Is it true that $n(\ln n+\ln \ln n-1+ \frac{\ln\ln n-2}{\ln n} + \frac{-\ln^2 \ln n+6\ln \ln n -11.5}{2 \ln^2 n}) \leq p_n \leq n(\ln n+\ln \ln n-1+ \frac{\ln\ln n-2}{\ln n} + \frac{-\ln^2 \ln n+6\ln \ln n -10.5}{2 \ln^2 n})$
? also is there a ref, is this inequality known ?
Unconditionally the sharpest known bounds are (Due to Peierre Dusart) as follows:
For $n \ge 3$ $$ p_n \ge n\ln n + n\ln\ln n - n + \frac{n\ln\ln n - 2.1n}{\ln n} $$ and for $n \ge 688383$ $$ p_n \le n\ln n + n\ln\ln n - n + \frac{n\ln\ln n - 2n}{\ln n}. $$
All such explicit upper and lower bounds will however eventually converge to the asymptotic expansion of the $n$-th prime which was derieved by Cipolla (1902).
Ref: Theorem 2.1 of https://arxiv.org/pdf/1011.1667.pdf