pr.probability – Relationship between $L_1$ and $L_2$ distances of two Gaussian Mixture models

Given two Gaussian mixture models with
begin{equation}
begin{aligned}
f(x) &=sum_{k=1}^{K} pi_{k} mathcal{N}left(x mid mu_{k}, sigma_{k}right), \
g(x) &=sum_{i=1}^{N} lambda_{i} mathcal{N}left(x mid mu_{i}, sigma_{i}right),
end{aligned}
end{equation}

I want to compute their $L_1$ distance,
begin{equation}
delta_{1}(f,g) = int_{x} |f(x) – g(x)|.
end{equation}

However, the $L_1$ distance cannot be computed directly, and I use the squared $L_2$ distance instead, i.e.,
begin{equation}
delta_{2}(f,g) = int_{x} (f(x) – g(x))^2.
end{equation}

The closed-form expression of $L_2$ distance is available (c.f. Distance between two Gaussian mixtures to evaluate cluster solutions), but I want to know: does there exist any relationship between $L_1$ distance and $L_2$ distance in this case?