We study two kinds of conjectural bounds for the prime gap after the kth prime p_k: (A) p_k+1 < p_k^1+1/k and (B) p_k+1 - p_k < log²p_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 → 1 as k → ∞ .