Let $\mathcal A_{s}$ be the set of sequences $X=(x_m)_{m \in I}$ where $I=\{1,2,...,n\}$ with $n \ge 2$ and possibly $n =\infty$ is an index set with $x_1=0$, $x_2=s>0$ and $x_m>x_{m-1}$ for $m,m-1 \in I.$ In addition, we require that the sequence $X$ satisfies $$ \sum_{m \in I} e^{-\gamma x_m}<\infty \text{ for all }\gamma>0.$$

We then fix $x_2=s>0$ (as $x_1=0$ is always fixed) and define $\gamma(X,\lambda)>0$ for $\lambda \in (0,x_2)$ by the condition $$ \sum_{m \in I} (\lambda-x_m) e^{-\gamma(X,\lambda) x_m}=0.$$ Based on some examples, I then claim that for $\lambda \in (0,s)$ $$\max_{X\in \mathcal A_s} \sum_{m \in I} e^{-\gamma(X,\lambda) x_m} $$ is attained for the sequence $X=(0,s)$, i.e. $I=\{1,2\}$ such that

$$\max_{X\in \mathcal A_s} \sum_{m \in I} e^{-\gamma(X,\lambda) x_m} = 1 + e^{-\gamma((0,s),\lambda)s}.$$