The mean of the reciprocal in a Cauchy-Stieltjes family

If the distribution of X belongs to a Cauchy-Stieltjes Kernel (CSK) family concentrated on the positive real line, we determine the expectation of the reciprocal of X as a function of the expectation m of X. This function characterizes the CSK family. Some examples and characterizations of quadratic and cubic CSK families are given.