upper bound on sum of exponential functions Let $$\sum_{k=1}^{N} |a_k| = A$$ where $A$ is some constant. I am looking for an upper bound on the sum 
$$\sum_{k=1}^{N} e^{|a_k|},$$ independent of $N$, but may depend on $A$. That is, I am looking for something like
$$\sum_{k=1}^{N} e^{|a_k|} \le f(A).$$ 
We can find the upper bound
$$\sum_{k=1}^{N} e^{|a_k|} \le Ne^{A}.$$ 
but I do not want this dependence on N.
Does anybody have any ideas? From some reason I can't seem to figure this one out. Something like this
this would work, but I'm not sure its correct.

Assume they are sorted,
so
$a_k \ge a_{k+1}
$.
Then
$\sum_{k=1}^n e^{a_k}
=e^{a_1}\sum_{k=1}^n e^{a_k-a_1}
\le n e^{a_1}
$
with equality iff
all the $a_k$ are equal.
This is pretty simple-minded,
but to do more
requires knowledge about the $a_k$.
