Using rearrangement inequalities prove the following inequality:
Let $a,b,c$ be positive real numbers satisfying $abc=1$. Prove that
$$ab^2+bc^2+ca^2 \geq a+b+c.$$
Thanks 🙂
Inequality – Proving ab^2 + bc^2 + ca^2 ? a + b + c
inequality
inequality
Using rearrangement inequalities prove the following inequality:
Let $a,b,c$ be positive real numbers satisfying $abc=1$. Prove that
$$ab^2+bc^2+ca^2 \geq a+b+c.$$
Thanks 🙂
Best Answer
To reiterate my comment above, Rearrangement Inequality needs some ordering in the variables, and hence cannot be applied here as the given inequality isn't symmetric wrt a,b,c.
To give a simple proof by AM-GM just note that $$a^2c + a^2c + ab^2 \ge 3 \sqrt[3]{a^5b^2c^2} = 3a$$ so adding the two other similar inequalities we get $ab^2 + bc^2 + ca^2 \ge a + b + c$.