By using a Lidstone interpolation, Green’s function and Montogomery’s identity, we prove a new generalization of Steffensen’s inequality. Some related inequalities providing generalizations of certain results given in (J. Math. Inequal. 9(2):481–487, 2015) have also been obtained. Moreover, from these inequalities, we formulate linear functionals and describe their properties. © 2018, The Author(s).