Question: Legendre`s conjecture states that there is always a prime between $n^2$ and $(n+1)^2$ for every natural $n$.

It is natural to create following generalization:

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> Is it true that for every $\varepsilon \in (0,1]$ there exists natural number $n\_0(\varepsilon)$ such that for every $n \geq n\_0(\varepsilon)$ there is always a prime between $n^{1+\varepsilon}$ and $(n+1)^{1+\varepsilon}$?
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I do not know if some results from number theory do not allow this generalization to be true but if it is not known is it true or not what is known, if anything, about this generalization of Legendre`s conjecture?

Answer: 
As mentioned in the remarks, this conjecture is equivalent to the statement that if $p$ is a prime, then the next prime satisfies $p'\ll\_\delta p+p^\delta$ for any $\delta>0$. The parameter $\delta$ corresponds to the OP's $\epsilon/(1+\epsilon)$. For $\delta>1/2$ (which corresponds to $\epsilon>1$) this follows from Legendre's conjecture and also from the Riemann Hypothesis, for $\delta>21/40$ (which corresponds to $\epsilon>21/19$) it follows unconditionally from the work of Baker-Harman-Pintz. For arbitrary $\delta>0$, the statement follows from Cramér's conjecture. The bottom line is: we don't know the truth yet.
This seems to be about as strong as [Cramer's conjecture.](https://www.wikiwand.com/en/Cram%C3%A9r%27s_conjecture)