Recently, there has been an increased awareness of the one-sided nature of many econometric testing problems. For testing Ho : 0 = 0 against H a : 0 * 0 where 0 is px1, SenGupta and Vermeire (1986) introduced the class of locally most mean powerful (LMMP) unbiased tests. They are constructed to maximize the mean curvature of the power hypersurface in the neighbourhood of 0 = 0. Our interest is in testing H 0 against H a : 0 1 0, ..., 0 0 with at least one strict inequality. We show how LMMP critical regions can be constructed and note that they suggest a new form for the Lagrange multiplier test in one-sided testing problems. Applications considered in the context of the linear regression model include joint one-sided testing for non-zero regression coefficients, autoregressive disturbances, heteroscedastic disturbances, random regression coefficients and variance components.