Ley $t:=eta$. Then

$$f(t)=frac{P(Gge K)}{P(B ge K)},$$

where $G$ is a random variable with the binomial distribution with parameters $N,q_Gt$ and $B$ is a random variable with the binomial distribution with parameters $N,q_Bt$; here we must assume that $q_B>0$ and $tin(0,1/g_G)$, so that $q_Gt$ and $q_Bt$ are in the interval $(0,1)$.

The random variables $G$ and $B$ have a monotone likelihood ratio (MLR): for each $xin{0,dots,N}$,

$$frac{P(G=x)}{P(B=x)}=CBig(frac{1-q_Gt}{1-q_Bt}Big)^{N-x},$$

which is decreasing in $tin(0,1/g_G)$; here, $C$ is a positive real number which does not depend on $t$.

It is well known that the MLR implies the MTR, the monotone tail ratio. Thus, the desired result follows.