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# Jensens Inequalty

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

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• #72386
Akash Arjun
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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:

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

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

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

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

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