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:

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

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