I would like to show the function $\bar{\gamma}\left(\alpha,\delta,\mu,N\right)$ decreases in $\delta$, where
$\bar{\gamma}\left(\alpha,\delta,\mu,N\right)\equiv\chi+\sqrt{\chi\left(\chi+\frac{2\alpha\delta\left(N\mu\left(2-\delta\mu\right)+\alpha\left(N\left(1-\delta\mu\right)-1\right)\right)+\mu\left(4-\delta^2\mu^2\right)N}{\alpha^2\delta^2\mu\left(N+1\right)}\right)}$, where $\chi=\frac{\delta\mu\left(N+1\right)}{2\left(N-1+\delta\mu\right)}$, $N>2$, $\alpha, \mu, \delta>0$, $\alpha\delta<1$, $\mu\delta<1$.