I am trying to show that the following inequalties hold $$\exp[\frac{k^2}{b^2 t}] \log ^{\alpha_2 -1}(t) \left(\frac{1}{2} \log (t) -\alpha_2\right) \leq \frac{k}{\sqrt{2 \pi b^2}} \leq \exp[\frac{k^2}{b^2 t}] \log ^{-\alpha_1 -1}(t) \left(\alpha_1 + \frac{1}{2} \log (t)\right),$$
where $k$, $b$ are positive constants and $\alpha_1 > 1$ and $\alpha_2 > -4$ and $t$ is the time variable.
I am concerned if the inequalities will hold for all values of $t$, but it seems difficult to isolate $t$.
Obviously the standard approach will be to find the inflection points for both the lower and the upper bound and isolate the maximum and the minimum values, but that again requires isolating $t$ which seems really hard.
How can one prove such a problem.
Your help is greatly appreciated.