Let $p_n$ denote the $n$th prime number and $\phi:\mathbb{N}\to\mathbb{N}$ an increasing function, with $\phi(n)>n$.
Is it true that $p_{\phi(n+1)}-p_{\phi(n)}<p_{\phi(n)}$ for infinitely many $n$ ?
Is it true that $p_{\phi(n+1)}-p_{\phi(n)}> p_{\phi(n)}$ for infinitely many $n$ ?