Properties of convex function with Lipshitz continuous gradient (Prof. Nesterov's textbook)

Question

I am reading the Prof. Nesterov's textbook:
Introductory lectures on convex optimization - a basic course p.57

I have problem in the following:

My question is how to get the green box?

I have trouble to get the green box from (2.1.6); particularly, how to put $L$ to denominator.

Note: To my question, you can assume (2.1.6) is known and just stat from "Further, let us fix $x_0\in R^n$".

Answer 1

It is a direct calculation. First inequality is followed from definition of $y^\ast$ For the second we use (2.1.6) :

$$ \phi \bigg(y- \frac{\phi'}{L}\bigg) -\phi (y) -\bigg(\phi'(y), - \frac{\phi'}{L}\bigg) \leq \frac{L}{2}\| - \frac{\phi'}{L}\|^2 $$