Is there a constant $\alpha$ such that: $P_{n+1} < P_n.\left(\frac{n+1}{n}\right)^\alpha$?

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 $$P_n=18361375334787046697$$

As $$P_n$$ is asymptotically $$n\log n$$, your question is equivalent to the following. Is it true that $$P_{n+1}-P_n\ll\log n?$$ In other words, is it true that the actual gap between primes is always at most a constant times the average (expected) gap? The answer is "no" by a 1931 result of Westzynthius. For the best known lower bound (for infinitely many prime gaps) see the paper of Ford-Green-Konyagin-Maynard-Tao.