  Let $T_{\epsilon}$ be the lifespan for the solution to the Schr\"odinger
equation on $\mathbb{R}^d$ with a power nonlinearity $\lambda |u|^{2\theta/d}u$
($\lambda \in \mathbb{C}$, $0<\theta<1$) and the initial data in the form
$\epsilon \varphi(x)$. We provide a sharp lower bound estimate for
$T_{\epsilon}$ as $\epsilon \to +0$ which can be written explicitly by
$\lambda$, $d$, $\theta$, $\varphi$ and $\epsilon$. This is an improvement of
the previous result by H.Sasaki [Adv. Diff. Eq. 14 (2009), 1021--1039].
