Jensens Inequalty

Home Forums Math Olympiad, I.S.I., C.M.I. Entrance Jensens Inequalty

Tagged: , ,

Viewing 2 posts - 1 through 2 (of 2 total)
  • Author
    Posts
  • #72386
    Akash Arjun
    Participant

    help me solve this problem

    #72421
    Saumik Karfa
    Moderator

    take $f(x)=\frac {x}{1-x}$

    the graph shows that the function is concave within $(1,\infty)$.

    take $\lambda_i =\frac 1n$

    Therefore, by jensen’s inequality:

    $\Rightarrow \displaystyle\sum_{i=1}^n \lambda_i f(x_i) \leq f\bigg(\displaystyle\sum_{i=1}^n \lambda_i x_i\bigg)$

    $\Rightarrow \frac 1n \cdot \displaystyle\sum_{i=1}^n \frac{x_i}{1-x_i} \leq f\bigg(\frac{\displaystyle\sum_{i=1}^n  x_i}{n}\bigg)$

    $\Rightarrow \frac 1n \cdot \displaystyle\sum_{i=1}^n \frac{x_i}{1-x_i} \leq \frac{\frac{\displaystyle\sum_{i=1}^n  x_i}{n}}{1-\frac{\displaystyle\sum_{i=1}^n  x_i}{n}}$

    $\Rightarrow \displaystyle\sum_{i=1}^n \frac{x_i}{1-x_i} \leq n \cdot \frac{\displaystyle\sum_{i=1}^n  x_i}{n-\displaystyle\sum_{i=1}^n  x_i} $

Viewing 2 posts - 1 through 2 (of 2 total)
  • You must be logged in to reply to this topic.