I have divided the Problem into two parts, a) and b):
a) Let $a$, $b$ and $c$ be the side lengths of a triangle with a perimeter of $4$. I need to prove that
$a^2 + b^2 + c^2 + abc < 8$.
b) And is there a real number $d<8$ such that for each triangle with the perimeter $4$, the inequality equation
$a^2 + b^2 + c^2 + abc <d$
is valid?
I tried to use Heron's Formula somehow for the first inequality equation.
$s \, = \, \frac{a+b+c}{2}$
and
$F_{\triangle} = \sqrt{s(s-a)(s-b)(s-c)}$
Now we have:
$2s=4$ so that $s=2$
Thus:
$F_{\triangle} = \sqrt{2(2-a)(2-b)(2-c)}$
Maybe we can exchange $F_{\triangle}$ somehow?
Has anyone another approach or an idea how to continue with Heron's formula?
Thx