Questions on proving and manipulating inequalities.

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2
votes
3answers
32 views

Find the general values of $x$ satisfying the trigonometric equation

Find the general values of $x$ satisfying $$\frac{\tan^2 x \sin^2 x}{1-\sin^2 x \cos2x}+\frac{\cot^2 x \cos^2 x}{1-\cos^2 x \cos2x}+\frac{2\sin^2 x}{\tan^2 x+\cot^2 x}=\frac{3}{2}$$ It seems to me ...
2
votes
3answers
35 views

Intuition behind Chebyshev's inequality

Is there any intuition behind Chebyshev's inequality or is that only pure mathematics? What strikes me is that any random variable (whatever distribution it has) applies to that. $$ ...
0
votes
2answers
53 views

Prove that $\frac{8}{5}\le 2a+b\le 8$

Let $a,b,c,d,e$ be real numbers such that $$\begin{cases} a+b+c+d+e=8\\ a^2+b^2+c^2+d^2+e^2=16 \end{cases}$$ Prove that: $$\dfrac{8}{5}\le 2a+b\le 8$$
1
vote
1answer
29 views

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $ I tried $x^2-|5x-3|-x<2$ , case $1$ , $x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\ $ ...
8
votes
0answers
63 views
0
votes
2answers
29 views

Calculus: simpler way of showing that derivative is negative?

I want to show that $\frac{1-(1-\beta)^N}{\beta}$ is strictly decreasing in $\beta$ for $\beta \in (0,1)$ and $N \geq 2$. My approach so far is as follows: I take the derivative with respect to ...
2
votes
2answers
234 views

Proving the an expression is larger than a simplified quadratic

Let p and q be positive real numbers. Prove that $$ (p + 2)(q+2)(p+q) \ge 16pq $$ Any explanation/answer would be extremely helpful. Thanks : )
-1
votes
1answer
28 views

Two parameters inequality problem

Find $p \gt 0$ for which $f(x) =x\ln p - p\ln x$ is $\ge0$ for $x \gt 0$ They also say $p$ is real number.
0
votes
1answer
22 views

Simple case of Young's inequality

I have a question concerning Young's inequality stated as follows: $||a∗b||_{ℓ_q}≤||a||_{\ell_1}||b||_{ℓ_q},~~~~ 1≤q≤∞$. Here you can find something on $\ell_q\big(\mathbb{Z}\big)$: Young's ...
1
vote
4answers
59 views

Prove the inequality $e^x \geq x^e$ for $x > 0$

Prove that $e^x \ge x^e$ for $x \gt 0$ I applied the natural logarithm to simplify the function and I get $$\frac{x}{\ln x}\ge e$$ How to solve these types of problems?
4
votes
1answer
76 views

Prove this inequality $\frac{1}{1+a}+\frac{2}{1+a+b}<\sqrt{\frac{1}{a}+\frac{1}{b}}$

Let $a,b>0$ show that $$\dfrac{1}{1+a}+\dfrac{2}{1+a+b}<\sqrt{\dfrac{1}{a}+\dfrac{1}{b}}$$ It suffices to show that $$\dfrac{(3a+b+3)^2}{((1+a)(1+a+b))^2}<\dfrac{a+b}{ab}$$ or ...
1
vote
1answer
37 views

Inequality using integrals and absolute values

Let $u,v$ be continous functions in $[a,b]$ a compact interval and let $c > 0$. Suppose that $\forall x\in [a,b]$, the following inequality is true: $$|u(x)-v(x)|\leq c\int^x_a|u(t)-v(t)|dt$$ ...
5
votes
0answers
21 views

Lower bound on absolute value of determinant of sum of matrices

I needed to find a lower bound on $|\det(A+B)|$ where $|.|$ is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But ...
0
votes
1answer
36 views

Prove that:$\sum_{i=1}^n\frac{1}{x_i}-\sum_{i<j}\frac{1}{x_i+x_j}+\sum_{i<j<k}\frac{1}{x_i+x_j+x_k}-\cdots+(-1)^{n-1}\frac{1}{x_1+\ldots+x_n}>0.$?

Is there someone show me how do I prove this , I guess this inequality hold only if $x=0$ . Let $x_1,x_2,\ldots,x_n>0$. Prove that ...
3
votes
1answer
34 views

How to prove the following inequality $|\prod_{i=1}^{i=n}a_i-\prod_{i=1}^{i=n}b_i| < n\delta$?

The constraints are $0 \le a_1,a_2....a_n,b_1,b_2....b_n \le 1$. $|a_i-b_i|< \delta$ for all $1 \le i \le n $ How do I go about proving the following $$|\prod_{i=1}^n a_i-\prod_{i=1}^n b_i| ...
6
votes
3answers
411 views

Some trouble with the induction

Prove, that for any positive integer $n \geqslant 2$ we have the inequality $$ \frac{ 4^n }{ n+1 } < \frac{ (2n)! }{ (n!)^2 }.$$ For $n=2$ the inequality is true. Directly just take and ...
0
votes
0answers
19 views

Order of 'Strength' of inequalities

There have been times when we solve an inequality and we get the reverse sign of inequality. The reason is quite simple- we did not choose a strong inequality. So my question is- Is there an order of ...
0
votes
3answers
68 views

Prove sum of $\sin$ of angles is greater than $\sin$ of sum of angles

It seems that $\displaystyle \sum_{x_i \in X} \sin\left(x_i\right) \geq \sin\left(\sum_{x_i \in X} x_i\right)$ where $X$ is a set of angles where $\displaystyle \sum_{x_i \in X} x_i \leq \pi$ radians ...
5
votes
2answers
35 views

Doubt with Absolute Value Inequality

Problem: Find all values of $x$ for which $\dfrac{|x-2|}{x-2}>0$ My incorrect attempt: Using the definition the Modulus, $|x-2|=x-2$ for all $x\ge2$ and $|x-2|=-x+2$ for all $x\le2.$ ...
0
votes
1answer
33 views

Finding a metric such that $\Phi$ becomes a contraction

Let $$ f:\:[0,1]\rightarrow\mathbb{R};\\ \Phi: \mathcal{F}([0,1], \mathbb{R}) \rightarrow \mathcal{F}([0,1], \mathbb{R}),\\ f\mapsto\Phi(f):=Φf := \left(x\mapsto\left\{ ...
1
vote
1answer
37 views

Help with a summation+inequality problem.

I need help in solving for all possible x values for the below inequality: (Note: $x \in N)$ $$\sum^x_{k=1}\frac{k^2+k+1}{k(k+1)(k+1)!} \leq \frac{599}{600}$$ I think the series is telescopic; I'm ...
1
vote
0answers
19 views

search on a split data structure

I have the following problem: Part 1: Lets say I have n items in a data structure and I want to search for them. I know that a subset of my data $r \cdot ...
0
votes
0answers
21 views

An inequality of probability [on hold]

Suppose $X$ is a random variable. $E(X^2)=1$, $E|X|\ge a>0$. Let $\lambda\in[0,1]$, prove that $$P(|x|\ge\lambda a)\ge(1-\lambda)^2a^2.$$ I have tried to use Chebyshev's inequality, but I didn't ...
-1
votes
0answers
23 views

Quadratic recurrence inequality

I have the recurrence relation: $r_{k+1} \leq r_k^2+ (1/2)r_k \quad (k =1,2,\ldots)$, where each $r_k$ is non-negative and $r_1<1$. I have the following questions in this regard: A simple plot ...
-1
votes
1answer
28 views

an inequality for multiplication of cubic numbers

I need to know is there a positive constant $c$ such that $$ |a_1 \cdots a_n|^3 \leq c\big(|a_1|^3+\cdots+|a_n|^3\big), $$ where $a_{i}\neq 0$ ? I tried geometric and arithmetic inequalities but I ...
1
vote
2answers
41 views

Concluding three statements regarding $3$ real numbers.

$\{a,b,c\}\in \mathbb{R},\ a<b<c,\ a+b+c=6 ,\ ab+bc+ac=9$ Conclusion $I.)\ 1<b<3$ Conclusion $II.)\ 2<a<3$ Conclusion $III.)\ 0<c<1$ Options By ...
6
votes
2answers
125 views

Find the minimum value of $A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}$

Let $a, b$ and $c$ three positive real numbers such that $a+b+c=3$. Find the minimum value of $$A=\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{2-c^3}{c}.$$ Here is my attempt. By symmetry we can assume that ...
5
votes
1answer
66 views

Inequality with reciprocals of $n$-variable sums

Let $a_1,a_2,\ldots,a_n$ be positive real numbers. Is it always true that $$\sum_{i=1}^n\frac{1}{a_i}-\sum_{1\leq i<j\leq n}\frac{1}{a_i+a_j}+\sum_{1\leq i<j<k\leq ...
2
votes
3answers
41 views

Reasoning about numbers close to two other numbers $a,b$ (inequalities)

Let $a < b$ and $0 < \varepsilon < (b - a)$ and let $x, y \in \mathbb R$ be such that $$ | x - a | < \frac{(b - a) - \varepsilon}{2}, \qquad | y - b | < \frac{(b - a) - ...
0
votes
1answer
30 views

An inequality involving logarithmic function and floor function [on hold]

How do I show that $${\log_2(\log_2(x))}\geq\lfloor{\ln(\ln(x))}\rfloor.$$
0
votes
3answers
56 views

Proving that $|a-b|≤|a|+|b|$ [on hold]

Can someone prove this to me: $$|a-b|≤|a|+|b|$$ I am in 8th grade and I have this for my homework. Thanks for helping.
3
votes
2answers
109 views

Show that $\frac{(x^2 + y^2 )}{4} \leq e^{x+y-2}$

Show that \begin{equation} \frac{x^2 + y^2}{4} \leq e^{x+y-2} \end{equation} is true for $x,y \geq 0$. As far, I have prove that \begin{equation} x^2 + y^2 \leq e^{x}e^{y}\leq e^{x+y} ...
10
votes
0answers
72 views
+50

Inclusion-exclusion-like fractional sum is positive?

Let $A_1,A_2,\ldots,A_n$ be finite nonempty sets. Is it true that $$\sum_{i=1}^n\frac{1}{|A_i|}-\sum_{1\leq i<j\leq n}\frac{1}{|A_i\cup A_j|}+\sum_{1\leq i<j<k\leq n}\frac{1}{|A_i\cup ...
1
vote
1answer
129 views

If $a,b,c>0, a+b+c=3$, minimize $\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{3-c^3}{c}$ [duplicate]

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Find the minimum value of the expression $A= \frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{3-c^3}{c}$ I tried solving it, but I got nothing
2
votes
1answer
41 views

Minkowski's inequality

Minkowski's inequality for sums states $$\left(\sum_{j=0}^\infty |a_j+b_j|^2 \right)^{1/2} \le \left(\sum_{j=0}^\infty |a_j|^2 \right)^{1/2}+\left(\sum_{j=0}^\infty |b_j|^2 \right)^{1/2} $$ for ...
1
vote
2answers
60 views

Minimum value of cosA+cosB+cosC in a triangle ABC

I have used jensen's inequality but couldn't move on.
1
vote
5answers
82 views

Proving that $\frac{1}{a^2}+\frac{1}{b^2} \geq \frac{8}{(a+b)^2}$ for $a,b>0$

I found something that I'm not quite sure about when trying to prove this inequality. I've proven that $$\dfrac{1}{a}+\dfrac{1}{b}\geq \dfrac{4}{a+b}$$ already. My idea now is to replace $a$ with ...
4
votes
2answers
112 views

How to prove this maximum is $\frac{\sqrt{3}}{5}$

let $a,b,c>0$, such $ab+bc+ac=1$,show that $$\dfrac{1}{3a+5b+7c}+\dfrac{1}{3b+5c+7a}+\dfrac{1}{3c+5a+7b}\le\dfrac{\sqrt{3}}{5}$$ since dear Mac sir,he solve with inequality ...
2
votes
3answers
35 views

$\frac{1+\sqrt x}{1-\sqrt x} \leq e^{Cx}\frac{1+x}{1-x}$

I try find a constant $C$ such that $$\frac{1+\sqrt x}{1-\sqrt x} \leq e^{Cx}\frac{1+x}{1-x}$$ Maybe is not posible
1
vote
0answers
22 views

The decay of the Fourier coefficients of the disjoint union of arcs

Given $N$ disjoint arcs $\{I_{\alpha}\}_{\alpha=1}^{N}\subset\mathbb{T} $,set $f=\displaystyle\sum_{\alpha=1}^{N}\chi_{I_{\alpha}}$ show that $$\sum_{|v|>k}|\hat{f}(v)|^2\le\dfrac{CN}{k}$$ This ...
0
votes
4answers
50 views

How to isolate $x$ in the inequality $3^{x+2}<5^{x-1}$

For example, \begin{align} 0.4^x &\gt 0.9 \\ \left(\frac{2}{5}\right)^x &\gt \left(\frac{9}{10}\right) \\ \left(\frac{2}{5}\right)^x &\gt \left(\frac{2}{5}\right)^{\log_{2/5} ...
5
votes
3answers
140 views

Prove that $A_{100} \gt 14$ where $A_{n}=A_{n-1}+\frac{1}{A_{n-1}}$ and $A_1=1$

I tried attempting the question, and the best upper bound I could obtain was $1+\ln{98}$. I tried using $A_{n}\le n$ to form a harmonic series, but that wasn't strong enough. Any help would be ...
3
votes
1answer
52 views

Complex Inequalities

Let $\mathbb{H}=\{z\in \mathbb{C} | \ \Im(z)>0\}$ and $f:\mathbb{H} \to \mathbb{H}$ analytic. Prove that for every $z_1, z_2 \in \mathbb{H}$, it must happen that $$ ...
2
votes
1answer
38 views

Proving $1-\cos(k)\geq\frac{2}{\pi^{2}}k^{2}$ for $k\in(-\pi,\pi)$

I am trying to prove the following: $$1-\cos(k)\geq\frac{2}{\pi^{2}}k^{2}\quad\hbox{for}\quad k\in(-\pi,\pi]$$ So far, I have tried using some Maclaurin expansion arguments, but when that didn't ...
0
votes
1answer
47 views

3 variable symmetric inequality

Show that for positive reals $a,b,c$, $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\geq \frac{3a^3+3b^3+3c^3}{2a^2+2b^2+2c^2}$ What I did was WLOG $a+b+c=1$ (since the inequality is homogenous) ...
1
vote
4answers
28 views

How to express a=8 versus b=4?

I want to know how to express in English a and b. a is larger(=greater, bigger) than b as four. (right?) b is smaller(=less) than a as four. (right?) thank you for reading my question.
2
votes
5answers
446 views

Find the maximum possible area of a certain right triangle

I want to find the maximum possible area of a right triangle with hypotenuse $=10$. My approach so far: let $x,y$ be the lengths of the two sides adjacent to the right angle; then $$100=x^2+y^2$$ ...
0
votes
1answer
19 views

How are the following inequalities concluded based on this first one?

$$I-\frac{\epsilon}{3} \leq s(f,T) \leq \underline{I} \leq \overline{I}\leq S(f,T) \leq I+ \frac{\epsilon}{3}$$ from this, the following is concluded, but how? $$1.\ \ \ 0 \leq |I-\underline{I}|\leq ...
-3
votes
0answers
31 views

Very difficult problem about inequality [closed]

Let $$ \Omega_1 = \left\{ (a,b,a+b,d)\in \mathbb{R}^4 \big| a>0, b>\frac{d^2}{2} \right\}, $$ $$ \Omega_2 = \left\{ (a,b,a-b,d)\in \mathbb{R}^4 \big| a>0, b<-\frac{d^2}{2} \right\}, $$ ...
4
votes
2answers
40 views

Given $a_1 \ge \cdots \ge a_n$ and $b_1 \ge \cdots \ge b_n$, then show $\sum a_ib_{\pi(i)}$ is maximum when $\pi=id$.

Suppose $a_1 \ge \cdots \ge a_n$ and $b_1 \ge \cdots \ge b_n$ are two sequences of positive real numbers. Then show $\sum a_ib_{\pi(i)}$ is maximum when $\pi=id$. Here, $\pi \in S_n$. I ...