Some Families of Integral, Trigonometric and Other Related Inequalities

In this lecture, we propose to survey several interesting recent developments on improvements and generalizations of what is popularly known as Steffensen’s integral inequality. Moreover, as a by-product of the investigation presented here, we show how one can correct an error in a recent generalization of Steffensen’s inequality. We also present a brief survey of some old and new inequalities associated with trigonometric functions. These include (among other results) a weighted and exponential generalization of what is popularly known as Wilker’s inequality and a substantially improved version of the Sándor-Bencze conjectured inequality. References to many related developments in recent years are also provided for the interested reader.