Allaart's 2011 inequality can potentially be generalized by replacing the binary digit sequence with a more general sequence of positive real numbers. However, for such a generalization to hold, certain conditions would need to be imposed on this sequence. One crucial condition could be that the sequence is non-decreasing but not excessively so, possibly ensuring that each term is not too far ahead of the previous one. This constraint could be formulated as ensuring that each term in the sequence is not significantly larger than the preceding term, possibly by a factor of a constant greater than or equal to 2. Additionally, the sequence may need to be bounded to prevent divergence or erratic behavior in the inequality.