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$$P\_{n+1} < P\_n \cdot \left(\frac{n+1}{n}\right)^\alpha$$


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 $P\_n=18361375334787046697$



Is there a constant $\alpha$ such that: $P_{n+1} < P_n.\left(\frac{n+1}{n}\right)^\alpha$?
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](https://arxiv.org/abs/1412.5029).


