Problem Prove that – P.S. It is straightforward holder, but what about a solution by Cauchy or AM-GM ? Solution By, AM-GM we have , Write up analogous inequalities, and add to get – which re-arranges to the desired question 🙂

Problem Prove that for positive reals with sum 1 , Proof Homogenize the inequality to – Since the inequality is cyclic, let Substitute – and let , The inequality by the above substitution is transformed to – By AM-GM inequality , therefore the inequality is true with equality which gives us the equality cases –

Problem Let be nonnegative numbers, no two of which are zero. Prove that: Proof It is equivalent to – Therefore , Let , So, and also note that – thus the inequality is proved.

Problem For positive reals prove that – Proof Let then the inequality is equivalent to – Obviously we have that – (since ) Also by AM-GM , multiplying these two we get the result with equality holding when

Problem Let . Prove that the following inequality holds –    My Solution  Set , and similarly that for any , where, we thus have that –  Thus, Therefore we have ,  Thus it is sufficient to prove that – By Holders’ Inequality , Equality occurs when Hence our proof is completed.

Problem 1 Show that positive reals such that,, Problem 2 Prove that the following inequalities hold for all  a,b,c – sides of a triangle Problem 3 Prove that for positive reals positive reals, Problem 4.  Prove that for positive reals a,b,c the following inequality holds good – Problem 5. Prove that for positive reals the […]

Problem Let be positive real numbers. Prove that Proof Let therefore , we have to prove that – which is true by C-S 🙂 Equality hold for $x=y=z$ .

Problem Let  . Prove that: Proof By C-S , Thus it is sufficient to prove that (after removing in the denominators)- Let , The inequality is transformed into – By AM-GM , Thus it is sufficient to prove that – By Schur’s Inequality we know that – Thus it is left to prove that – […]

Problem If where . Prove that: Proof By AM-GM, Thus , Set : the inequality beomes – the first inequality is C-S, the second inequality is equivalent to – which is true by AM-GM. Equality occurs for or

1) For positive reals with sum 1 , show that – 2) For positive reals with sum 3 , and the sum of any 2 greater than 1, show that – 3) For positive reals with sum 3 , show that – 4) For positive reals such that – Prove that – Proof For the first […]