I know that if we want to normalize variable $V$, we should do the following: $(V-\min)/(\max - \min)$
So, for Normalizing Joint Entropy $H(x,y)$ , this should be correct:
$$ \frac{H(x,y)-\max(H(x),H(y))}{H(x)+H(y)-\max(H(x),H(y))} \tag{1} $$
The $\max(H(x),H(y))$ is the Lower Bound, so the whole fraction is equal to:
$$ \frac{\min[H(y|x),H(x|y)]}{\min[H(x),H(y)]} \tag{2}\label{2} $$
The question I have here, is how can I prove that (2) equals to the following:
$$ \min[H(x|y)/H(x) , H(y|x)/H(y) ]\tag{3}\label{3} $$
Can anyone help me prove this? How can we get from \eqref{2} to \eqref{3} ?