I want to show that the argmin of the below integral is at $p,q =\eta$.
${1\over \beta(k(p+q),k(1-p-q)) \ \beta(kp,kq)} \iint\limits_{x+y \leq 1,\ x,y \geq \gamma} {x^{kp-1} y^{kq-1} (1-x-y)^{k(1-p-q)-1}dxdy}$
such that $p+q \leq 1$ and $p,q \geq \eta$ and $k\eta \geq 1$.
I tried minimizing the inner integral point wise but it doesn't work because the argmin varies for different $x,y$'s. Can anyone help?