Read about inequality | 23 Discussions | Page 1

  1. M

    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. J

    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

    A Deriving an inequality from a paper

    Hi, I am studying a paper by Yann Bugeaud: 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. W

    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. A

    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. E

    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. E

    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. G

    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. F

    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. dumbdumNotSmart

    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. ubergewehr273

    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. D

    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. D

    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. A

    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. N

    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. A

    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 =...
  20. Rectifier

    Inequality satisfaction

    Homework Statement $$x+\frac{16}{\sqrt{x}} \geq 12$$ How do I show that only x>0 satisfies the inequality above. Homework Equations The Attempt at a Solution I have not made a lot of progress here. I tried the following: $$x+\frac{16}{\sqrt{x}} - 12 \geq 0$$ I tried to multiply with $$...
  21. A

    Teacher told to set absolute value inequality to equal 0

    So I was helping my sister on homework and there was this problem: 2 abs(2x + 4) +1 > or equal to -3 teacher told her to ignore the -3 and just set it equal to zero. Soo should you? This question got me confused. can't you just go about solving, bringing the 1 to the left and then dividing by 2...
  22. E

    Prove that for a,b,c > 0, geometric mean <= arithmetic mean

    Homework Statement Let ## a,b,c \in \mathbb{R}^{+} ##. Prove that $$ \sqrt[3]{abc} \leq \frac{a+b+c}{3}. $$ Note: ## a,b,c ## can be expressed as ## a = r^3, b = s^3, c = t^3 ## for ## r,s,t > 0##. Homework Equations ## P(a,b,c): a,b,c \in \mathbb{R}^{+} ## ## Q(a,b,c): \sqrt[3]{abc} \leq...
  23. A

    |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