Thinking about the approximation of
$$ \ln \sum_{i=1}^{N}\theta_{i}x_{i}, $$
where $\{\theta_i\}$ are positive constant that satisfies $\sum_{i=1}^{N}\theta_i= 1$ (i.e., weights) and $\{x_{i}\}$ are some positive variables. By Jensen's inequality, we have:
$$ \ln \sum_{i=1}^{N}\theta_{i}x_{i} = \sum_{i=1}^{N}\theta_{i}\ln x_{i} + F(\mathbf{x},\mathbf{\theta}), $$
where $F(\mathbf{x},\mathbf{\theta}) \geq 0$. Then, what is the representation of $F(\mathbf{x},\mathbf{\theta})$? In other words, how can we approximate $\ln \sum_{i=1}^{N}\theta_{i}x_{i}$ up to the second-order and third-order (or higher-order using Landau’s $o$-notation)?
Thanks,