New inequalities for some classes of convex functions

This study consists of five chapters. In the first section, the history and development of the concept of inequality and convexity and the studies on this subject are given, and it is stated in which areas the concept of convexity is encountered in daily life. In the second section, basic definitions and concepts are introduced, definitions, properties and theorems related to convex function classes are given. In the third chapter, the inequalities known in the literature, previously studied Hermite-Hadamard type inequalities, means, midpoint and trapezoidal formulas are presented together with their applications. In the fourth section, by obtaining complex trapezoidal equations for n-order differentiable functions, perturbed trapezoidal integral inequalities have been constituded the convex function, s-convex function, tgs-convex function, log-convex function, s-log-convex function, strongly convex function, strongly s-convex function, strongly tgs-convex function, strongly log-convex function with this equation and the relationship between these inequalities has been examined. The applications of these inequalities have been examined together with the means, and their applications are given by using the midpoint and trapezoidal formulas under certain reductions. It has been revealed that more sensitive solutions can be found under the inequalities obtained with strongly convex function classes of the given error functions. Finally, in the fifth chapter, the results of this study and suggestions are given.