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Consider the problem of finding optimal bounds on the expected value of piecewise polynomials over all measures with a given set of moments. This is a special class of generalized Tchebycheff inequalities in probability theory. We study this problem within the framework of conic programming. Relying on a general approximation scheme for conic programming, we show that these bounds can be numerically computed or approximated via semidefinite programming. We also describe some applications of this class of generalized Tchebycheff inequalities in probability, finance, and inventory theory.