e^\pi)})}\ge \pi^{(e^{(\pi^e)})}.$ How we can prove these ...
-2
votes
0answers
33 views

Hölder inequality application to show that f=1

I want to proof that if $f \in L^{1}_{\mu}(\mathbb{R}), f > 0$ continuous, satisfies $(\int_\mathbb{R} f(x)d\mu)^{3} \le \int_\mathbb{R} f(x)^{3sin^{2}(x)}d\mu * (\int_\mathbb{R}f(x)^{\frac 32cos^{...
5
votes
1answer
100 views

$\lfloor x\rfloor \cdot \lfloor x^2\rfloor = \lfloor x^3\rfloor$ means that $x$ is close to an integer

Suppose $x>30$ is a number satisfying $\lfloor x\rfloor \cdot \lfloor x^2\rfloor = \lfloor x^3\rfloor$. Prove that $\{x\}<\frac{1}{2700}$, where $\{x\}$ is the fractional part of $x$. My ...
1
vote
3answers
49 views

Real values of a function involving the Lambert $W(x)$ function

I have the following function: $$y=-\dfrac{W\left(-\ln(k)\right)}{\ln(k)}$$ where $W(x)$ is the Lambert $W$ function defined as the solution of the equation: $$x=W(x)e^{W(x)}$$ If $k\in\mathbb{R}$ ...
1
vote
1answer
35 views

A simple proof in the form of an inequality

Proof that for all $a, b$ are elements of $\mathbb{R}$ : $(a+b)^2\geq 4ab$. Does it satisfies after doing some simple arithmetic to say that $(a-b)^2\geq 0$? Or do I need to go over all the cases ...
2
votes
0answers
44 views

If $L > 1$ is an odd almost perfect number with $\omega(L)=6$, then $L$ must be divisible by $3$.

Edited July 15 2016 Let $\mathbb{N}$ denote the set of positive integers. Let $\sigma = \sigma_{1}$ denote the (classical) sum-of-divisors function. Let $I(x) = \dfrac{\sigma(x)}{x}$ denote the ...
3
votes
0answers
64 views

Upperbound of the ratio of column sums of an integer matrix

Suppose $X_{n \times n}$ is a positive integer matrix where $n\geq 2$. The element in the $i_{th}$ row and $j_{th}$ column of the matrix $X$ is defined as $x_{i,j}$. Now, consider $S_{j,j+1}=argmax_{...
0
votes
0answers
23 views

system of inequalities in Mathematica

I am trying to solve the following system of inequalities in Mathematica but the output is just the same system of inequalities. I need to get the expressions for x, y in terms of the pi indexed ...
4
votes
1answer
88 views

Bounds on a system of coupled ODEs

Suppose we have a $1$-dimensional differential inequality $$\frac{dx}{dt} \leq x - x^3 $$ We can apply the Comparison principle to claim that if $y(t)$ is the solution to $\frac{dy}{dt} = y - y^3$, ...
1
vote
1answer
43 views

compare $x^{y^{z}}$ and $y^{x^{z}}$ (help)

the problem goes like this : let $(x,y,z) \in \mathbb{R^3}_{+}$ and $0<x<y<z$ compare $ x^{y^z}$ ;$x^{z^y}$; $y^{x^z}$ ; $y^{z^x}$ ; $z^{y^x}$ ; $ z^{x^y}$ how can we compare the values ?
2
votes
0answers
56 views

olympiad-type inequality

Prove that for any $x_1,\dots,x_n>0$ $$ {\root{n}\of{\prod _{k=1}^{n}\ \sum_{t=1}^{k}\ \frac{1}{t^2\cdot\sqrt[t]{x_1\cdot\ldots\cdot x_t}} }} \ \cdot\ \sum _{k=1}^{n}\frac{\sum_{j=1}^{k}\sum_{i=1}^...