Throughout this paper we are concerned with the problem of estimating a real parameter when the loss function is such that the Bayes estimate exists, is unique, and satisfies a simple Equation, (1.5). If the estimate is unbiased (in the general sense of Lehmann ) we show under weak conditions that it must satisfy another Equation, (1.14). The main result of Section 1, Theorem 1.3, shows that, in general, these two equations are incompatible unless the Bayes risk is 0. This extends Theorem 11.24 of  which states that in estimation with quadratic loss, unbiased Bayes estimates have Bayes risk 0. Some counter-examples at the end of the section indicate the limits of this incompatibility result.
"A Note on Bayes Estimates." Ann. Math. Statist. 38 (6) 1907 - 1911, December, 1967. https://doi.org/10.1214/aoms/1177698625