Let $X_n$ be a chi-squared random variable with $n$ degrees of freedom. What are the sharpest known lower bounds on the tails of its distribution? Specifically, I am looking for the lower bounds in the form:

$$P(X_n-n\geq f_1(x)\sqrt{n}+g_1(x))\geq \exp(-x)$$ $$P(n-X_n\geq f_2(x)\sqrt{n}+g_2(x))\geq \exp(-x)$$

for some $f_1(x)>0$, $f_2(x)>0$, and $g_1(x)\geq0$, $g_2(x)\geq0$.

Basically, I am wondering if the lower-bound "equivalent" of the following upper bounds from Massart and Laurent (see Lemma 1 on page 1325) exist:

$$P(X_n-n\geq 2\sqrt{xn}+2x)\leq\exp(-x)$$ $$P(n-X_n\geq 2\sqrt{xn})\leq\exp(-x)$$

I tried Cramer-Chernoff Theorem (i.e. Theorem 1 in here), but the lower bound isn't sharp enough...

Let $X_n$ be a chi-squared random variable with $n$ degrees of freedom. What are the sharpest known lower bounds on the tails of its distribution? Specifically, I am looking for the lower bounds in the form:

$$P(X_n-n\geq f_1(x)\sqrt{n}+g_1(x))\geq \exp(-x)$$ $$P(n-X_n\geq f_2(x)\sqrt{n}+g_2(x))\geq \exp(-x)$$

for some $f_1(x)>0$, $f_2(x)>0$, and $g_1(x)\geq0$, $g_2(x)\geq0$.

Basically, I am wondering if the lower-bound "equivalent" of the following upper bounds from Massart and Laurent (see Lemma 1 on page 1325) exist:

$$P(X_n-n\geq 2\sqrt{xn}+2x)\leq\exp(-x)$$ $$P(n-X_n\geq 2\sqrt{xn})\leq\exp(-x)$$

I tried Cramer-Chernoff Theorem (i.e. Theorem 1 in here), but the lower bound isn't sharp enough...