Inequalities for convex functions and some of their applications

In this thesis, firstly, we collect and classify various proofs and generalizations of Jensen's inequality, which is a well-known inequality in convex analysis, and the associated Hermite-Hadamard inequality. Then, using Riemann integral sums, we give a new proof of Hermite-Hadamard inequality. We combine Hermite-Hadamard inequality with the Hölder's inequality to obtain a new inequality for the product of convex fuctions. Finally, using the Jensen's and Hermite-Hadamard inequalities, we prove numerous inequalities for the class of functions whose derivatives are convex.