Topic: Large deviations, in particular: Sanov's theorem

Let $\Sigma$ be a Polish space and $M_1(\Sigma)$ be the space of probability measures in $\Sigma$. Prove that
$$\left\lVert\nu-\mu\right\rVert_{var}^2\leq 2\textbf{H}(\nu|\mu), \ \ \ \ \mu,\nu\in\textbf{M}_1(\Sigma).\tag{*}$$
A proof of (*) can be based on the observation that
$$3(x-1)^2\leq(4+2x)(x\log x-x+1),\ \ \ \ x\in[0,\infty),$$
the fact that $\left\lVert\nu-\mu\right\rVert_{var}=\left\lVert f-1\right\rVert_{L^1(\mu)}$ if $\nu <<\mu$ and $f=\frac{d\nu}{d\mu}$, and Schwartz's inequality.


notes:
$\textbf{H}(\nu|\mu)=\left\{ \begin{array}{rcl} \int_\Sigma f\log f\ d\mu & if\ \ \nu\ll\mu\ and\ \ f=\frac{d\nu}{d\mu}\\ \infty & otherwise \end{array}\right.$
side note: $\int_\Sigma f\log f\ d\mu=\int_\Sigma \log f\ d\nu$

$\left\lVert\alpha\right\rVert_{var}=\sup\bigg\{\int\phi\ d\alpha:\phi\in C_b(\Sigma;\mathbb{R})\ with\ \left\lVert\phi\right\rVert_{C_b}\leq 1\bigg\}$
is the (total) variation norm (that is the definition in large deviation book written by jean-dominique deuschel and daniel w. stroock)