Generalizations of Aczel's inequality and Popoviciu's inequality

In this paper, we give new generalizations of Aczel's inequality and Popoviciu's inequality, which has broad applications to mathematical analysis. We improve a number of algebraic inequalities and obtain some new ones. Furthermore, we also point out that a new generalization of Aczel's inequality by Mitrinovic (2), Wang (7) and Kuang (8) is not true in general. In the end, we give an analogue of Popoviciu's inequality.