An inequality on prime gaps

Bertrand's postulate gives that $p_k-p_{k-1}\le p_{k-1}.$ A result of Baker, Harman, & Pintz can be used to improve this to $p_k-p_{k-1}\ll p_{k-1}^{0.525}.$

It is conjectured that $p_k-p_{k-1}\ll \log^2 p_{k-1},$ perhaps with a constant as small as $2e^{-\gamma}\approx1.1229\ldots.$