In the attempt of proving a large deviations result, the following quantity pops up:
$$H(\delta,h):= \frac{\frac{1}{2\alpha}(2\alpha-8\delta)+ \frac{4}{\alpha}\frac{1}{1-\gamma}\delta^{1-\gamma}}{h^{(\delta-\log(\delta)\delta)\frac{4}{\alpha}}} $$
where $\alpha \in (\frac{1}{4},\frac{1}{2})$ is fixed, $\gamma= 1-\frac{1}{\log_{h}(e)}$.
The result that I would need is: "there exists some vaules of $e>h>1$ and $\delta=\delta(h)>0$ such that $H(\delta_0,h)<1$ for all $\delta_0<\delta$.
Unfortunately, I believe this is not true, but I could not come up with a good proof. Can anyone help me understand if this is true or not and how to prove it? Thanks