ناهيد قراخانلو

‎One of the major areas of research in Operator Theory an​d Matrix Analysis is operator inequalities‎. ‎Some inequalities such as Young an​d Heinz inequalities have became powerful tools in mathematics‎. ‎In this thesis‎, ‎we consider the joint convexity an​d concavity of some known trace functions without using the classical definition of operator convexity an​d concavity‎. ‎We present further improvements of the reverse Young an​d Heinz inequalities for positive real numbers‎. ‎We use these modified inequalities to obtain the operator version of the improved reverse Young an​d Heinz inequalities‎. ‎Finally‎, ‎we introduce the notions of three parameter operator geometric mean an​d operator entropies‎. ‎Then we obtain some operator inequalities for them which are refinements an​d generalizations of some existing inequalities‎.