Integral inequalities for quantum integrals and their applications

This thesis consists of five main parts. In the first part, q-Hermite-Hadamard inequality and kinds of this inequality have been proved and quantum predictions for q-midpoint type inequalities for convex and quasi convex functions have been obtained. In addition, it is shown that q=1 gives the classical results. In the second part, Hermite-Hadamard inequality and the types of this inequality have been proved on (p,q)-calculus. At the same time, quantum estimations for the convex and quasi convex functions (p,q)-midpoint inequalities have been obtained. Also, previous studies of the classical analysis were obtained in the case of q=1 and p = 1. In the third part, a new definition is given to quantum integrals with a new perspective and the properties of the new q-integral definition have been proved. Furthermore, q-Hermite-Hadamard integral inequalities were obtained with the help of this new q-integral. In the fourth part, the inequalities of Young, Hölder and Minkowski have been proven for q-integral. Additionally, quantum estimations for Ostrowski type inequalities were obtained. In the last part, q-Gamma-Beta functions were redefined using q-integral. Also, the properties, results and the relationship between these functions were investigated.