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Four sets of bounds on E (X^ - 1), E (X^ - 2) and var (X^ - 1) are computed in terms of = E (X), ² = E (X -) ², and a > 0 for which P (X a) = 1. The four sets correspond to X unrestricted, X symmetric at, X symmetric and unimodal at, and X unimodal at m. For X unimodal, bounds on var (X^ - 1) are omitted since they appear to have no simple closed form.
Robert Lew (Tue,) studied this question.