Given $f:\mathbb{R^n}\to\mathbb{R}$ is a convex function, satisfying $$f(y) \leq f(x) + \langle \nabla f(x), y-x \rangle +\frac{L}{2} \vert \vert x - y \vert \vert ^2$$ show that $$f(y) \geq f(x) + \langle \nabla f, y - x \rangle + \frac{1}{2L}\vert \vert \nabla f(x) - \nabla f(y) \vert \vert_*^2$$ where $\vert \vert . \vert \vert_*$ is the dual norm of $\vert \vert . \vert \vert$.
I tried rearranging the first equation to reduce it to a primal problem and get to the second inequality using its dual problem. But I'm stuck as I can't understand how the dual norm will come into the picture. Thanks for your help.