Means that minimize relative error,and an associated integral equation

Determinations are made of the means that minimize various relative errors in the sense that the harmonic mean of a and b minimizes the traditional relative error on a, b]. The general problem for averaged relative error leads to a nonlinear integral equation for which we prove existence and uniqueness results, as well as constructive solution procedures. The inverse problem of finding the measure of relative error corresponding to a given mean is also analyzed in detail. These studies shed light on both new and previously known inequalities for specific means.