How Do I Solve This Inequality? $\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a} \geq ab + bc + ca $ [duplicate]

It is given that a, b and c are positive real numbers. Prove that: $\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a} \geq ab + bc + ca $

Solutions involve the following and the summation of:
$\frac{a^3}{b}+\frac{b^3}{c}+bc \geq 3ab \\ \frac{a^3}{b}+\frac{c^3}{a}+ab \geq 3ac \\ \frac{b^3}{c}+\frac{c^3}{a}+ac \geq 3bc$.

However, I am not entirely sure as to how the above is reached. Thanks.

Solution 1:

By AM-GM inequality:

\begin{align} \frac{a^3}b + \frac{b^3}c+bc \ge 3\left( \frac{a^3}b\frac{b^3}{c}bc\right)^\frac13=3ab \end{align}

Now, you can add those inequalities and manipulate a little to reach your desired inequality.