If $p$, $q$, $r$ and $s$ are four sides of a quadrilateral, then find the minimum value of $\frac{p^2+ q^2 + r^2}{s^2}$ with logic

Question

If $p$, $q$, $r$ and $s$ are four sides of a quadrilateral then find the minimum value of $\frac{p^2+ q^2 + r^2}{s^2}$ with logic.

Please help me with this.

Answer 1

By the triangle inequality and by C-S we obtain: $$\frac{p^2+q^2+r^2}{s^2}>\frac{p^2+q^2+r^2}{(p+q+r)^2}=\frac{(1+1+1)(p^2+q^2+r^2)}{3(p+q+r)^2}\geq\frac{(p+q+r)^2}{3(p+q+r)^2}=\frac{1}{3}.$$ The equality does not occur, but easy to see that $\frac{1}{3}$ is an infimum.