Inequalities involving q-integrals for convex function classes

Convex functions and inequality theory are two fields that has a close relationship, they have been attracted attentşon of many researchers and several studies have been presented. Especially, integral inequalities for convex functions have been dealed intensively and interested with the problem of finding bounds related to mean value of convex functions. In these studies that have been presented in this field, different derivative and integral operators such as Caputo derivative, Riemann-Liouville integrals, conformable fractional derivatives and k-local fractional derivatives have been used. In this thesis, firstly a general investigation has been performaed on the concept of convexity and the kinds of convexity. Later, analysis of quantum calculus have been presented and emphasized on q-derivative, q-integral and properties of them. After, it is referred to inequalities and results that include q-integrals on the classes of convex functions. Finally, classical integral inequalities such as Steffensen inequality, Grüss inequality, Chebysev inequality and Ostrowski inequality have been given. Besides, two new integral inequalities for m-convex functions have been obtained via q-integrals. Keywords: q-derivative, q-integral, quantum calculus, convex function, integral inequalities.