Review on young's inequality

Abstract

Young's inequality is a nice inequality which we are using in various concept of Mathematics. Some of its applications are envisaged for the development of proofs of other theorems and results. The main object of this project is to review and discuss such type of concepts and show its different kinds of proofs and applications. Here we develop the similar kinds of the inequality in different types of spaces ,i.e., finite dimensional as well as infinite dimensional spaces. In the beginning we start with the statement of Young's inequality which is already discussed by the Mathematician for the euclidian-space and Lebesgue space. We have extended this ideas to the abstract Banach spaces and studied its application by changing various condition and assumptions. In this sequel we have proved reverse Young's inequality and Fenchel- Young's inequality. Also we have investigated the affect of product and convolution of two functions on the Young's inequalities.