Is $f\left( x \right)=\log \left( \sum_i \beta_i e^{-\alpha_ix} \right)$ a convex function where $\beta_i,\alpha_i\in \mathbb{R}$?
2026-03-26 20:26:02.1774556762
Convexity of log sum function
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If there are no restrictions on $\beta_i$ and $\alpha_i$ then there are counter-examples, e.g. $$ g(x) = 2 e^{0 x} - 2 e^{-1x} + 1e^{-2x}.$$ We have $g(x)>0$ for $x\in \mathbb{R}$ with a minimum at $x=0$, but the second derivative of $f(x) \equiv \log(g(x))$ changes sign near $x=0$.