Let $y=\sqrt{j}z$ and $a=2y$, then by definition,

(4.89)

$$\widehat{F}=-\frac{a^{2}}{8}-(t_{0}(a))^{2}+at_{0}(a)+h\log(t_{0}(a))$$

and

(4.90)

$$\widehat{U}=-\frac{a^{2}}{8}-(s_{0}(a))^{2}-as_{0}(a)+h\log(s_{0}(a)).$$

We show that $\widehat{F}$ is increasing in $a$ and $\widehat{U}$ is decreasing in $a$. Indeed,

(4.91)

$\displaystyle\frac{d\widehat{F}}{da}=-\frac{a}{4}+t_{0}(a)+\Big(-2t_{0}(a)+a+\frac{h}{t_{0}(a)}\Big)t_{0}^{\prime}(a)=\sqrt{\frac{h}{2}+\frac{a^{2}}{16}}\geq\frac{a}{4}.$

So the monotonicity of $\widehat{F}(z)-\eta z$ immediately follows when $z>2\eta$. Similarly, the monotonicity of $\widehat{U}+\eta z$ follows from the computation

(4.92)

$\displaystyle\frac{d\widehat{U}}{da}=-\frac{a}{4}-s_{0}(a)+\Big(-2s_{0}(a)-a+\frac{h}{s_{0}(a)}\Big)s_{0}^{\prime}(a)=-\sqrt{\frac{h}{2}+\frac{a^{2}}{16}}\leq-\frac{a}{4}.$

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