The proof is based on Lemma 4.6. First, we discuss the case $h=0$ and $j\in\mathbb{Z}_{+}$. Direct computations give that $t_{0}=y$ and $s_{0}=0$, then by definition we have that $\widehat{F}(z)=\frac{jz^{2}}{2}$ and $\widehat{U}(z)=-\frac{jz^{2}}{2}$. Therefore, (4.79) immediately follows.

Next, we prove the case $j\in\mathbb{Z}_{+}$ and $h>0$. We notice that

(4.82)		$\displaystyle t_{0}-s_{0}$	$\displaystyle=\frac{a}{2}$
(4.83)		$\displaystyle t_{0}^{2}+s_{0}^{2}$	$\displaystyle=\frac{a^{2}}{4}+h$
(4.84)		$\displaystyle t_{0}s_{0}$	$\displaystyle=\frac{h}{2},$

by elementary calculations,

	$\displaystyle F(t_{0})+U(s_{0})$	$\displaystyle=-(t_{0}^{2}+s_{0}^{2})+a(t_{0}-s_{0})+h\log(t_{0}s_{0})$
(4.85)			$\displaystyle=\frac{a^{2}}{4}-h+h\log(\frac{h}{2})=jz^{2}-h+h\log(\frac{h}{2}).$

Immediately we have that

(4.86)

$$e^{\widehat{F}(z)+\widehat{U}(z)}\leq e^{-h+h\log(\frac{h}{2})}.$$

Combining (4.86) and Lemma 4.5,

(4.87)

$\displaystyle\frac{e^{\widehat{F}(z)+\widehat{U}(z)}}{\mathcal{W}(z)}\leq\frac{e^{-h+h\log(\frac{h}{2})}}{\sqrt{j\pi}2^{-h}\Gamma(h+1)}\leq C_{0}.$

This proves the lemma. ∎