  We study two kinds of conjectural bounds for the prime gap after the k-th
prime $p_k$: (A) $p_{k+1} < (p_k)^{1+1/k}$ and (B) $p_{k+1}-p_k < \log^2 p_k -
\log p_k - b$ for $k>9$. The upper bound (A) is equivalent to Firoozbakht's
conjecture. We prove that (A) implies (B) with $b=1$; on the other hand, (B)
with $b=1.17$ implies (A). We also give other sufficient conditions for (A)
that have the form (B) with $b\to1$ as $k\to\infty$.
