I am aware of the theorem that $p_{n+1} - p_n \leq n^{0.535}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit for all sufficiently large number to explicit, is it known that $ p_{n+1}- p_n \leq c_0 n^\alpha $ for all $n \geq 1$ and for small $c_0$, lets say $c_0 \leq 2$ and $\alpha \leq 0.55$ ?
Any ref or known explicit numbers $c_0 ,\alpha $ would be great.
Thanks