Question

Prove that $\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, where $a,b,c$ are different positive real numbers.

Answer 1

Using A.M and G.M inequalities

$\frac{ab}{c^3}+\frac{bc}{a^3}>2\frac{b}{ac}\dots \dots \left(1\right)$

$\frac{bc}{a^3}+\frac{ca}{b^3}>2\frac{c}{ab}\dots \dots \left(2\right)$

$\frac{ca}{b^3}+\frac{ab}{c^3}>2\frac{a}{bc}\dots \dots \left(3\right)$

Adding all three equations, we get

$\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}>\frac{b}{ac}+\frac{c}{ab}+\frac{a}{bc}\dots \dots \left(4\right)$

Similarly, $\frac{b}{ac}+\frac{c}{ab}>\frac{1}{a}$

$\frac{c}{ab}+\frac{a}{bc}>\frac{1}{b}$

$\frac{a}{bc}+\frac{b}{ac}>\frac{1}{c}$

From above equations, we get

$\frac{b}{ac}+\frac{a}{bc}+\frac{c}{ab}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\dots \dots \left(5\right)$

From $\left(4\right)$ and $\left(5\right),$

$\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$