  The maximization of Nash welfare, which equals the geometric mean of agents'
utilities, is widely studied because it balances efficiency and fairness in
resource allocation problems. Banerjee, Gkatzelis, Gorokh, and Jin (2022)
recently introduced the model of online Nash welfare maximization for $T$
divisible items and $N$ agents with additive utilities with predictions of each
agent's utility for receiving all items. They gave online algorithms whose
competitive ratios are logarithmic. We initiate the study of online Nash
welfare maximization \emph{without predictions}, assuming either that the
agents' utilities for receiving all items differ by a bounded ratio, or that
their utilities for the Nash welfare maximizing allocation differ by a bounded
ratio. We design online algorithms whose competitive ratios only depend on the
logarithms of the aforementioned ratios of agents' utilities and the number of
agents.
