Most of the concentration bounds for chi squared are centered at it's mean. I am wondering if there are known exponential bounds on the probability of being near zero, meaning $ \mathbb{P}(\chi^2_k \leq \varepsilon \mathbb{E} \chi^2_k ) \leq ?? $
I have tried using the Payley-Zygmund inequality, but this only gives an inverse polynomial bound (in terms of k).