Conjecture:
There exists at least one prime number $p_{m}$ : $ap_{n} < p_{m} < (a+1)p_{n}$, $\forall$ $a \in \mathbb{N}$ and $\forall$ $p_{n}$ $\in \mathbb{P} $ if $(a+1)p_{n} < p_{n+1}^2$ .
Is there a name for this conjecture, and has it been proven or disproven?
Bertrand's postulate says that there is always a prime between $n$ and $2n$ for any $n$. So for $a=1$, your conjecture is a special case of this (and you don't need that $(a+1)p_n<p_{n+1}^2$ hypothesis). Note that at the end of that article, it says it has only recently been proven that there is always a prime between $2n$ and $3n$, and between $3n$ and $4n$, so your conjecture is true for $a=2$ and $a=3$ as well (and again, without needing that extra hypothesis). I don't know about the general case (or if there is a name for it), but you might want to start by seeing if the techniques developed for those latter cases can be adapted to prove what you want.