Prime number function inequality conjecture

Question

In my intuitive random search for conjectures I found $$n\neq 4\implies p_{n^2}\leq p_n^2<p_{4n^2}$$ It's tested for $n<1000$. I've looked at it the point of view of PNT but haven't the skills to prove or disprove it. Could there be counter examples out of reach of my conjecture seeking routines?

Answer 1

This follows trivially from PNT.

For the left inequality. Since $p_n \sim n \log n$ hence $p_{n^2} \sim 2n^2 \log n$ but $p_n^2 \sim n^2 \log^2 n$ hence for some $n > n_0$, the growth rate of $p_n^2$ will exceed that of $p_{n^2}$.

For the right inequality, $p_n^2 \sim n^2 \log^2 n$ but $p_{4n^2} \sim 4n^2 \log 4n^2 = 4n^2(2\log 2 + 2\log n)$ so clearly this inequality will eventually fail.