So I have a function:
$\phi(P) = \lvert (1-\alpha_2)^P - (1-\alpha_1)^P\rvert$
It is easy to show that $\phi$ is increasing between $0$ and $P_M$, where
$P_{M} = \lvert \ln\bigg(\frac{\ln(1-\alpha_2)}{\ln(1-\alpha_1)}\bigg)/\ln\bigg(\frac{1-\alpha_1}{1-\alpha_2}\bigg)\rvert$
and decreasing afterwards.
Assuming $1 > \alpha_1 > \alpha_2$ > 0, if $P_{M} \leq 1$, how could I prove the following?
$\alpha_1 > 1-\frac{1}{e} > 0.63$
I have stuck here for hours. I think it is a easy step to make it. However, I couldn't figure it out.
Any idea is much appreciated. Thanks!