given the sum of a finite sequence of real numbers $x_i$'s, find the $\sum_{i=1}^{N} e^{x_i}$

Question

Let $\sum_{i=1}^{N} x_i $$=$$ 1 $ then what could one say about $\sum_{i=1}^{N} e^{x_i} $$=$$ ? $

Answer 1

Since $e^x \ge e^{1/N} + e^{1/N}(x-1/N)$, $$\sum_{i=1}^N e^{x_i} \ge N e^{1/N}$$ This inequality is tight (it is an equality in the case where all $x_i = 1/N$). Any value $\ge N e^{1/N}$ is possible, if $N > 1$ (take $x_1 \to \infty$ to make it arbitrarily large).