>> Is there a constant $\alpha$ such that:

$$P_{n+1} < P_n \cdot \left(\frac{n+1}{n}\right)^\alpha$$

Or 

$$\lim_{n\to\infty}\frac{\ln\frac{P_{n+1}}{P_n}}{\ln\frac{n+1}{n}} < +\infty$$

Where $P_n$ is $n$-th prime number.

In the table [The 80 known maximal prime gaps]( https://en.wikipedia.org/wiki/Prime_gap#Numerical_results):

* $\alpha < 33.3$ with $P_n=1693182318746371$

* $\alpha < 35.77$ with $P_n=18361375334787046697$