Improving Jensen-type inequalities via the Taylor interpolation formula

Abstract

The main objective of this paper is to establish some general improvements of the Jensen inequality for the classes of absolutely and completely monotonic functions.
The key role in this work is played by the Taylor interpolation formula. Besides the improvement of the  Jensen inequality, we also derive more accurate superadditivity and monotonicity relations
for the Jensen functional.  As an application, we obtain improved versions of  power mean inequalities and  the
H\"{o}lder inequality. Finally, we  obtain more accurate form of the  Lah-Ribari\v c inequality for the aforementioned classes of functions. In particular, by using the developed method, we also get a non-trivial lower bound for the non-weighted Jensen functional.