If $ k\geq \dfrac {n - 5 + \sqrt {(n - 1)(n + 7)}}{4(n - 1)}$, then
$$ \prod \limits_{cyc}\left(k + \frac {a_1}{a_2 + a_3 + ... + a_n}\right)\ge \left(k + \frac 1{n - 1}\right)^n$$
Here the function $f(x)=\ln\left(k+\dfrac{x}{s-x}\right)$ is not convex where $s=a_1+a_2+\cdots+a_n$. So i can not apply Jensen.
I found this inequality proved here but i understood nothing. Can you please tell me how to approach this.
