What is the upper bound of $\sum_{n=1}^{N}d_{n}2^{-\frac{b}{a_n}}$ where $a_n, d_n$ is non negative values?
Or can i find the solution $b$ satisfying following constraint ? $\sum_{n=1}^{N}d_{n}2^{-\frac{b}{a_n}}=C$
where $C$ is positive constant.
Using the following arithmetic-geometric inequality, I can find the lower bound. $\sum_{n=1}^{N}p_n x_n\geq\prod_{n=1}^{N}{x_n}^{p_n}$
But, I don't have any idea or mathematical theory to find upper bound.
Update: I modify the question.