In this thesis , we introduce the semi doubly stochastic operator S, which is between doubly stochastic operators and column stochastic operators , so as to apply to characterized operator S on l^1 such that Sf is majorized by f for every f in l^1 . We present some classes of majorization preservers on l^1 under semi doubly stochastic operators . Moreover , as an application of our result in quantum physics , the convertibility of pure states of a composite system by local operations and classical communication has been considered . Then by using semi doubly stochastic operator, the notion of majorization is extended to the space of all integrable functions on a sigma-finite measure space. Also, we consider stochastic operators or Markov operators . We show that , on finite-dimensional spaces , any stochastic operator can be approximated by a sequence of stochastic integral operators. We collect a number of results for vector-valued functions on L^1 , simplifying some proofs found in the literature . In particular , matrix majorization and multivariate majorization are related in R^n . In R , these are also equivalent to convex function inequalities .