1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Conditions on Complex Inequality

  1. Mar 12, 2013 #1
    1. The problem statement, all variables and given/known data
    Find constraints on [itex] a,b,c \in \mathbb{R} [/itex] such that [itex] \forall w_1,w_2,w_3 \in \mathbb{C} [/itex],

    (1) [itex] x = |w_1|^2(1-c) + a|w_2|^2 + c|w_1+w_3|^2 + |w_3|^2(b-c) \ge 0 [/itex] and

    (2) [itex] x=0 \Rightarrow w_1=w_2=w_3=0 [/itex].


    2. Relevant equations



    3. The attempt at a solution
    I believe the solution is [itex] a>0, c = \mathbb{R_{+}} \setminus \{1\}, b>c [/itex], but I am not sure if there are stronger bounds. How can I know for sure?
     
    Last edited: Mar 12, 2013
  2. jcsd
  3. Mar 12, 2013 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You seem to have made sure that each term is not negative - which certainly guarantees the sum is not negative. Though note: c ≤ 1 means that (1 - c) ≥ 0 doesn't it so do you need the lower bound on c?

    Is it possible for some terms to be negative and still end up with x not negative?
    i.e. if you allow the constraints on a,b,c, to depend on the |wi| etc. (where i is in {1,2,3}.)

    Though I'm guessing that conditions (1) and (2) have to be satisfied simultaneously ... i.e. need to see your reasoning.

    note: ##w_1=w_2=w_3=0## certainly means the ##x=0## no matter what a,b,c are. But that's not what condition (2) says is it?
     
    Last edited: Mar 12, 2013
  4. Mar 12, 2013 #3
    Well as you note I chose a and b so that the terms involving them are positive. c can be any positive real except 1. It can be positive and less than 1 obviously. It can't equal 1 because then w_1 can be any complex number. It can be greater than 1 because the sum of the first and third terms of x are then positive. If c is not bounded below by 0, then for large w_3, the sum of the first and third terms of x are negative.

    As to your note, no, that's not what condition (2) says.
     
  5. Mar 12, 2013 #4

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    OK - if every term has to be, individually, non-negative (and that "for all" kinda suggests this) then I don't see tighter constraints. There is this feeling there is something left out isn't there - but that's the only thing that springs to my mind.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Conditions on Complex Inequality
  1. Complex inequality (Replies: 7)

  2. Complex inequality (Replies: 4)

Loading...