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 twocomponent 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?

1$\begingroup$ "Both the Bhattacharyya and Helling distances does not admit closedform expressions when dealing with mixture model" from ieeexplore.ieee.org/document/6460482 or www2.sonycsl.co.jp/person/nielsen/… $\endgroup$– Steve HuntsmanOct 30 '18 at 0:23
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_0f_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 $j=0,1$.