This question is a follow-up to my comment to the answer to this question. Writing $g_{n}:=p_{n+1}-p_{n}$, and as all numbers between $p_{n}$ and $p_{n+1}$ are composite, one has $j(p_{n})=O(\log^{2}p_{n})$ from the result of Iwaniec. As one has the obvious inequality $\max_{a<x}(f(a)g(a))\leq \max_{a<x}f(a)\times\max_{b<x}g(b)$, with $f$ and $g$ both taking positive values on the integers, can one hope to get a weak form of Cramer's conjecture?