Questions on proving and manipulating inequalities.

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4
votes
3answers
91 views

Is it true that $\sin x > \frac x{\sqrt {x^2+1}} , \forall x \in (0, \frac {\pi}2)$?

Is it true that $$\sin x > \dfrac x{\sqrt {x^2+1}} , \forall x \in \left(0, \dfrac {\pi}2\right)$$ (I tried differentiating , but it's not coming , please help)
-2
votes
1answer
32 views

Maximum number of positive integers $x\neq y$ such that $\frac{xy}{100}\leq|x-y|$

I've been trying to solve the next problem but I have no idea of how to find the solution: Find the largest number of positive integers in such a way that any two of them $x$ and $y$ ($x\neq y$) ...
-5
votes
1answer
38 views

Select group of real numbers $x$ [on hold]

Select group of real numbers $x$, satisfy the inequality $$\frac{4x^2}{(1-\sqrt{2x+1})^2}< 2x+9$$ help guys!!
0
votes
1answer
17 views

Polynomial inequalities of the form $C P_2 \leq P_1 \leq D P_2$

Let $P_1$ and $P_2$ be polynomials in $\mathbb{R} [x_1, \ldots, x_n]$ of the same degree. Under what conditions are there $C,D \in \mathbb{R}$ so that $C P_2 \leq P_1 \leq D P_2$ (as functions)? ...
3
votes
2answers
25 views

Inequality: $\sum_{i} \frac1{\alpha_i} \ge n^2$

$\alpha_1, \ldots, \alpha_n$ are positive reals whose sum does not exceed one. It is required to prove that: $$\sum_{i} \frac1{\alpha_i} \ge n^2$$ I would show my work, but I am certain that it does ...
6
votes
1answer
43 views

$\frac{MA}{BC}+\frac{MB}{CA}+\frac{MC}{AB}\geq \sqrt{3}$

Given ∆$ABC$ and $M$ is an interior point.Prove that: $\dfrac{MA}{BC}+\dfrac{MB}{CA}+\dfrac{MC}{AB}\geq \sqrt{3}$ When does equality holds?
2
votes
1answer
52 views

Sum of quotients

Assume $0<x_i\leq y<z$ for $i=1\ldots,n$. Is there an easy argument to show $$\frac{x_1}{y}+\sum_{i=1}^{n-1} \frac{x_{i+1}}{x_i}+\frac{z}{x_n}\geq n+\frac{z}{y}?$$ For $n=1$ the statement is ...
4
votes
4answers
157 views

Is My Proof that $\pi^e < e^{\pi}$ Valid? [duplicate]

The other day, a math teacher at my college gave me a challenge problem: Prove that $$\pi^e < e^{\pi}$$ without using a calculator. The next day, I found a valid proof, but I used a log table ...
0
votes
2answers
25 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
1
vote
1answer
22 views

Bounds for double exponential integrals

I understand that the double-exponential integral $$ F(a,b,C) := \int_{C}^\infty \exp(-a \exp(b x)) \, dx \quad \text{(with $a,b>0$ and $C \geq 0$)} $$ can in general not be solved in closed-form. ...
0
votes
0answers
32 views

What are the numbers in an inequality called?

Summand is to addition what multiplicand is to multiplication, but what is the terminology for the quantities of an inequality, such as 1<4? My best guess is simply "quantity" for both parts of ...
2
votes
0answers
36 views

“Triangle” inequality for integrals

I have got two questions: 1) Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be any continuous function. Let $\Gamma$ be a piecewise smooth curve on $\mathbb{R}^2$. The following inequality holds: ...
0
votes
2answers
41 views

Largest integer $x$ that satisfies $\dfrac{4x+19}{x+5}<\dfrac{4x-17}{x-3}$

Find the largest integral $x$ that satisfies $\dfrac{4x+19}{x+5}<\dfrac{4x-17}{x-3}$ I tried $ \dfrac{4x+19}{x+5} < \dfrac{4x-17}{x-3}\\~\\ (4x+19)(x-3)<(4x-17)(x+5)\\~\\ x<-7 ...
0
votes
0answers
17 views

Establishing consistency

I need to establish the (weak) consistency of an estimator of the mean, $T=a+b\bar{X}$. I tried to apply Chebyshev's inequality, but I couldn't do much because the parameter that subtract in the ...
2
votes
1answer
52 views
+50

Do more generalizations of Schur's inequality exist?

I meet this following problem If $$n\ge 3,\sum_{i=1}^{n}\left(\prod_{j\neq i}(a_{i}-a_{j})\right)\ge 0$$ where $a_{i}$ are real numbers. when $n=3$, it is Schur's inequality so which $n$ ...
0
votes
1answer
16 views

Replacing a fractional quantity by one in inequalities

I was looking through proofs for the product law for limits and I stumbled upon a very clear one and managed to follow all the steps involving algebra and limits, but, near the end, in the proof, a ...
1
vote
2answers
56 views

solve $\sqrt{x+7}<x$ for $x\in \mathbb{R}$

solve $\sqrt{x+7}<x$ I tried $\sqrt{x+7}<x\\ x+7<x^2\\ x^2-x-7>0\\ x\in \left(-\infty, \dfrac{1-\sqrt{29}}{2}\right) \cup \left( \dfrac{1+\sqrt{29}}{2},+\infty\right) $ I m not ...
3
votes
1answer
51 views

A Real Matrix, its Kernel and Image

This is an old exam problem: For an $m \times n$ real matrix $A$, define $\ker A = \{x \in \mathbb{R}^n \mid Ax=0 \}$ and $\operatorname{Im} A = \{Ax \mid x \in \mathbb{R}^n \}$. Show that for all $b ...
3
votes
3answers
37 views

solve $|x-6|>|x^2-5x+9|$

solve $|x-6|>|x^2-5x+9|,\ \ x\in \mathbb{R}$ I have done $4$ cases. $1.)\ x-6>x^2-5x+9\ \ ,\implies x\in \emptyset \\ 2.)\ x-6<x^2-5x+9\ \ ,\implies x\in \mathbb{R} \\ 3.)\ ...
3
votes
2answers
27 views

Check proof of some simple inequality

Can you check please my proof of this inequality? It's all right?
-1
votes
1answer
60 views

Proof inequality using Lagrange Multipliers

Is it possible: $a,b,c$ are non-negative real numbers for which holds that $a+b+c=3.$ Prove the following inequality: $$ 4\ge a^2b+b^2c+c^2a+abc $$ Is it possible using Lagrange Multipliers. I ...
8
votes
3answers
94 views

Prove that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x} \geq 3$ for $x,y,z>0$

By considering that $$\frac{x}{y}+\frac{y}{x} \geq 2$$I can show that $$\frac{x}{y}+\frac{y}{x}+\frac{x}{z}+\frac{z}{x}+\frac{y}{z}+\frac{z}{y} \geq 6$$ But how would one go from here to prove the ...
0
votes
2answers
28 views

squaring both side for an absolute inequalites on only one side

this is about squaring both side for an absolute inequalities on only one side problem. For example: $|6-2x|< x+4$ when solved both by squaring both sides and by defining it$ -(x+4)<6-2x< ...
1
vote
2answers
81 views

Proving inequalities using Calculus

In general how do you prove inequalities using calculus, I believe it is using maxima or minima right? For example $$a^2b+b^2c+c^2a \le 3, \qquad a,b,c \ge 0,\quad a+b+c=3.$$ How would you use ...
2
votes
1answer
76 views

Graham's Number versus another large number

I recently read this article about the most damage you can do in a single turn in Magic the Gathering. According to the current version of the deck, that damage is about a) $2 \rightarrow 17 ...
0
votes
3answers
38 views

complete the proof for this statement

$$\forall x \in \mathbb{R}, x \neq 0 \implies \frac{1}{x^2\:+3}\:<\:\frac{4}{5}\: $$ I thought of doing the contrapositive but not sure what to do next. $$ \frac{1}{x^{2\:}+3}\:\ge ...
2
votes
3answers
49 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 ...
3
votes
5answers
288 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
62 views

Prove that $\frac{8}{5}\le 2a+b\le 8$ [on hold]

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: $$\dfrac{8}{5}\le 2a+b\le 8$$
3
votes
1answer
35 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\\ $ ...
17
votes
2answers
143 views
0
votes
2answers
34 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
293 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
26 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 ...
2
votes
4answers
64 views

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

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
2answers
98 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
42 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: $$|u(x)-v(x)|\leq c\int^x_a|u(t)-v(t)|dt$$ ...
6
votes
0answers
27 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 ...
3
votes
1answer
74 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
423 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
20 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
71 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
40 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
38 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
22 views

An inequality of probability [closed]

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
25 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 ...