Upper bounds for prime gaps related to Firoozbakht's conjecture

We study two kinds of conjectural bounds for the prime gap after the k-th prime $p_k$: (A) $p_{k+1}<(p_k{)}^{1+1\mathrm{/}k}$ 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 $b\to 1$ as $k\to \mathrm{\infty }$.