for the operator: $\Delta A(t) = A(t+1)-A(t)$, let :
$$ \Delta C = \sum^{T-1}_{t=0} [~~H(\pi(t+1)~|~\mu(t+1))~~ - ~~H(\pi(t)~|~\mu(t+1))~~] $$ Where $H (\pi(t+1)|\mu(t+1)) = \sum^n_{i=1}\pi_i(t) \log \frac{\pi_i(t)}{\mu_i(t+1}$ is the relative entropy.
This implies that: $$ C(t) = H(\pi(t)~|~\mu(t+1)) $$
Now, I am trying to show that equating the first order taylor expansion of $C$ as a function of s to a quantity $\alpha(t)$ should yield the following: $$ s:= \frac{\alpha(t)~~\big/~~|~\nabla H(\pi(t)| \mu(t+1)\bullet v)~|~~}{|\mu(t+1) - \pi(t)|} $$ where $v=\frac{\mu(t+1) - \pi(t)}{|\mu(t+1) - \pi(t)|}$
Now: $$ C(s) = C(0) + C'(0) ~s\\ =C'(0) ~s $$ then equating to $\alpha(t)$:
$$ s = \alpha(t) / C'(0) $$
and I've shown that $$ \nabla H = \big(\frac{\partial H}{\partial \pi_1},....,\frac{\partial H}{\partial \pi_n} \big) = (1,...,1) + \big(\log \frac{\pi_1(s)}{\mu_1(s+1)},...., \big) $$
I cannot get that value of s though with this working, some intermediate questions. I'm not looking for an entire solution but how would the vector norms come into play at all?