Abstarct: In the first part of this thesis we briefly state some basic facts and definitions about bounded operators on Hilbert spaces,C^*-algebras and their positive elements. Then we show that the symmetrized product AB + BA of two positive operators A and B is positive if and only if f(A+B) ≤ f(A)+ f(B) for all non-negative operator monotone functions f on [0,∞) and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition f ∘ g of an operator convex function f on [0,∞) and a non-negative operator monotone function g on an interval (a,b) is operator monotone and present some applications.