Birkhauser New Perspectives on the Theory of Inequalities for Integral and Sum

This book provides new contributions to the theory of inequalities for integral and sum, and includes four chapters. In the first chapter, linear inequalities via interpolation polynomials and green functions are discussed. New results related to Popoviciu type linear inequalities via extension of the Montgomery identity, the Taylor formula, Abel-Gontscharoff's interpolation polynomials, Hermite interpolation polynomials and the Fink identity with Green's functions, are presented. The second chapter is dedicated to Ostrowski's inequality and results with applications to numerical integration and probability theory. The third chapter deals with results involving functions with nondecreasing increments. Real life applications are discussed, as well as and connection of functions with nondecreasing increments together with many important concepts including arithmetic integral mean, wright convex functions, convex functions, nabla-convex functions, Jensen m-convex functions, m-convex functions, m-nabla-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace transform and exponentially convex functions, by using the finite difference operator of order m. The fourth chapter is mainly based on Popoviciu and Cebysev-Popoviciu type identities and inequalities. In this last chapter, the authors present results by using delta and nabla operators of higher order. From the Back Cover This book provides new contributions to the theory of inequalities for integral and sum, and includes four chapters. In the first chapter, linear inequalities via interpolation polynomials and green functions are discussed. New results related to Popoviciu type linear inequalities via extension of the Montgomery identity, the Taylor formula, Abel-Gontscharoff's interpolation polynomials, Hermite interpolation polynomials and the Fink identity with Green's functions, are presented. The second chapter is dedicated to Ostrowski's inequality and results with applications to numerical integration and probability theory. The third chapter deals with results involving functions with nondecreasing increments. Real life applications are discussed, as well as and connection of functions with nondecreasing increments together with many important concepts including arithmetic integral mean, wright convex functions, convex functions, nabla-convex functions, Jensen m-convex functions, m-convex functions, m-nabla-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace transform and exponentially convex functions, by using the finite difference operator of order m. The fourth chapter is mainly based on Popoviciu and Cebysev-Popoviciu type identities and inequalities. In this last chapter, the authors present results by using delta and nabla operators of higher order. About the Author Dr. Nazia Irshad is associated with Dawood University of Engineering and Technology as an Assistant Professor of Mathematics. Her field specialization is in the Mathematical Inequalities and Applications. Dr. Asif Raza Khan is an Assistant Professor and HEC Approved PhD Supervisor at Department of Mathematics, University of Karachi. His field of research is Mathematical Inequalities and Applications, Convex Analysis, Time Scale and Numerical Integration. He is the author of a monograph entitled "General Linear Inequalities and Positivity: Higher Order Convexity". Dr. Faraz Mehmood has a PhD from the University of Karachi. He has been working at Dawood University of Engineering and Technology since November 2009 and presently serving his duties there as an Associate Professor of Mathematics. His field of research is Mathematical Inequalities and Applications, Convex Analysis, Time Scale and Numerical Integration. Prof. Dr. Josip Pečaric is a Croatian mathematician. He has recently retired as a professor of mathematics from the Faculty of Textile Technology at the U