A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that $$\sum _{\mathrm{c}\mathrm{y}\mathrm{c}}{\left(\frac{a(b+c-a)}{bc}\right)}^k\ge 2+{\left(\frac{2r}{R}\right)}^k,k>1,$$ with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.