For two continuous probability distributions $F,G$ and their densities, $f,g$, the (squared) Hellinger distance/affinity is given by $d^2\_H(F,G)=1-\int\_{\mathbb{R}} \sqrt{fg}~dx$. Suppose that $f,g$ are two-component mixture densities with mixing probability $\pi$, such that
$$
f(x)=\pi f\_0(x)+(1-\pi)f\_1(x)\\
g(x)=\pi g\_0(x)+(1-\pi)g\_1(x)
$$
The square root term is fairly complicated given this model. Are there any references that tackle the Hellinger distance between these mixture distributions?



References for Hellinger distance/affinity involving mixture distributions
For $f=\pi f\_0+(1-\pi)f\_1$, $g=\pi g\_0+(1-\pi)g\_1$, and $\pi\in(0,1)$, 
we have 
\begin{equation}
\frac{\partial^2}{\partial\pi^2}\sqrt{fg}= -\frac{\left(f\_1 g\_0-f\_0 g\_1\right){}^2}{4 (fg)^{3/2}}\le0.
\end{equation}
So, $\sqrt{fg}$ is concave in $\pi$ and hence $d^2\_H(F,G)$ is convex in $\pi\,$: 
$$
d^2\_H(F,G)\le\pi d^2\_H(F\_0,G\_0)+(1-\pi)d^2\_H(F\_1,G\_1)
$$
for $\pi\in[0,1]$, 
where $F\_j,G\_j$ are the distributions with densities $f\_j,g\_j$, respectively, for $j=0,1$. 


