Quote by maris205
Is there any conclusion about the gap of these primes? Some conjectures show the upper bound of primes gaps before p is g(p)< ln(p)^2 or g(p)<p^(1/2) (If RH is true).

RH would imply g(p)=O((log p)*p^(1/2)), not p^(1/2).
I don't know whose conjecture your g(p)< ln(p)^2 is refering to? There's Cramer's that says the lim sup of g(p)/log(p)^2 is 1 but this doesn't imply your inequality.
But here we just consider the primes between p and p^2, which are filtered by determinate primes. So I want to know whether we could get some easy conclusions about their gaps by elementary number theory. For example, could we prove the gap of the primes between p and p^2 is less than p? My calculation shows it’s true.

What's a "determinate prime"? Knowing the largest gap between p and p^2 is p implies the largest gap less than p is p^(1/2) (just consider a prime larger than sqrt(p) and apply this again, and repeat). Since this is quite a large leap from current results (and stronger than what RH implies) you're going to have to provide something stronger than "My calculation show's it's true" before I come near believing you can prove this.