Thinking about the prime number theorem, I wondered whether it is known that there is some constant $c$ such that $π(x+y) ≤ π(x) + c·y/\ln(y)$ for every integers $x,y > 1$. I read that experts believe $π(x+y) ≤ π(x) + π(y)$ fails for some $y$, since it fails for $y = 3159$ if the k-tuple conjecture holds, but it is just barely false, so I am curious if it is known to be true if the inequality is relaxed by a constant factor. If so, is it also known that $π(x+y) ≤ π(x) + π(y) + c·\!\sqrt{y}·\ln(y)$ for some constant $c$? I simply do not know how to search for such conjectures, and neither Wikipedia nor Wolfram seem to state any results that would affirm or refute these two conjectures easily, so any references would be appreciated!
(After a year, I've now posted it on MO.)