This paper mainly focuses on in-depth research on inequalities on symmetric cones. We will further analyze and discuss the inequalities we developed on the second-order cone and develop more inequalities. According to our past research in dealing with second-order cone inequalities, we derive more inequalities concerning the eigenvalue version of Young’s inequality and trace a version of an inverse Young inequality and its applications. These conclusions align with the results established for the positive semidefinite cone, which is also a symmetric cone. It is of considerable help to the establishment of inequalities on symmetric cones and the analysis of their derivative algorithms.