Minimum and maximum value related to the sides of the quadrilateral.

Question

If $a$, $b$, $c$ and $d$ are the sides of the quadrilateral then find the minimum value of $$\frac{a^2+b^2+c^2}{d^2}.$$

I have tied by the inequality $a+b+c>d$, but it doesn't work.

Answer 1

The maximum does not exist: $a=b=c\rightarrow+\infty$.

The minimum does not exist.

Indeed, by C-S $$\frac{a^2+b^2+c^2}{d^2}=\frac{(1+1+1)(a^2+b^2+c^2)}{3d^2}\geq\frac{(a+b+c)^2}{3d^2}>\frac{d^2}{3d^2}=\frac{1}{3}.$$ Since for $a=b=c\rightarrow\frac{d}{3}$ we have $\frac{a^2+b^2+c^2}{d^2}\rightarrow\frac{1}{3}$, we obtain that $\frac{1}{3}$ is an infimum of our expression.