In this thesis, we first state and prove a reverse to the Young inequality in the set of complex numbers which we call the inverse Young inequality , (abbreviated IYI). After that applying operator monotone and operator convex functions, the IYI will be proved to hold for eigenvalues of positive definite operators acting on a finite dimensional complex Hilbert space. Then, using this, we obtain an operator form of IYI which can be considered as a complement to an excellent result of Ando. Continuing our progress, we then apply a result of Hirzallah and Kittaneh as well as a necessary and sufficient condition for the case of equality in the scalar Young inequality, to present a necessary and sufficient condition for equality in the operator form of the IYI. Moreover, based on the complex representation of the quaternion operators, we state and prove a quaternion form of the IYI and its case of equality. We bring also some of the applications of operator inequalities, especially of trace inequalities in quantum mechanics. Finally we point out some of our future works we aim to do in some weeks later. Key words : operator inequality, operator monotone function, operator convex function, quaternion operators, Young inequality, inverse Young inequality, operator mean, spectral theorem, eigenvalues, spectrum, ltr"