1. ### Find the set of points that satisfy:|z|^2 + |z - 2*i|^2 =< 10

Hello everyone, I've been struggling quite a bit with this problem, since I'm not sure how to approach it correctly. The inequality form reminds me of the equation of a circle (x^2 + y^2 = r^2), but I have no idea how to be sure about it. Would it help just to simplify the inequality in terms...
2. ### I Need help with a proof involving points on a quadratic

Summary: Given three points on a positive definite quadratic line, I need to prove that the middle point is never higher than at least one of the other two. I am struggling to write a proof down for something. It's obvious when looking at it graphically, but I don't know how to write the...
3. ### A Deriving an inequality from a paper

Hi, I am studying a paper by Yann Bugeaud: http://irma.math.unistra.fr/~bugeaud/travaux/ConfMumbaidef.pdf on page 13 there is an inequality (16) as given below- which is obtained from - , on page 12. How the inequality (16) is derived? I couldn't figure it out. However one of my...
4. ### I Joint PDF and its marginals

Hi all, I was wondering if there exist any theorems that allow one to relate any joint distribution to its marginals in the form of an inequality, whether or not ##X,Y## are independent. For example, is it possible to make a general statement like this? $$f_{XY}(x,y) \geq f_X (x) f_Y(y)$$...
5. ### Boundary of an inequality

Homework Statement I would like to know how the boundary of the inequality change when the origin of the coordinate system changes. Homework Equations The original inequality is[/B] $$r_0 \le x^2+y^2+z^2 \le R^2$$ I would like to know the boundary of the following term, considering the...
6. ### I Spivak's proof of Cauchy Schwarz

I was browsing through Spivak's Calculus book and found in a problem a very simple way to prove the cauchy schwarz inequality. Basically he tells to substitute x=xᵢ/[√(x₁²+x₂²)] and similarly for y (i=1 and 2), put into x^2 + y^2 >= 2xy. Add the two cases and we get the result. The problem is...
7. S

### B Surjective/injective operators

Hi, I found in Kreyszig that if for any ##x_1\ and\ x_2\ \in \mathscr{D}(T)## then an injective operator gives: ##x_1 \ne x_2 \rightarrow Tx_1 \ne Tx_2 ## and ##x_1 = x_2 \rightarrow Tx_1 = Tx_2 ## If one has an operator T, is there an inequality or equality one can deduce from this...
8. ### Bounding p-norm expression

problem statement: need to show: ||w||_p^2+||u||_p^2-2(||u||_p^p)^{\frac{2}{p}-1}\Sigma_i(u(i)^{p-1}w(i)) can be bounded as a function of ||w-u||_p^2 where p\in[2,\infty) work done: the expressions are equal for p=2, and i suspect that...
9. B

### B Inequality of absolute values

Hi there, I'm having trouble understanding this math problem: |x| + |x-2| = 2 The answer says its: 0<=x<=2 I understand you need different "cases" in order to solve this. For example, cases for when x is less than 0, when x-2 is less than 0, etc. Thanks, blueblast
10. H

### I Bell's Inequality is only valid for non-negative numbers

The Bell Inequality tests are only valid for positive numbers, which is reasonable because counts and probabilities cannot be negative. CHSH generates a negative number, which means CHSH experiments are invalid. Bell's Inequality can be violated by having a negative value. For example...
11. ### I Bell test where observers never were in a common light cone

Hi. I wonder if following thought experiment (which is most probably impossible to be put into practice) could have any implications concerning interpretations of QM. Consider five parties A, B, C, D and E, lined up in that order and with no relevant relative motion. No pair of them have ever...
12. ### A question involving inequality

Homework Statement If a,b,c,d,e>1 then prove that a^2/(c-1)+b^2/(d-1)+c^2/(e-1)+d^2/(a-1)+e^2/(b-1)=>20 The Attempt at a Solution Given a,b,c,d,e are roots of a polynomial equation of a degree 5 then x^2/(x-1)+x^2/(x-1)+x^2/(x-1)+x^2/(x-1)+x^2/(x-1)=>20 5 x^2/(x-1)=>20 x^2/(x-1)=>4...
13. ### Complex Conjugate Inequality Proof

Homework Statement $$\left | \frac{z}{\left | z \right |} + \frac{w}{\left | w \right |} \right |\left ( \left | z \right | +\left | w \right | \right )\leq 2\left | z+w \right |$$ Where z and w are complex numbers not equal to zero. 2.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$...
14. ### A problem about logarithmic inequality

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...
15. ### Discover the form of real solution set

Homework Statement ##|4^{3x}-2^{4x+2}*3^{x+1}+20*12^x*3^x| > 8*6^x(8^{x-1}+6^x)## For some numbers ##a, b, c, d## such that ##-\infty < a <b < c <d < +\infty ## the real solution set to the given inequality is of the form ##(-\infty, a] \cup [b, c] \cup [d, +\infty)## Prove it by arriving at...
16. ### Find the sets of real solutions

[b[1. Homework Statement [/b] ##|4^{3x}-2^{4x+2}*3^{x+1}+20*12^x*3^x| \ge 8*6^x(8^{x-1}+6^x)## The sets containing the real solutions for some numbers ##a, b, c, d,## such that ##-\infty < a < b < c < d < +\infty## is of the form ##(-\infty, a] \cup [b, c] \cup [d, +\infty)##. Prove it by...
17. ### Stuck on Proof by induction of 2^n>n^3 for all n>=10

Homework Statement Using the principle of mathematical induction, prove that for all n>=10, 2^n>n^3 Homework Equations 2^(n+1) = 2(2^n) (n+1)^3 = n^3 + 3n^2 + 3n +1 The Attempt at a Solution i) (Base case) Statement is true for n=10 ii)(inductive step) Suppose 2^n > n^3 for some integer >=...
18. ### Proof sin x < x for all x>0.

Hello all, I want to prove the following inequality. sin(x)<x for all x>0. Now I figured that I put a function f(x)=x-sin(x), and show that it is increasing for all x>0. But this alone doesn't prove it. I need to show we have inequality from the start. I can't show that lim f(x) as x->0 is...
19. ### Adding increasing fractions without averaging numerators

I'm interested in the following inequality (which may or may not be true) Theorem 1: ##( \sum_{i=1}^n \frac{a_i} {n}\ )( \sum_{i=1}^n \frac{1} {b_i}\ ) > \sum_{i=1}^n \frac{a_i} {b_i}\ ## Where ##n ≥ 2, a_1 < a_2 < ... < a_n## and ##b_1 < b_2 < ... < b_n##. My attempt at a proof: 1) When n =...

23. ### |1/(2+a)|<1, a?

Hi I'm trying to solve this inequality |1/(2+a)| < 1. 1/(2+a) < 1 ∨ 1/(2+a) > -1 1 < 2+a a > -1 and 1 > -2-a 3 > -a a > -3 I know that the boundaries are -∞ < a < -3 ∨ -1 < a < ∞ What have I done wrong? thanks in advance