The study of inequalities reflects a general tendency in modern and contemporary mathematics which may be explained by its intimate connection with the many and various aspects of the discipline and with its applications in science. There are now numerous applications of inequalities in a wide variety of fields, from mathematical physics and biology to information theory and economics. It is also clear that the impact in applications will grow even more spectacularly in the future due to the wide accessibility of the new results to various scientists who use the Internet as a primary source of information.
Taking into account the 21st Century new principle: If you are not online, then you do not exist the talk will review in its first part the impact, achievements and failures for the last 14 years of the international web-based Research Group of Mathematical Inequalities & Applications (RGMIA), that comprises more than 1300 members from all over the world. In the second part, some new reverses of the celebrated Jensen’s integral inequality for convex functions with applications to continuous functions of self-adjoint operators in Hilbert spaces and fdivergence measures in information theory are presented.