Suppose $\{a_n\}_{n\geq 1}$ is a sequence of nonnegative real numbers. Define sequence \begin{align} b_n = \frac{\sum_{i = 1}^n a_i}{n}. \end{align}
Prove the following conjecture:
There exists a universal constant $C>0$ such that for any $n$ positive integer, any $\epsilon>0$ and any sequence $\{a_n\}$ of nonnegative real numbers, we have \begin{align} \sum_{i = 1}^n a_i \ln\left ( \frac{a_i}{b_i} \right ) \mathbf{1}(b_i\leq \epsilon) & \leq C n\epsilon \cdot \ln(n+1) . \end{align}
We adopt the convention that $0 \ln(0/0) = 0$.
Partial converse: it is clear that one cannot relax the RHS upper bound. Consider the sequence $\{a_n\}$ such that $a_i = 0$, $1\leq i\leq m$, and $a_i = 1, i\geq m+1$. Take $n = \sup\{j: b_j \leq \epsilon\}$.