sum of exponential vs exponential of sums

Question

Let $x_1,\cdots,x_k\in (0,1]$ are $k$ reals such that $\sum_{i=1}^kx_i > 1/2$. Since $f(x)=\exp(-x)$ is convex function, from Jensen's inequality, we have that $\sum_{i=1}^k\exp(-x_i) \geq k.\exp(-\frac{\sum_{i=1}^kx_i}{k})$. Is there any reverse inequality, meaning can $\sum_{i=1}^k\exp(-x_i)$ be upper bounded by some expression involving $\exp(-\sum_{i=1}^kx_i)$?