For the normally distributed r.v. $\xi$ there is a rule of $3\sigma$ which says that $ \mathsf P\{\xi\in (\mu-3\sigma,\mu+3\sigma)\}\geq 0.99. $
Clearly, this rule not necessary holds for other distributions. I wonder if there are lower bounds for $ p(\lambda) = P\{\xi\in (\mu-\lambda\sigma,\mu+\lambda\sigma)\} $ regardless of the distribution of real-valued random variable $\xi$. If we are focused only on absolute continuous distributions, a naive approach is to consider the variational problem $ \int\limits_{\int\limits xf(x)\,dx - \lambda\sqrt{\int\limits x^2f(x)\,dx-(\int\limits xf(x)\,dx)^2}}^{\int\limits xf(x)\,dx + \lambda\sqrt{\int\limits x^2f(x)\,dx-(\int\limits xf(x)\,dx)^2}} f(x)\,dx \to\inf\limits_f $ which may be too naive. The other problem is that dsitributions can be not necessary absolutely continuous.
So my question is if there are known lower bounds for $p(\lambda)$?