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 http://mathoverflow.net/questions/191391/entropy-dominance, one criteria for negative result is given.


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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, $H(q)-H(p)>H(s)-H(r)$?