Matloob Anwar

New Means of Cauchy's Type

We will introduce new means of Cauchys type defined, for example, as in the case when , . We will show that this new Cauchys mean is monotonic, that is, the following result. Theorem. Let , such that , . Then for , one has . We will also give some related comparison results.

On Some Improvements of the Jensen Inequality with Some Applications

An improvement of the Jensen inequality for convex and monotone function is given as well as various applications for mean. Similar results for related inequalities of the Jensen type are also obtained. Also some applications of the Cauchy mean and the Jensen inequality are discussed.

Cauchy Means of the Popoviciu Type

We discuss log-convexity for the differences of the Popoviciu inequalities and introduce some mean value theorems and related results. Also we give the Cauchy means of the Popoviciu type and we show that these means are monotonic.

On Logarithmic Convexity for Ky-Fan Inequality

We give an improvement and a reversion of the well-known Ky-Fan inequality as well as some related results.

Jensen Functionals on Time Scales for Several Variables

We define Jensen functionals and concerned generalized means for several variables on time scales. We derive properties of Jensen functionals and apply them to generalized means. In this setting, we obtain generalizations, refinements, and conversions of many remarkable inequalities.