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!

Using Jensen's Inequality

  1. Apr 4, 2006 #1
    Let [itex]x \in \mathbb{R}^n[/itex] and

    [tex]u_0>0, \qquad \int\limits_\Omega u_0(x) dx =1, \qquad E(t)=\int\limits_\Omega u(x,t)u_0(x)dx[/tex]

    Im having trouble proving the following inequality

    [tex]\int\limits_\Omega \frac{u_0(x)}{(1+u(x,t))^2}dx \ge \dfrac{1}{(1+E)^2}. \qquad \hbox{(1)}[/tex]

    I know i have to use Jensen's inequality

    [tex] f\left(\frac{1}{|\Omega|}\int\limits_\Omega u dx \right) \le \frac{1}{|\Omega|}\int\limits_\Omega f(u) dx [/tex],

    where [itex]f(u)[/itex] is convex.

    But in order to use it to prove (1), I need to rewrite the left hand side of the equation or use a previous inequality right?

    There is where im stuck. Can anybody give me a sugestion pls?
     
    Last edited: Apr 4, 2006
  2. jcsd
  3. Apr 4, 2006 #2
    Is it just me or nobody can see the TeX?
     
  4. Apr 4, 2006 #3
    I cannot either.
     
  5. Apr 6, 2006 #4
    Well, first of all, it would be nice if someone tell me why the TeX doesnt work. Second of all, i got it, so nevermind.

    [tex]\int[/tex]

    [tex]\Omega[/tex]

    [tex]\omega[/tex]

    no \int???? nice.....
     
    Last edited: Apr 6, 2006
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?