Inequality — 400 discussions

The Rearrangement Inequality states that for two sequences ##{a_i}## and ##{b_i}##, the sum ##S_n = \sum_{i=1}^n a_ib_i## is maximized if ##a_i## and ##b_i## are similarly arranged. That is, big numbers are paired with big numbers and small numbers are paired with small numbers. The question...

Mod note: Moved from a technical math section, so missing the homework template. This is for an Intro to Analysis course. It's been a very long time since I've taken a math course, so I do not remember much of anything. ============= Here is the problem:For the inequality below, find all values...

Homework Statement By choosing the correct vector b in the Schwarz inequality, prove that (a1 + ... + an)^2 =< n(a1^2 + ... +an^2) Homework Equations Schwarz inequality The Attempt at a Solution since the answer key says that a1 = a2 = ... = an, i tried plugging in values, but i am not...

Homework Statement Find coefficients a,b>0 such that a||x||∞≤||x||≤b||x||∞.Homework EquationsThe Attempt at a Solution No idea how to get started. Help will be appreciated.

Prove, that \[3(x^2y^2+x^2z^2+y^2z^2)-2xyz(x+y+z) \leq 3,\: \: \: \forall x,y,z \in \left [ 0;1 \right ]\]

Hi! I would like to know if my solution for this inequality is corect or not:

Question 1: (a) Show that the complex number i is a root of the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0 (b) Find the other roots of this equation Work: Well, I thought about factoring the equation into (x^2 + ...) (x^2+...) but I couldn't do it. Is there a method for that? Anyways the reason I...

hi! i need help for this inequality 1. ##a\in\mathbb{N}*~and~ \frac{a}{a+1}<\frac{a+1}{a+2}<\frac{a+2}{a+3}## show that : ##\frac{1}{2}*\frac{4}{5}*...*\frac{2005}{2006}*\frac{2008}{2009}<\frac{1}{12}## Here i have stoped. Please tell me if is corect what i have done so far and how to continue ...

Homework Statement Let a,b and c be lengths of sides in a triangle, show that √(a+b-c)+√(a-b+c)+√(-a+b+c)≤√a+√b+√c The Attempt at a Solution With Ravi-transformation the expressions can be written as √(2x)+√(2y)+√(2z)≤√(x+y)+√(y+z)+√(x+z). Im stuck with this inequality. Can´t find a way to...

$x,y,z,w>0$ prove: $(1+x)(1+y)(1+z)(1+w)\geq (\sqrt[3]{1+xyz}\,\,\,)(\sqrt[3]{1+yzw}\,\,\,)(\sqrt[3]{1+zwx}\,\,\,)(\sqrt[3]{1+wxy}\,\,\,)$

Let $a,\,b,\,c$ and $d$ be non-negative real numbers such that $a + b + c + d = 2$. Prove that $ab(a^2+ b^2 + c^2) + bc(b^2+ c^2+ d^2) + cd(c^2+ d^2+ a^2) + da(d^2+ a^2+ b^2) ≤ 2$.

Currently revising for my A-Level maths (UK), there is unfortunately no key in the book; Given the triangle with sides a,b,c respectively and the area S, show that ab+bc+ca => 4*sqrt(3)*S I have tried using the Ravi transformation without luck, any takers?

Let $a,\,b$ and $c$ be non-negative real numbers such that $a+b+c=1$. Prove that $$\sum_{cyclic}\sqrt{4a+1} \ge \sqrt{5}+2$$.

I was talking to my professor and she said that $(ln n)^a < n$ for all values of $a$. Is this true or was I misunderstanding?

A few weeks ago I created a discussion titled "How does Bell's inequalities rule out realism." Essentially my question was pertaining to how does removing realism retain locality and not violet Bell's inequality. Someone answered with the this, I'm not really happy with this explanation, but...

Let $a,\,b$ and $c$ be the sides of a triangle and $x,\,y$ and $z$ are real numbers such that $x+ y+ z = 0$. Prove that $a^2yz +b^2xz+c^2xy\le 0$.

Prove $$\frac{10}{\sqrt{11^{11}}}+\frac{11}{\sqrt{12^{12}}}+\cdots+\frac{2015}{\sqrt{2016^{2016}}}\gt \frac{1}{10!}-\frac{1}{2016!}$$

I have this inequality: $$ \frac{n^3}{n^5 + 4n + 1} \le \frac{1}{n^2}$$ for all $n \ge 1$ I get that $$ \frac{1}{n^5 + 4n + 1} \le \frac{1}{n^2}$$ but how do I guarantee that when $n^3$ is in the numerator, this inequality holds? Is this for any numerator greater than 1? Also, why must $n$...

Prove $$\tan x+\tan y+\tan z\ge \sin x \sec y+\sin y\sec z+\sin z \sec x$$ for $x,\,y,\,z\in \left(0,\,\dfrac{\pi}{2}\right)$.

Given that $a,\,b$ and $c$ are positive real numbers. Prove that $$\frac{a^3+b^3+c^3}{3abc}+\frac{8abc}{(a+b)(b+c)(c+a)}\ge 2$$.

Given that $x,\,y$ and $z$ are positive real numbers such that $xy + yz + zx = 3xyz.$ Prove that $x^2y + y^2z + z^2x\ge 2(x + y + z) − 3$.

I have $$-1 \le \cos\left({2x}\right) \le 1 $$ If everything is squared, it goes to $$0 \le \cos^2\left({2x}\right) \le 1 $$ and I'm not sure how $(-1)^2$ turns into $0$

I understand Bell's inequality, and I can see how removing locality can produce the observed statistical correlations. However something that I often read is that eradicating realism can also generate the correlation observed in entanglement. I don't see how a particle not having definite...

prove the following $a,b,c,d>0$ prove :$\sqrt {b^2+c^2}+\sqrt {a^2+c^2+d^2+2cd}>\sqrt {a^2+b^2+d^2+2ab}$

Given that ##v_0## and ##v_f## are positive variables related by the equation where ##g## and ##\alpha## are positive constants. Can you show that ##v_f<v_0## for all positive values of ##v_0## using a non-graphical method? Physically, ##v_0## and ##v_f## are the initial and final speeds (at...

In the derivation of triangle inequality |(x,y)| \leq ||x|| ||y|| one use some ##z=x-ty## where ##t## is real number. And then from ##(z,z) \geq 0## one gets quadratic inequality ||x||^2+||y||^2t^2-2tRe(x,y) \geq 0 And from here they said that discriminant of quadratic equation D=4(Re(x,y))^2-4...

Let $a,\,b$ and $c$ be real numbers such that $a+b+c=1$, prove that $$\frac{1}{3^{a+1}}+\frac{1}{3^{b+1}}+\frac{1}{3^{c+1}}\ge \left(\frac{a}{3^a}+\frac{b}{3^b}+\frac{c}{3^c}\right)$$.

Let $a,\,b$ and $c$ be positive real numbers, prove that $$\frac{a}{2a+b+c}+\frac{b}{a+2b+c}+\frac{c}{a+b+2c}\le \frac{3}{4}$$.

Homework Statement part c Homework Equations The Attempt at a Solution Jm+2=m+2-1/m+2 Jm=m+1/m+2 Jm hence Jm+2<Jm should i expend Jm+2 Jm+1 Jm to the term J0 then compare them? why the inequality is <= but not <? should i use M.I to proof it??[/B]

For positive reals $a,\,b,\,c$, prove that $$\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}\gt 2$$.

Homework Statement If a,b,c are positive real numbers such that ##{loga}/(b-c) = {logb}/(c-a)={logc}/(a-b)## then prove that (a) ##a^{b+c} + b^{c+a} + c^{a+b} >= 3## (b) ##a^a + b^b + c^c >=3## Homework Equations A.M ##>=## G.M The Attempt at a Solution Using the above inequation, I am able...

Let the reals $a, b, c∈(1,\,∞)$ with $a + b + c = 9$. Prove the following inequality holds: $\sqrt{(\log_3a^b +\log_3a^c)}+\sqrt{(\log_3b^c +\log_3b^a)}+\sqrt{(\log_3c^a +\log_3c^b)}\le 3\sqrt{6}$.

Prove that $x^2 + y^2+ z^2\le xyz + 2$ where the reals $x,\,y,\, z\in [0,1]$.

Homework Statement Solution set of the inequality (cot-1(x))2 -(5 cot-1(x)) +6 >0 is? Homework EquationsThe Attempt at a Solution Subs cot-1(x)=y We get a quadratic inequality in y. y2-5y+6>0 (y-2)(y-3)>0 Using the wavy curve method, the solution set is...

Given that a, b, c are positive reals and not all equal, show that $\dfrac{a^3 + b^3 + c^3}{a^3b^3c^3}\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$

I am trying to solve this inequality without using a factor table. The problem $$ \frac{x+4}{x-1} > 0 $$ The attempt at a solution As I can see ##x \neq 1##. I want to muliply both sides of the expression with x-1 to get rid of it, from the fraction. But before that, I have to consider two...

trying to soak in this paper. https://arxiv.org/abs/gr-qc/0205035 The following statement is found early on: "The violation of Bell’s inequality proves that any realistic interpretation of quantum theory needs a preferred frame." Whether anyone agrees or disagrees I'd appreciate a sketch of...

What is the smallest n such that \lg {n\choose0.15n} + 0.15n \geq {112}

I have attached two images from my textbook one of which is a diagram and the other a paragraph with which I am having problems. The last sentence mentions that due to violation of 2nd law we cannot convert all the heat to work in this thermodynamic cycle. However what is preventing the carnot...

Prove that for all real numbers $x$, we have $$\left(2^{\sin x}+2^{\cos x}\right)^2\ge2^{2-\sqrt{2}}$$.

For real numbers $$0\lt x\lt \frac{\pi}{2}$$, prove that $\cos^2 x \cot x+\sin^2 x \tan x\ge 1$.

Hello! (Wave) Suppose that $k$ football matches are being done and a bet consists of the prediction of the result of each match, where the result can be 1 if the first group wins, 2 if the second group wins, or 0 if we have tie. So a bet is an element of $\{0,1,2 \}^k$. I want to show that...

Hello, Solve inequality x^2+2ix+3<0 where i^2=-1

Prove, with no knowledge of the decimal value of $\pi$ should be assumed or used that $$1\lt \int_{3}^{5} \frac{1}{\sqrt{-x^2+8x-12}}\,dx \lt \frac{2\sqrt{3}}{3}$$.

Prove that $$\frac{\sin^3 x}{(1+\sin^2 x)^2}+\frac{\cos^3 x}{(1+\cos^2 x)^2}\lt \frac{3\sqrt{3}}{16}$$ holds for all real $x$.

if $a,b,c$ are positive real numbers show that $(1+a)^7(1+b)^7(1+c)^7 > 7^7 a^4b^4c^4$

Prove that, for all real $a,\,b,\,c$ such that $a+b+c=3$, the following inequality holds: $\log_3(1+a+b)\log_3(1+b+c)\log_3(1+c+a)\le 1$

Prove $$\frac{a-\sqrt{bc}}{a+2b+2c}+\frac{b-\sqrt{ca}}{b+2c+2a}+\frac{c-\sqrt{ab}}{c+2a+2b}\ge 0$$ holds for all positive real $a,\,b$ and $c$.

Hey! :o I want to show that if $G$ is finite and $f:G\rightarrow H$ is a group epimorphism then $|\text{Syl}_p(G)|\geq |\text{Syl}_p(H)|$. I have done the following: Since $f:G\rightarrow H$ is a group epimorphism, from the first isomorphism theorem we have that $H$ is isomorphism to $G/\ker...

$n\in N,n\geq 2$ prove: $ \sum_{1}^{n}(\dfrac{1}{2n-1}-\dfrac{1}{2n})>\dfrac {2n}{3n+1}$