Let $y=-jz^{n}$, then it is straightforward that

(5.150)

$\displaystyle\frac{d\widehat{F}(y)}{dy}=\frac{1}{2}+t_{0}(y)+F^{\prime}(t_{0}(y))\cdot\frac{dt_{0}(y)}{dy}=\frac{1}{2}+t_{0}(y)=\frac{1}{2}\sqrt{1+\frac{4Q}{-y}}\geq\frac{1}{2}.$

This implies that, as $z\geq\eta^{\frac{2}{n}}$,

(5.151)

$\displaystyle\frac{d(\widehat{F}(z)+\eta z^{\frac{n}{2}})}{dz}=\frac{d\widehat{F}(y)}{dy}\cdot(-nj\cdot z^{n-1})+\frac{n\cdot\eta}{2}\cdot z^{\frac{n}{2}-1}\leq-\frac{n}{2}\cdot z^{\frac{n}{2}-1}(j\cdot z^{\frac{n}{2}}-\eta)\leq 0.$

By similar calculations, one can also obtain that $\widehat{G}(z)-\eta\cdot z^{\frac{n}{2}}$ is increasing as $z\geq\eta^{\frac{2}{n}}$.

∎