Consider $f(t)=\left(\sum_{i=1}^n b_i e^{ta_i}\right)^A$, where $a_i\in R_+$, $0\leq b_i\leq 1, \sum_i b_i=1$, $t$ - parameter, for $i=1, \ldots, n$ and $A \in R_+$.
I would like to bound from above the k-th derivative of $f$ at $t=0$, as follows: $$ f^{(k)}\leq C(A,k)\|a\|_1. $$ So far, I have been able to get bound on the r.h.s with $C(A, k)=A^k$, but I need a tighter bound than that.