Suppose $P(f<a)\leq \exp(-\frac{(a-\mu)^2}{2s^2\sigma^2 })$ where $f$ is random variable and $s$ is parameter, and $\sigma^2,\mu$ are variance and mean of $f$.
Is it possible to further upper bound it by $\exp(-\frac{a^2}{2})$, under the condition $s^2\prec \frac{\mu^2}{\sigma^2}$?