Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\frac{1}{\lambda}x_n-c\leq y_n\leq \lambda x_n+c$, and $\inf\left\{a\in \mathbb{R}\mid \sum_{n=1}^{+\infty}e^{-ax_n}<+\infty\right\}=\inf\left\{b\in \mathbb{R}\mid \sum_{n=1}^{+\infty}e^{-by_n}<+\infty\right\}=1$.
Consider a series: $P(a,b)=\sum_{n=1}^{+\infty}e^{-ax_n-by_n}$.
For each $a\in \mathbb{R}$, denote $\varphi(a)=\inf\left\{b\in\mathbb{R}\mid P(a,b)<+\infty\right\}$. It can be shown that $-\infty<\varphi(a)<+\infty$ for all $a\in \mathbb{R}$.
If $\varphi(a)=1-a$ for all $a\in [0,1]$, can we deduce $\varphi(a)=1-a$ for all $a\in \mathbb{R}$?