Questions on proving and manipulating inequalities.

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7
votes
5answers
379 views
0
votes
1answer
32 views

If $d_1(x,y)$ and $d_2(x,y)$ are metrics, prove that $d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$ is a metric.

$$d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$$ The first three properties are trivially proven. The triangle inequality, not so much. I tried using the triangle inequalities that apply to $d_1$ and $d_2$, ...
24
votes
4answers
1k views

Prove that $\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$

show that $$\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$$ and I found $$LHs-RHS=0.017\cdots$$ I have post this interesting problem Prove $(\dfrac{2}{5})^{\frac{2}{5}}<\ln{2}$ can someone suggest ...
0
votes
2answers
29 views

Explanation for 2 inequalities with same solutions.

While studying, I read that $(x-a)^{2k \pm 1}f(x)>0$ has the same solution as $(x-a)f(x)>0$. I do not get why this is true. Could someone please explain why these two inequalities have the same ...
3
votes
2answers
80 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 $n$ ...
1
vote
2answers
22 views

Inequalities for f(x) is always positive

Given that $f(x)=4x^2-1$ Find the range of values of $x$ so that $f(x)$ is always positive. My attempt, $4x^2-1>0$ $4x^2>1$ $x^2>\frac{1}{4}$ $x>\pm\frac{1}{2}$ So ...
4
votes
0answers
27 views

Show that $p \in \left[\frac{4^m}{\sqrt{2m}},\frac{4^m}{\sqrt{2m+1}}\right]$

If the number of ways in which $m$ identical apples can be put in $2m$ boxes, so that no box contains more than one apple, is $p$, prove that $$p \in ...
0
votes
2answers
59 views

Solving this inequality with integral

We have function $f:\mathbb{R}-\{2 \}\to\mathbb{R}$ $$f(x)=\frac{x^2}{x-2}$$ Show that $8\le\int\limits _3^4f\left(x\right)dx\le9$ I solved the definite integral and got $\int\limits ...
3
votes
2answers
511 views

When does the equality hold in the triangle inequality? [duplicate]

Hi guys could you please help me on this question I'm confused. question: when does the equality hold in the triangle inequality: my attempt : $|x + y| \leq |x| + |y|$ this implies $(|x+y|)^2 = ...
3
votes
3answers
3k views

Equality holds in triangle inequality iff both numbers are positive, both are negative or one is zero

How do we show that equality holds in the triangle inequality $|a+b|=|a|+|b|$ iff both numbers are positive, both are negative or one is zero? I already showed that equality holds when one of the ...
1
vote
1answer
23 views

Bound of the Complex Expression

Here, $x$, $y$ and $\alpha$ are all complex numbers such that $|x|<\epsilon$ and $|y|<\epsilon$. Now what would be upper bound of the following expression: $|\frac{\alpha+y}{x y - \alpha}|$? ...
0
votes
2answers
30 views

Proving Lower bounds on an Approximately Linear Function

We are looking for a lower bound on the function, $\frac{1.31}{e^{\frac{1.31}{x+1}} - 1}$ for $x \geq 2$. This function seems to behave linearly. We believe that the following statement holds: ...
1
vote
2answers
34 views

Squaring both sides of an inequality: attempt to prove a general rule

I have attempted to produce a proof of the intuitive rule for squaring inequalities, according to which, given any two numbers x and y and regardless of their sign, 1) if |x| < |y| then ...
0
votes
4answers
33 views

If $d(x,y)$ is a metric, how does the following inequality apply?

I'm interested if someone can formally type out why this is. I thought it was trivial, but the professor wanted a more detailed explanation: $${d(x,y)\over {1+d(x,y)}}\leq ...
3
votes
3answers
37 views

Let $A \subset \mathbb Z^3$ / $|A| < \infty$. Prove that: $|A| \le \sqrt{|A_x| |A_y| |A_z|}$

Here is the problem statement word by word: $1)$ Prove that if $a_{ij}$, $b_{jk}$ and $c_{ki}$ are non-negative reals with $1 \le i,j,k \le n$, then: $$\sum_{i,j,k = 1}^n \sqrt{a_{ij} \times ...
3
votes
2answers
45 views

this inequality $\prod_{cyc} (x^2+x+1)\ge 9\sum_{cyc} xy$

Let $x,y,z\in R$,and $x+y+z=3$ show that: $$(x^2+x+1)(y^2+y+1)(z^2+z+1)\ge 9(xy+yz+xz)$$ Things I have tried so far:$$9(xy+yz+xz)\le 3(x+y+z)^2=27$$ so it suffices to prove that ...
0
votes
4answers
56 views

Show $ \lim_{n\rightarrow \infty} 2^{-1/\sqrt{n}}=1$

I am tasked with proving the following limit: $$ \lim_{n\rightarrow \infty} 2^{-1/\sqrt{n}}=1$$ using the definition of the limit. I think I have done so correctly. I was hoping to have someone ...
1
vote
1answer
370 views

What are the similarities and differences in solving equations and inequalities?

What are the similarities and differences in solving equations and inequalities?
1
vote
0answers
50 views

Is the Schwarz inequality a special case of the Cauchy-Schwarz inequality?

Given two vectors $\mathbf{x},\mathbf{y}$ in $\mathbb{R}^n$, we all know that:$$\left | \mathbf{x}\cdot\mathbf{y} \right | \le \left \| \mathbf{x} \right \| \cdot\left \| \mathbf{y} \right \|$$ ...
2
votes
2answers
119 views

$\frac{AB}{A'B'}+\frac{BC}{B'C'}+\frac{CA}{C'A'} \geq 4 \left(\sin{\frac{A}{2}}+\sin{\frac{B}{2}}+\sin{\frac{C}{2}}\right). $

Let be a circle inscribed in the triangle $\triangle ABC$ wiht the center $I$. The intersection of the circle with $AI$ is $A'$, with $BI$ is $B'$ and with $CI$ is $C'$. Prove that: ...
0
votes
0answers
20 views

Norm estimate in fractional-order periodic Sobolev spaces

Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ${P}$-periodic functions on the line with norm \begin{equation*} \| u ...
3
votes
4answers
106 views

Taking limits on each term in inequality invalid?

So this inequality came up in a proof I was going through. $$c - 1/n < f(x_n) \leq c$$ Where $c$ is a real number, $f(x_n)$ is the image sequence of some arbitrary sequence being passed through a ...
6
votes
3answers
113 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)
0
votes
1answer
56 views

How do I prove this nice inequality $x+3\sqrt[3]{xy^2}\geq4\sqrt{xy} $?

Let $x,y\geq0$. Prove that: $$ x+3\sqrt[3]{xy^2}\geq4\sqrt{xy} $$ Note: It's seems easy but when I tried to show it I went to complicated formula.
0
votes
0answers
75 views

Show a function defined by summation is increasing, another is decreasing

Problem: For real numbers $x\ge1$ and $k>0$, let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as follows. $f(x) = -\frac{1}{x}+\sum_{n=1}^{\infty}\frac{1}{(nk+x)^2}$ , ...
0
votes
1answer
26 views

Value of discriminant in inequality

Find $m\in \mathbb{R}$ so that for any $x\in \mathbb{R}$ the inequality is true: $$\left(m+2\right)e^{-2x}+2\left(m+2\right)e^{-x}+m>0$$ I tried substituting $e^{-x}$ with t, so that I'd have a ...
3
votes
10answers
387 views

Prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$

I'm trying to prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$ I've first written down $-5\lt a-b \lt5$ and have tried to add different things from all sides of the ...
6
votes
3answers
177 views

Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$

Use the mean value theorem to prove that if $0 \leq x \leq 1$, then $$x\sqrt{1-x^2} \leq \sin x \leq x$$ The theorem guarantees the existence of a point, but not an inequality, so I don't know how to ...
3
votes
2answers
175 views

Concentration inequality for the median

Most concentration inequalities talk about deviation of the sample mean from the population mean. Is there a result bounding the probability of deviation of the sample median from the median of the ...
0
votes
1answer
24 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)? ...
-2
votes
1answer
38 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
45 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!!
6
votes
1answer
53 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?
3
votes
2answers
28 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 ...
3
votes
1answer
59 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
178 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 ...
9
votes
5answers
526 views

Improving bound on $\sqrt{2 \sqrt{3 \sqrt{4 \ldots}}}$

An old challenge problem I saw asked to prove that $\sqrt{2 \sqrt{3 \sqrt{4 \ldots}}} < 3$. A simple calculation shows the actual value seems to be around $2.8$, which is pretty close to $3$ but ...
19
votes
4answers
193 views

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 ...
4
votes
3answers
164 views

Absolute Value inequality help: $|x+1| \geq 3$

Find the solutions to the inequality: $$|x+1| \geq 3$$ I translate this as: which numbers are at least $3$ units from $1$? So, picturing a number line, I would place a filled in circle at the ...
2
votes
1answer
291 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 ...
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
23 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. ...
38
votes
10answers
3k views

Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator?

$1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$. I got this question in an Application of Derivatives test. I think log might be used here to compare the values, but even then the values ...
0
votes
0answers
18 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 ...
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
43 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 ...
2
votes
1answer
84 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 ...
1
vote
2answers
120 views

How to prove this special inequality: $\sum_{cyc} \frac1{xy+z}>3$ for $\sum_{cyc} xy=1$?

Let $x,y,z>0$ s.t. $$\underset{cyc}{\sum} xy=1$$ Prove that $$\underset{cyc}{\sum}\frac{1}{xy+z}>3$$ I see that when $x=1,y\rightarrow1,z\rightarrow0$ $$\underset{cyc}{\sum}\frac{1}{xy+z} ...