My Math Forum How can I prove this inequality?

 Algebra Pre-Algebra and Basic Algebra Math Forum

 March 13th, 2019, 05:54 AM #1 Newbie   Joined: Mar 2019 From: USA Posts: 4 Thanks: 0 How can I prove this inequality? Let $a_1,a_2,...,a_n$ positive real numbers, where $n$ is an integer greater or equal to 3. Let $a_1+...+a_n=S_1$ and $a_1^2+...+a_n^2=S_2$. Prove that $$\sum_{i=1}^{n}\displaystyle \frac{S_1-a_i}{S_2-a_i^2}\geq n \cdot \displaystyle \frac{S_1}{S_2}$$ My trial. The inequality is in fact Chebyshevâ€™s inequality: $$\left( \sum_{i=1}^{n}\displaystyle \frac{S_1-a_i}{S_2-a_i^2}\right)\cdot\left( \sum_{i=1}^{n}\left(S_2-a_i^2\right)\right) \geq n \cdot \left( \sum_{i=1}^{n}\left(S_1-a_i\right)\right)$$ The only problem is that it is impossible to prove that the sequences are inverse ordered.. so another method is required. If the inequality is true, please help me with a proof. If the inequality is not true, please give an example.

 Tags inequality, prove

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post StillAlive Calculus 5 September 2nd, 2016 11:45 PM walter r Real Analysis 2 January 15th, 2014 02:24 AM Albert.Teng Algebra 2 July 15th, 2013 04:06 AM Albert.Teng Algebra 4 July 13th, 2012 11:13 AM Albert.Teng Algebra 7 April 17th, 2012 04:53 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top