Consider n data samples {x1, ⋯, xn} such that o<L⩽xi⩽U < ∞. Let K = U/L; then it is shown that independent of n a lower bound on the ratio of the geometric mean to the arithmetic mean of the data samples is given by [lnK/(K − 1)]K{(1/lnK)−1/(K−1)}. This bound is useful in acoustic signal processing since it limits the amount of deviation that can be attributed to averaging logarithms vice taking the logarithm of the average of data samples. Both methods are currently in use at facilities specializing the processing of acoustic data. For a K of 10 dB, for example, the geometric mean is less than 1.5 dB below the arithmetic mean.
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H. Elizabeth Cox (1966) studied this question.