There are at least: $\big\lfloor\sqrt{p_{n}}\big\rfloor(p_{n}-1)-|p_{n}-2n|+1$, primes less than $p_{n}^{2}$, where $p_{n}$ is the $n$-th prime?

Question

There are at least: $\big\lfloor\sqrt{p_{n}}\big\rfloor(p_{n}-1)-|p_{n}-2n|+1$, primes less than $p_{n}^{2}$, where $p_{n}$ is the $n$-th prime?

Is this true or false? If true, how does one prove it?

I made this observation just an hour ago and I checked it thoroughly and carefully. The accuracy often seems very promising as far as I checked. Again I have no idea as to how to approach a proof.