In this article, a nearly unbiased test for relevant difference is proposed. This test works under heterogeneous population variances as well as under the homoscedasticity assumption. Its rejection region is limited by functions  that depend on the empirical variances. h is stated implicitly by a partial differential equation, an exact solution of which would provide a test that is exactly similar at the boundary of the null hypothesis of non-equivalence. h is approximated by Taylor series up to third powers in the reciprocal number of degrees of freedom. This suffices to obtain error probabilities of the first kind that are very close to a nominal level of  at the boundary of the null hypothesis. In the case of heterogeneous population variances, they range between 0.0499 and 0.0501 if there are more than 12 data points in each group.