Say you have positive $\{a_i\}_{i=1}^n$ and you have $p_i=\frac{a_i}{\sum_{i=1}^na_i}$, then assume you have a $C$ such that $C<2a_n\ll\sum_{i=1}^na_i$ (that is $C$ is not very large), then define $q_i=\frac{a_i}{C+\sum_{i=1}^na_i}$ and $q_{n+1}=\frac{C}{C+\sum_{i=1}^na_i}$. Does Shannon entropy of $q$ dominate entropy of $p$?

Take $C=a_n+\log_2^ka_n$ for any fixed $k$. Now does Shannon entropy of $q$ dominate entropy of $p$ after certain $n$?

There are two cases to consider. Case $(1)$ $a_{i+1}=a_i+O(\log^ka_i)$ Case $(2)$ $a_{i+1}=a_i+O(a_i)$.

When can one expect Shannon entropy to be dominant in sequences of these types? Clearly in this post Entropy dominance, one criteria for negative result is given.

The following property also seems true if $m+1\leq n$. If $p$ correspond to $\{a_i\}_{i=1}^{m}$, $q$ correspond to $\{a_i\}_{i=1}^{m+1}$, $r$ correspond to $\{a_i\}_{i=1}^{n}$ and $s$ correspond to $\{a_i\}_{i=1}^{n+1}$ where $a_i$ satisfy properties above, then is it true that, $H(q)-H(p)>H(s)-H(r)$?