  We present new lower and upper bounds for the compression rate of binary
prefix codes optimized over memoryless sources according to two related
exponential codeword length objectives. The objectives explored here are
exponential-average length and exponential-average redundancy. The first of
these relates to various problems involving queueing, uncertainty, and lossless
communications, and it can be reduced to the second, which has properties more
amenable to analysis. These bounds, some of which are tight, are in terms of a
form of entropy and/or the probability of an input symbol, improving on
recently discovered bounds of similar form. We also observe properties of
optimal codes over the exponential-average redundancy utility.
