Multidimensional Gru¨ ss-Cˇ ebysˇev and-Trapezoid-type Inequalities

Abstract

During the last two decades many researchers have given considerable attention to the famous inequalities (1), (3) and (4) associated to the names of Cˇ ebysˇev, Gru¨ss and Trapezoid. In view of the usefulness of these inequalities and their applications, many authors have investigated a large number of new multidimensional, Gru¨ss, Cˇ ebysˇev and Trapezoid type inequalities. Some of these results provide simple and elegant extensions of the inequalities (1), (3) and (4) and have a wider scope of applicability. These results did not just add new objects of study, but also brought new insights and techniques to handle such inequalities. This chapter deals with a number of new multidimensional inequalities discovered by various investigators, which claim their origin to the well-known inequalities in (1), (3) and (4). Some applications are given to illustrate the usefulness of certain inequalities.