How to approach $\sum_{i=1}^m e^{\zeta_i}$ ($\zeta_i\leq0$)

Question

I would like to estimate (from above) the following sum $$\sum_{i=1}^m e^{\zeta_i}\quad \quad \zeta_i\leq0$$ The problem is that I do not know $\zeta_i$! (But I know $\sum_{i=1}^m \zeta_i$...) How would you handle this problem?

Answer 1

A tight bound can be found by solving the following optimization problem $$\max \sum_{i=1}^{m} e^{\zeta_i}\\{s.t.\\\sum_{i=1}^{m} \zeta_i=S\\\zeta_i\le 0}$$Now let $ \zeta_i\ne 0,i\in I,|I|=n$. Then we must maximize $$m-n+\sum_{i\in I}e^{\zeta_i}$$subject to$$\sum_{i\in I}\zeta_i=S$$ which is obtained when $\zeta_k=S,\zeta_i=0,i\ne k$therefore

$$\sum_{i=1}^{m} e^{-\zeta_i}\le m-1+e^{\sum_{i=1}^{m}\zeta_i}$$

Answer 2

Since all $\zeta_i\leq 0$, all you can say is that

$$\sum_{i=1}^m e^{\zeta_i} \le \sum_{i=1}^m 1=m.$$

If , for example, all $\zeta_i= 0$, then $\sum_{i=1}^m e^{\zeta_i}=m.$