Inequality Definition and 1000 Threads

For all positive integers m > n, prove that : \operatorname{lcm}(m,n)+\operatorname{lcm}(m+1,n+1)>\frac{2mn}{\sqrt{m-n}}

a,b,c >0 , prove that : $(1+\dfrac {a}{b})(1+\dfrac {b}{c})(1+\dfrac {c}{a})\geq 2(1+\dfrac {a+b+c}{\sqrt[3]{abc}})$

I have worked my way though the proof of the Cauchy Schwarz inequality in Rudin but I am struggling to understand how one could have arrived at that proof in the first place. The essence of the proof is that this sum: ##\sum |B a_j - C b_j|^2## is shown to be equivalent to the following...

I am stuck at the inequality proof of this convext set problem. $\Omega = \{ \textbf{x} \in \mathbb{R}^2 | x_1^2 - x_2 \leq 6 \}$ The set should be a convex set, meaning for $\textbf{x}, \textbf{y} \in \mathbb{R}^2$ and $\theta \in [0,1]$, $\theta \textbf{x} + (1-\theta)\textbf{y}$ also belong...

Hi, so here is my question that I am totally stumped on. for all real values of x and y, show that |x|+|y|≥ √(x^2+y^2 ) and find the real values of x and y in which equality holds. I sort of thought I could do the second part, but it confuses me with two pronumerals and how to get rid...

Deduce from the simple estimate that if 1<\sqrt{3}<2, then 6<3^{\sqrt{3}}<7. Hi members of the forum, This problem says the resulting inequality may be deduced from the simple estimate, but I was unable to do so; could anyone shed some light on how to deduce the intended result? Thanks in...

Let C[-pi,pi] be the set of continuous function from [-pi,pi] to C. Endow this with usual inner product (<f,g>= integral from -pi to pi of f multiplied by g conjugate, and let ||.|| be the corresponding norm). Let h(n) be Fourier coefficent of fNow, |h(n)|<_ 1/2pi( ||f||.||e^int||) by schwarz...

$ T_n=\left(1-\dfrac{1}{3^2} \right)+\left(1-\dfrac{1}{5^2} \right)+\left(1-\dfrac{1}{7^2} \right)+\cdots+\left[1-\dfrac{1}{(2n+1)^2} \right]$ prove: $ \sqrt{\dfrac{n+1}{2n+1}}<T_n<\sqrt{\dfrac{2n+3}{3n+3}}$

Homework Statement Find constraints on a,b,c \in \mathbb{R} such that \forall w_1,w_2,w_3 \in \mathbb{C} , (1) x = |w_1|^2(1-c) + a|w_2|^2 + c|w_1+w_3|^2 + |w_3|^2(b-c) \ge 0 and (2) x=0 \Rightarrow w_1=w_2=w_3=0 .Homework Equations The Attempt at a Solution I believe the solution is...

Homework Statement |4 + 2r - r^2| <1 Homework Equations 4 + 2r - r^2 = (r - (1+ √5) ) (r - (1 - √5)) The Attempt at a Solution I tried to use the roots but no use. How should I proceed?

What is the best way to prove it?

Homework Statement The task is to find all solutions of the following inequality: x+3^x <4 But I was trying to find a solution for this problem in general: x+a^x < b Homework Equations n/a The Attempt at a Solution a^x < b-x \text{log}_a(a^x) < \text{log}_a...

let z,w be complex numbers. Prove: 2|z||w| <_ |z|^2 + |w|^2

is clausius inequality correct for negative temperature?, if you see the proof of it in positive temperature its not correct.

Homework Statement Show that 2-norm is less equal to 1-norm But I've found this proof http://img825.imageshack.us/img825/5451/capturaklt.jpg Which basically shows that if p=1 and q=2 then 1-norm is less equal than 2-norm, i.e. the opposite hypothesis Homework Equations NoneThe Attempt at...

Let a, b, c be the lengths of the sides of a triangle. Prove that: $\sqrt{a+b-c}$+$\sqrt{b+c-a}$+$\sqrt{c+a-b}\leq\sqrt{a}+\sqrt{b}+\sqrt{c}$

Homework Statement Question 1: You find an old map revealing a treasure hidden on a small island. The treasure was buried in the following way: the island has one tree and two rocks, one small one and one large one. Walk from the tree to the small rock, turn 90 to the left and walk the same...

I have proven to n=3 an inequality that seems useful. x1n+...+xnn≥nx1...xn for all positive x. I'm sure this has been proven before. I'm not quite sure how to extend it from n=3 to for all n. I'm thinking induction, but that has proven challenging. Any hints?

Homework Statement Prove that: Homework Equations \sum_{k=1}^{n}x_{k}^{2}\geq \frac{1}{n}\left ( \sum_{k=1}^{n}x_{k} \right )^{2} The Attempt at a Solution I am not sure what to do to be honest. But it looks like the Cauchy–Schwarz inequality to me.

Homework Statement these two functions will give the same Fourier series? because when I write the graph they look the same? Homework Equations The Attempt at a Solution in the picture thank you

Homework Statement Question 3 here: http://www.stat.washington.edu/peter/395/samplemidterm.pdf Solution to it here: http://www.stat.washington.edu/peter/395/prmt.sln/sln.html By the way, I could use help soon since this is the practice exam for an exam I'm taking 7 hours from now...

Homework Statement Let E have finite outer measure. Show that if E is not measurable, then there is an open set O containing E that has finite measure and for which m*(O~E) > m*(O) - m*(E) Homework Equations The Attempt at a Solution This is what I did... m^*(O) = m^*((O \cap E^c) \cup...

$ a,b,c \in\mathbb{R}^+ , \,\,\text{Prove:}$ $\dfrac {c}{a+b} +\dfrac {a}{b+c} +\dfrac {b}{c+a}\geq \dfrac {3}{2}$

Homework Statement 3 - 3^(x+1) < 5^x - 15^x 3(1-3^x) < 5^x(1-3^x) Do I have to impose 1-3^x > 0 ? It results x<0 and x>log(5,3) but book has written 0 < x < log(5,3) where did I wrong ?

a,b,c,d > 0 , please prove : $ \sqrt{a+b+c+d} \geq \dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}{2}$

Homework Statement I need some help setting up this inequality: How accurate do the sides of a cube have to be measured if the volume of the cube has to be within 1% of 216 cm^3 Not very good with word problems and for some reason this course never deals with them until now? And this is...

Homework Statement The range of k for which the inequality ##k\cos^2x-k\cos x+1≥0## for all x, is a. k<-1/2 b. k>4 c. -1/2≤k≤4 d. -1/2≤k≤2Homework Equations The Attempt at a Solution I am not sure about how to begin with this. This seems to me a quadratic in cos(x) and here, the discriminant...

Problem: Let x and y be positive real numbers satisfying the inequality $\displaystyle x^3+y^3\le x-y$. Prove that $\displaystyle x^2+y^2\le 1$ . Hi all, I'm at my wit's end to prove the question as stated above, and I know it's obvious that $\displaystyle x-y>0$ and $\displaystyle...

Homework Statement Let ##a## and ##b## real numbers such that ##a>b>0##. Determine the least possible value of ##a+ \frac{1}{b(a-b)}## I took this example from page 3 of this paper Homework Equations In the article previously linked, explaining the example, the author writes...

Homework Statement If g(x)\ge 0, then for any constant ##c>0, r>0##: P(g(X)\ge c)\le \frac{E((g(X))^r)}{c^r} Homework Equations I know that E(g(X))=\int_0^\infty g(x)f(x)dx if g(x)\ge 0 where ##f(x)## is the pdf of ##X##. The Attempt at a Solution I tried following a similar...

I imagine that some topics and questions keep reappearing since it is hard to track through all past posts even with the query tool. So apologies if this has been covered before (as it probably has). I just want to check my intuitive understanding of the Bell experiment, having heard an...

Let f(x) \in C[a,b] and let f(x)>0 on [a,b]. Prove that \exp \Big(\frac{1}{b-a}\int_a^b \ln f(x) dx \Big)\leq \frac{1}{b-a}\int_a^b f(x) dx I have learned Gronwall's Inequality and Jensen's Inequality(and inequality deduced from it like Cauchy Schwarz Inequality) but i couldn't use them to...

Hey! I have this: 2(√(1-a^2 ))+ 2a How to determine the maximum value of this? I think good for this is Inequality of arithmetic and geometric means, but I don't know how use this, because I don't calculate with this yet. So, have you got any ideas? Poor Czech Numeriprimi... If you...

Hi, can someone give me a refference for the FKG inequality? that is, let f,g be two nondecreasing and nonegative measurable functions on a probability space.Then ∫fgdμ≥(∫fdμ)(∫gdμ)

Hi everybody, while doing a complex analysis exercise, i came to a strange inequality which i don't know how to interpretate. Suppose you have a sequence $\{a_j\}$ of positive real number. Let $\rho$ a positive real number. The inequality i found after some calculation is...

Hi all, I've been having a hard time trying to solve the following inequality: Prove that $\displaystyle \left(\log_{24}(48) \right)^2+\displaystyle \left(\log_{12}(54) \right)^2 >4$ I've tried to change the bases to base-10 log and relating all the figures (12, 24, 48, and 54) in terms of 2...

1. Another approach to the proof of the Cauchy-Shwarz Inequality is suggested by figure 16 (sorry, I don't have the image), which shows that, in ℝ2 or ℝ3, llproj_u{}vll ≤ llvll. Show that this inequality is equivalent to the Cauchy-Schwartz Inequality. 2. Cauchy-Schawrtz Inequality: lu • vl...

Homework Statement For every x in the interval [0,1] show that:j \frac{1}{4}x+1\leq\sqrt[3]{1+x}\leq\frac{1}{3}x+1 The Attempt at a Solution Well i subtracted 1 from all sides and divided by x and I got: \frac{1}{4}\leq\frac{\sqrt[3]{1+x}-1}{x}\leq\frac{1}{3} But now I need to find a...

Homework Statement \frac{2^{x+1}-3}{2^{x}-4}\leq1 Homework Equations The Attempt at a Solution When I go through it, I keep getting 2^{x}\leq-1 I don't think the answer is suppose to be complex... let me show you my work in a file. It's extremely embarrassing that I don't...

Hello there, Im studying QM with Shankar's book. I'm wrestling myself trough the linear algebra now and I have some questiosn. Let me start with this one: I have absolutely no idea where this is coming from or what does it mean. I don't know how to multiply a ket with an inner...

I am trying to solve the following inequality: x3 <= 7x3 - 24x2 + 6x - 10 I have worked it out as follows: 0 <= 6x3 - 24x2 + 6x - 10 10 <= 6x3 - 24x2 + 6x 10 <= 6x(x2 - 4x + 1) At this point, I'm not sure how to proceed and I'm not sure if the factoring on the last step was helpful. Any...

Homework Statement Solve the inequality -9 < 1/x A simple inequality, I can see the solution is just x < -1/9 but I can't prove it at all. The Attempt at a Solution -9 < 1/x -9x < 1 x > -1/9 Any helpful rules I am forgetting about inequalities? This was a problem in a review...

$$ \frac{\pi^2}{3} + 4\sum_{n = 1}^{\infty}\frac{(-1)^n}{n^2}\cos n\theta. $$ Let's look at $f\left(\frac{\pi}{2}\right) = \frac{\pi^2}{4}$. \begin{alignat*}{3} \frac{\pi^2}{3} + \sum_{n = 1}^{\infty}\frac{(-1)^n}{n^2}\cos\frac{n\pi}{2} & = & \frac{\pi^2}{3} + \left(\frac{-1}{4} + \frac{1}{16} -...

Hello I have a worked example where I have to maximize a function with an inequality constraint. The problem is worked out below. https://zgqqmw.sn2.livefilestore.com/y1pLc13HVWpA9dATZEzikySeSMBN2hn1mJCw71rJ5vvUJcr9W7KBPFkOz7HQEppa6EPbLi5yyAwDagh3ezF_7eyVL6tBK7q6ise/maxProbem.png?psid=1 I...

I just can't figure this out one question from my review test. I don't know hot to express it graphically or algebraically.

Homework Statement If n is a positive integer, prove that 2^n > 1+n\sqrt{2^{n-1}} Homework Equations The Attempt at a Solution I am thinking of applying AM GM HM inequality. But which numbers should I take to arrive at this inequality?

Solve the rational inequality (a-5)/(a+2) < -1.This is what I got so far: a-5/a+2 = -1 a-5= -a-2 0= -2a+3 subtract 3 from both sides: -3=-2a Divide by -2 3/2=a I know that the answer is (-2, 3/2), but I'm not sure where the -2 in the answer comes from. Thanks!

Homework Statement Let V be a real inner product space, and let v1, v2, ... , vk be a set of orthonormal vectors. Prove Ʃ (from j=1 to k)|<x,vj><y,vj>| ≤ ||x|| ||y|| When is there equality? Homework Equations The Attempt at a Solution I've tried using the two inequalities given to us in...

Homework Statement If a,b,c are the positive real numbers, prove that a^2(1+b^2)+b^2(1+c^2)+c^2(1+a^2) \geq 6abc Homework Equations The Attempt at a Solution With a little simplification L.H.S = (a^2+b^2+c^2)+(a^2b^2+b^2c^2+c^2a^2) Using A.M>=G.M \dfrac{a^2+b^2+c^2}{3} \geq...

Prove that if ##m > 1## such that there exists a ##c > 1## that satisfies $$cm < m^c$$ then for any ##k > c## $$km < m^k$$ holds. Prove this without using logarithms or exponents or calculus. Basically using the properties of real numbers to prove this. One attempt I have tried, but didn't...