Results 1 to 5 of 5

Math Help - Inequality

  1. #1
    PQR
    PQR is offline
    Newbie
    Joined
    Aug 2009
    Posts
    19

    Inequality

    For any a, b, c > 0 with abc = 1, prove that

    \frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3  (a+b)} \ge \frac{3}{2}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member bigwave's Avatar
    Joined
    Nov 2009
    From
    honolulu
    Posts
    580
    with abc = 1
    then a=1\ b=1 \ and \ c=1

    therefore

     <br />
\frac{1}{1^3(1+1)} + \frac{1}{1^3(1+1)} + \frac{1}{1^3(1+1)} \geq \frac{3}{2}<br />
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    11
    you don't have a proof, that's a particular case you just took.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    PQR
    PQR is offline
    Newbie
    Joined
    Aug 2009
    Posts
    19
    Quote Originally Posted by bigwave View Post
    with abc = 1
    then a=1\ b=1 \ and \ c=1

    therefore

     <br />
\frac{1}{1^3(1+1)} + \frac{1}{1^3(1+1)} + \frac{1}{1^3(1+1)} \geq \frac{3}{2}<br />
    If it only was that easy!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    11
    a(b+c)+b(a+c)+c(a+b)=2(ab+bc+ac)=2\left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right), and ab+bc+ac\ge 3\sqrt[3]{a^{2}b^{2}c^{2}}=3, now by Cauchy - Schwarz we have <br />
\left( \frac{1}{a^{3}(b+c)}+\frac{1}{b^{3}(a+c)}+\frac{1}  {c^{3}(a+b)} \right)2(ab+bc+ac)\ge \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right)^{2}, finally \left( \frac{1}{a^{3}(b+c)}+\frac{1}{b^{3}(a+c)}+\frac{1}  {c^{3}(a+b)} \right)\ge \frac{ab+bc+ac}{2}\ge \frac{3}{2}.\quad\blacksquare
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: January 11th 2011, 08:20 PM
  2. Replies: 3
    Last Post: December 12th 2010, 01:16 PM
  3. inequality
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: July 26th 2010, 04:34 AM
  4. inequality
    Posted in the Algebra Forum
    Replies: 4
    Last Post: July 24th 2010, 12:08 PM
  5. Inequality
    Posted in the Algebra Forum
    Replies: 0
    Last Post: October 8th 2009, 03:06 AM

Search Tags


/mathhelpforum @mathhelpforum