Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$?

I found only one reference, to "Über einige Affininvarianten konvexer Bereiche", but unfortunately it is in German.

Added: formula (12) there looks like desirable, but I'm not sure.