Inequality involving triangle and the sum $a^2+b^2+c^2$

Question

Weitzenbock's inequality states: given $S$ the area of a triangle, and $a,b,c$ the sides of the triangle, the following inequality holds: $$a^2+b^2+c^2\ge 4S\sqrt3$$ and the Gordon's inequality states: $$ab+bc+ca\ge4S\sqrt3$$ But we know that: $$a^2+b^2+c^2\ge ab+bc+ca,$$ so the question is: is it known some other sharper inequality for the sum: $a^2+b^2+c^2$ where $a,b,c$ are the triangle's sides?

Answer 1

Maybe the following?

$$2(ab+ac+bc)-a^2-b^2-c^2\geq4\sqrt3S$$