Questions on proving and manipulating inequalities.

learn more… | top users | synonyms (1)

1
vote
1answer
23 views

Show $|t|\leq \pi \Rightarrow |e^{it}-1|\geq 2|t|/ \pi$

I would like to show that show $|t|\leq \pi$ implies $ |e^{it}-1|\geq 2|t|/ \pi$. I tried to proof it geometrically but without success.
2
votes
1answer
73 views

Prove a real valued function is increasing

Prove $$-\frac{x(1-x^c)\ln x}{(1-x)^2}$$ $c>0$ is an increasing function on $[0,1]$. I have a relatively cumbersome proof sketched below. I would like to see other ideas, particularly simpler ...
1
vote
1answer
18 views

Supremum of (e^(i z t) - 1)/z

i'm new here, so i'm not sure if this is the right place to ask this question: I know that the following holds true: $$ \forall\, t \in \mathbb{R} \; \forall\,x\in\mathbb{R}\setminus\{0\} ...
4
votes
2answers
128 views

Inequality with $a+b+c+d = 2$

If $a+b+c+d = 2$, prove that $$\dfrac{a^2}{(a^2+1)^2}+\dfrac{b^2}{(b^2+1)^2}+\dfrac{c^2}{(c^2+1)^2}+\dfrac{d^2}{(d^2+1)^2}\le \dfrac{16}{25}$$ Also $a,b,c,d \ge 0$.
0
votes
1answer
27 views

How find this function $f(x)=\sqrt{x^2-2x}+\sqrt{2x^2-3x+3}$ range

let $x\in R$,and $$f(x)=\sqrt{x^2-2x}+\sqrt{2x^2-3x+3}$$ find the $f(x)$ range My idea: since $$x^2-2x\ge 0,2x^2-3x+3\ge 0$$ $$\Longrightarrow x\ge 2,or, x\le 0$$ I use mathmatices give this ...
2
votes
1answer
58 views

Proving the Kochen-Stone lemma using the Paley-Zygmund inequality

I am trying to understand a proof to a lemma by Kochen and Stone which appears here, using the Paley-Zygmund inequality. I'll repeat the proof in a detailed manner, and explain what bothers me about ...
1
vote
1answer
22 views

Question regarding Monoalphabetic Phi Test

I've been asked to prove the following system of inequalities; $$1 \ge \phi(T) \ge \frac{n-k}{k(n-1)}$$ Where $\phi(T) = \sum_{i=1}^{k} \frac{n_i (n_i -1)}{n(n-1)}$, $T =$ some text, $n = $ length ...
0
votes
0answers
40 views

Simple proof explanation - Possibly triangle inequality involved

I'd like some help with understanding the following statements...I saw it on the internet while searching for a proof, and I'd like to understand why its true: let $A$ be a diagonally dominant matrix ...
1
vote
0answers
61 views

Integral inequality with a function twice differentiable

Let $f:[0,1]\longrightarrow\mathbb{R}$ be a function twice differentiable with continous second derivative and $f(1)=f(0)$. The inequality: $$\int_{0}^{1}(f''(x))^2dx\geq ...
0
votes
0answers
17 views

Get a count of a set when a condition is met

This feels like an easy question, but I can't seem to hammer it down. Just like $\Sigma$ can be used to sum over several items, it there a way to count them? The simple example, is that I want to ...
0
votes
3answers
39 views

how to solve a inequality?

I have to prove the following inequality with induction from $(n-1)$ to $n$: $(n+1)! \geq 2^n$ I know the solution, but I can't figure out where the $2$ came from: $\begin{aligned}(n+1)! &= ...
1
vote
2answers
44 views

Prove that $a_n$>$b_n$

For a given real number $\alpha>0$, define $a_n=(1^{\alpha}+2^{\alpha}+\dots+n^{\alpha})^{n}$ and $b_n=n^n(n!)^{\alpha}$, for $n=1,2,...$ Prove that $a_n>b_n$ for all $n>1$ There were powers ...
1
vote
0answers
16 views

How find this $m$ value range?

if this two function $f(x)=e^{1-|x-m|}-emx^2(m>0)$ and $g(x)=x+1$ graph have a least one common point .Find the value range of the $m$. where $e$ is constant,$e=2.718\cdots$ my idea: let $c$ ...
3
votes
0answers
30 views

How prove this inequality $\sum_{cyc}\frac{1-2\sin{\frac{C}{2}}}{\sin{\frac{B}{2}}}\ge 0$

in $\Delta ABC$,prove or disprove $$\dfrac{1-2\sin{\dfrac{C}{2}}}{\sin{\dfrac{B}{2}}}+\dfrac{1-2\sin{\dfrac{A}{2}}}{\sin{\dfrac{C}{2}}}+\dfrac{1-2\sin{\dfrac{A}{2}}}{\sin{\dfrac{A}{2}}}\ge 0$$ My ...
0
votes
4answers
63 views

Elementary proofs of inequalities

I was just introduced into elementary proofs of inequalities, my text's explanation however feels incomplete. I did further research on the subject, my question is thus: Prove: If $0 < a < b$, ...
2
votes
3answers
124 views

Proving that if $\frac{m}{n}<\sqrt{2}$ then there exists $\frac{m'}{n'}$ such that $\frac{m}{n}< \frac{m'}{n'}<\sqrt{2}$

This is from Spivak. The problem is as follows: Prove that if $\frac{m}{n}<\sqrt{2}$ then there exists $\frac{m'}{n'}$ such that $\frac{m}{n}< \frac{m'}{n'}<\sqrt{2}$ So far, I have that ...
0
votes
0answers
31 views

Bounds of the solution space

I have a continuous function $f(x)=a x^2+b x+c$ that is defined on $]0,X[$. I know that the function values are bounded, $Fu <f(x) <Fl$ for all values of $x$. I want to find the bounds on the ...
1
vote
2answers
31 views

Let $-1<t<0$. Is there a $ t$ such that

Let $-1<t<0$. Is there $t$ such that for all positive integers $n>c$($c$ is a positive integer depends on $t$) $n^t - (n+1)^t > 1/n$
0
votes
1answer
34 views

Proving triangle inequality geometrically

If $\mathbf{u}$ and $\mathbf{v}$ are vectors, how can you prove geometrically that $$ \|\mathbf{u}+\mathbf{v}\|\leq\|\mathbf{u}\|+\|\mathbf{v}\| $$ I am aware of the proof that uses dot product but I ...
1
vote
1answer
18 views

Help with integral/logarithm inequality

I have to prove the following inequality: $1/(n+1) < \int_n^{n+1} 1/t$ $dt$ $<1/n$ I thought it would be easier to attack this via integration, so I get: $1/(n+1) <$ log $(n+1)-$ ...
0
votes
1answer
45 views

Caratheodory's theorem and outer measure

I'm trying to show that $$\lambda(A)=\lambda(A\cap E)+\lambda(A\cap E^c)$$ where $\lambda$ is an outer measure, $A\subset \mathbb{R}$, $E \subset \mathbb{R}$, and $E$ is an elementary set; that is, ...
2
votes
2answers
48 views

2x2 Matrix Inequality

Is it true that if I have a positive definite matrix $m = \left( \begin{smallmatrix} m_{11} & m_{12}\\ m_{21} & m_{22} \\\end{smallmatrix} \right)$ in $\mathrm{M}(\mathbb{C};2)$ the following ...
2
votes
1answer
38 views

I need help showing this inequality

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a twice differentiable function such that $f'>0$, $f''<0$, and $f(0)=0$. I need to show, that for every $x>0$: $\frac{f(x)}{f'(x)}>x$ Thanks ...
2
votes
0answers
36 views

When this inequality true?

If $a$ and $b$ are non-negative integers and $c$ and $d$ are non-negative real numbers, for what values is the following inequality true? $\log((a+b)!) - \log(a!b!) \ge(a+b) \log(c+d) - (a \log(c) ...
2
votes
1answer
45 views

Poincaré inequality for a subspace of $H^2(\Omega)$

Suppose that $\Omega\subset\mathbb{R}^d$ is a smooth, bounded, and connected domain. Let \begin{equation} H=\{u\in H^2(\Omega):\int_\Omega u(x) dx=0 ~\text{and}~ \nabla u\cdot v=0~ ...
6
votes
1answer
96 views

How find the largest numbers $\lambda(n)$ such $\sum_{k=1}^{n}|z_{k}|^2\ge\lambda(n)\cdot\min_{1\le k\le n}{|z_{k+1}-z_{k}|^2}$

Assmue that give the positive integer number $n$,Find the largerst the constant $\lambda(n)$,such for any complex $z_{1},z_{2},\cdots,z_{n}(z_{i}\neq 0,i=1,2,\cdots n)$,have ...
2
votes
1answer
63 views

What is the converse of the triangle inequality?

It's usual when presenting a theorem to also present its converse. Surprisingly, I've never seen the triangle inequality's converse stated. Triangle inequality: If the sides of a triangle are a, b, ...
1
vote
0answers
88 views

product of sums

This is a question which has puzzled me for a while. I would be very thankful if somebody can help me with it. Assume you have $S$ rectangles appearing in front of your screen one by one. Each ...
1
vote
3answers
57 views

Why is this 5 negative?

I'm struggling with the difference between a negative number and a positive number that is being subtracted. For example, when working on this inequality: $$ 2 - 5x \leq 7 $$ After subtracting 2 ...
0
votes
0answers
18 views

Bound of LogGamma and Log

It's true this inequality: $\log(\Gamma(a+1) - a * \log(b) > (\log(\Gamma(a1+1) - a1 * \log(b1)) + (\log(\Gamma(a2+1) - a2 * \log(b2))$ Where: $a = a1 + a2$ $b = b1 + b2$ and $a,b,a1,a2,b1,b2 ...
0
votes
0answers
51 views

Maximum area of quadrilateral of given perimeter.

Let $0\lt a\lt b$ (i) Show that among the triangles with base $a$ and perimeter $a + b$, the maximum area is obtained when the other two sides have equal length $b/2$. (ii) Using the ...
3
votes
4answers
63 views

Prove that $\frac{x \log(x)}{x^2-1} \leq \frac{1}{2}$ for positive $x$, $x \neq 1$.

I'd like to prove $$\frac{x \,\log(x)}{x^2-1} \leq \frac{1}{2} $$ for positive $x$, $x \neq 1$. I showed that the limit of the function $f(x) = \frac{x \,\text{log}(x)}{x^2-1}$ is zero as $x$ ...
10
votes
4answers
89 views

Prove $\sin^2(x)<\sin(x^2)$ for $0<x<\sqrt{\frac{\pi}{2}}$

I'm trying to prove $\sin^2(x)<\sin(x^2)$ for $0<x<\sqrt{\frac{\pi}{2}}$. Attempt: This is equivalent to showing $f(x)=\sin(x^2)-\sin^2(x) = ...
7
votes
1answer
106 views

About B. Ya Levin's proof that $|f(x)| \leq M$ implies $|f(x+iy)| \leq Me^{\sigma y}$

This question is about Theorems 1 through 3 on pages 37-38 of B. Ya Levin's Lectures on Entire Functions, available on Google Books. If you can't access the Google Books link there is also a ...
0
votes
0answers
54 views

Help with Gronwall's Inequality

I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
3
votes
1answer
50 views

If $|a|,|b|<1$, prove that $\frac{|a+b|}{|1+ab|}<1$.

So I've gotten as far as $\displaystyle\frac{|a+b|}{|1+ab|}<\frac{2}{|1+ab|}$ which is clearly wrong because it is greater than 1. What am I doing wrong? Is this question even true?
0
votes
2answers
30 views

Subtracting positive numbers from denominator in an inequality (with conditions).

When we have the following inequality: $$\frac{a}{b+c} \ge \frac{d}{e+c},$$ with $a,c,d \in \mathbb N_{\ge 0}$, $b, e \in \mathbb N_{\gt 0}$, $a \le b$ and $d \le e$ Then it seems to hold that ...
1
vote
2answers
52 views

How prove this $\ln{(x+\sqrt{x^2+1})}<\frac{x(a^x-1)}{(a^x+1)\log_{a}{(\sqrt{x^2+1}-x)}}$

let $0<a<1,x<0$,show that $$\ln{(x+\sqrt{x^2+1})}<\dfrac{x(a^x-1)}{(a^x+1)\log_{a}{(\sqrt{x^2+1}-x)}}$$ My idea: $$\Longleftrightarrow ...
0
votes
1answer
33 views

An unusual inequality

Problem: $(x_i)_{i=1}^n$ is a finite sequence of positive integers. Define $f\big(S\big)=\displaystyle \sum_{i\,\in\, S\,\subseteq\, [n]}x_i$, and suppose $f$ is injective. Prove that: ...
2
votes
2answers
56 views

Inequality: $x^2+y^2+z^2 \geq \sqrt{2}x(z+y)$

How can I prove the following inequality: $$x^2+y^2+z^2 \geq \sqrt{2}x(z+y)?$$ Thanks!
1
vote
2answers
54 views

Solving inequalities with fractions with unfactorable polynomials

So I've been cracking my head open trying to solve this inequality: $$\frac {x+1}{2-x} \le \frac {x}{3+x}$$ I've been taught you have to put all factors to one side of the inequality (leaving zero ...
1
vote
2answers
45 views

Prove $\frac{\sin(a)}{\sin(b)}<\frac{a}{b}<\frac{\tan(a)}{\tan(b)}$ for $0<b<a<\pi/2$ [duplicate]

not sure how to approach the following $\frac{\sin(a)}{\sin(b)}<\frac{a}{b}<\frac{\tan(a)}{\tan(b)}$ for $0<b<a<\pi/2$. Hints would be appreciated!
0
votes
3answers
43 views

Triangle inequality and its equality

How do I prove this? $$|x+y|=|x|+|y|\Leftrightarrow xy\geq0$$ I tried to use the triangle inequality, but I didn't get so far... Thanks!
2
votes
1answer
30 views

Subtracting positive numbers from denominator in an inequality.

When we have the following inequality: $$\frac{a}{b+c} \ge \frac{d}{e+c},$$ with $a,b,c,d,e \in \mathbb R_{\ge 0}$ Then it seems to hold that $$\frac{a}{b} \ge \frac{d}{e},$$ Is this correct? ...
0
votes
2answers
37 views

How do I conclude that $G(a)= a_1^\frac{1}{n},…,a_n^\frac{1}{n}$ must obtain its maximum when $a_1=…=a_n = U(a)$?

In the AM-GM Inequality, how do I conclude that $G(a)= a_1^\frac{1}{n},...,a_n^\frac{1}{n}$ must obtain its maximum when $a_1=...=a_n = U(a)$ and ($U(a)= \frac{a_1+...a_n}{n}$ is the arithmetic mean)? ...
1
vote
0answers
33 views

An indirect optimizing problem

we have $z_1=x_1y_1$ , $z_2=x_2y_2$ and $z=xy$. we know $x<x_1+x_2$ and $y<y_1+y_2$. Can we conclude minimizing $z_1$ and $z_2$ will lead us to minimum (or lesser values) of $z$? ...
1
vote
3answers
80 views

Express $|a+b|-|b|$ without absolute value signs

I am having trouble understanding what cases I need to evaluate. So far I've checked $a = b = 0$ and that results in the expression being equal to $0$. I've checked $0 \le b < a$ which results in ...
10
votes
5answers
338 views

How prove this $2(x^4+y^4+z^4)+2xyz+7\ge 5(x^2+y^2+z^2)$

let $x,y,z\ge 0$,show that $$2(x^4+y^4+z^4)+2xyz+7\ge 5(x^2+y^2+z^2)$$ my idea: let $$x+y+z=p,xy+yz+xz=q,xyz=r$$ since $$x^2+y^2+z^2=(x+y+z)^2-2(xy+yz+xz)=p^2-2q$$ and ...
3
votes
2answers
72 views

A summation identity which is for me hard to verify

I found a summation identity when I try to figure out the proof of an exercise in Apostol's Mathematical Analysis (Page 27, Exercise 1.26): $\ ~~~~$ If $a_1\geq a_2\geq \dots\geq a_n$ and $b_1\geq ...
2
votes
2answers
42 views

Maximum of linear combination

I have an range like this: $$x + 2y \leq 40$$ $$4x + 3y \leq 120$$ $$x \geq 0, y \geq 0 $$ I made an plot using wolfram alpha. Now I have a linear combination $$4x+5y$$ and I want to find the maximum ...