Let $\alpha$ and $\beta$ be real numbers such that $\beta > \alpha > 1$ and consider the exponential function $e^{-x}$. Suppose that the parameters $a$ and $b$ are in $[0,1]$ and satisfy the conditions
(1) $a+b = 1$;
(2) $b \geq \frac{\alpha}{\beta}$.
Is it possible to find $a$ and $b$ such that the inequality \begin{equation} a + b e^{-\beta x} \geq e^{-x} \end{equation} holds for $x > 0$ while satisfy the conditions (1) and (2)?
From Jensen's inequality we know that $a + b e^{-\beta x} \geq e^{-b \beta x}$. But $b \beta \geq \alpha > 1$ which makes $e^{-b \beta x} < e^{-x}$ so that it does not work here. Anyone has an idea?