含log(u)的当u→0

hbghlyj

$u\in(0,1)$, 求证\[f(u)=(u+1)^{3/2} \log (u)+2\log \left(\sqrt{u+1}-1\right)-3\sqrt{u+1} \log (u)<-\log4\]Source:
这帖计算$I_1$时, 代入0遇到的

hbghlyj

渐近展开

1. Asymptotic[(u + 1)^(3/2) Log[u] + 2 Log[Sqrt[u + 1] - 1] - 3 Sqrt[u + 1] Log[u], {u, 0, 2}]

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$f(u)\asymp-\frac{u}{2}+\frac{3 u^2}{16}-\log (4)+\frac{3}{4} u^2 \log (u)$

hbghlyj

把$(u+1)^{3/2} $换成1, 把$\sqrt{u+1} $也换成1
\begin{align*}f(u)&\asymp\log (u)+2\log \left(\sqrt{u+1}-1\right)-3\log (u)\\&=2\left(\log \left(\sqrt{u+1}-1\right)-\log u\right)\\&=2\log \left(\sqrt{u+1}-1\over u\right)\to2\log\frac12
\end{align*}