# What are the sharpest known tail bounds for χ2k\chi_k^2 distributed variables?

Let $X \sim \chi^2_k$ be a chi-squared distributed random variable with $k$ degrees of freedom. What are the sharpest known bounds for the following probabilities

and

where $\delta_1$ and $\delta_2$ are some functions. Pointers to relevant papers would be appreciated.

The Sharpest bound I know is that of Massart and Laurent Lemma 1 p1325.

A corollary of their bound is: