Jessen's functional, its properties and applications

Abstract

In this paper we consider Jessen's functional, defined by means of a positive isotonic linear functional, and investigate its properties. Derived results are then applied to weighted generalized power means, which yields extensions of some recent results, known from the literature. In particular, we obtain the whole series of refinements and converses of numerous classical inequalities such as the arithmetic-geometric mean inequality, Young's inequality and Hölder's inequality

On further refinements of the Jensen inequality and applications

“…This inequality is still of interest to numerous mathematicians. In particular, it has been proved in [6] that $$$$ {\displaystyle \begin{array}{cc}\hfill n\kern3pt \underset{1\le i\le n}{\min}\left\{{p}_i\right\}\left[{m}_s^s\left(\mathbf{x}\right)-{m}_r^s\left(\mathbf{x}\right)\right]\le & \kern0.2em {M}_s^s\left(\mathbf{x},\mathbf{p}\right)-{M}_r^s\left(\mathbf{x},\mathbf{p}\right)\hfill \\ {}\hfill \le & \kern0.2em n\kern3pt \underset{1\le i\le n}{\max}\left\{{p}_i\right\}\left[{m}_s^s\left(\mathbf{x}\right)-{m}_r^s\left(\mathbf{x}\right)\right].\hfill \end{array}} $$$$ …”

“…Clearly, the left inequality in () provides a refinement of (), while the right one represents a reverse expressed in terms of the corresponding non‐weighted means. For a comprehensive study of power means including refinements and generalizations, the reader is referred to monographs [2, 4], as well as to papers [6, 7] and the references cited therein.…”

“…By virtue of the monotonicity property, Krni ć et al [6] established the mutual bounds for  n (𝑓 , x, p) expressed in terms of the corresponding non-weighted functional; that is, they proved that…”

“…By virtue of the monotonicity property, Krnić et al [6] established the mutual bounds for $$$ {\mathcal{J}}_n\left(f,\mathbf{x},\mathbf{p}\right) $$$ expressed in terms of the corresponding non‐weighted functional; that is, they proved that $$$$ n\kern3pt \underset{1\le i\le n}{\min}\left\{{p}_i\right\}{\mathcal{I}}_n\left(f,\mathbf{x}\right)\le {\mathcal{J}}_n\left(f,\mathbf{x},\mathbf{p}\right)\le n\kern3pt \underset{1\le i\le n}{\max}\left\{{p}_i\right\}{\mathcal{I}}_n\left(f,\mathbf{x}\right), $$$$ where $$$ {\mathcal{I}}_n\left(f,\mathbf{x}\right) $$$ stands for the associated non‐weighted functional, that is, $$$$ {\mathcal{I}}_n\left(f,\mathbf{x}\right)=\frac{1}{n}\sum \limits_{i=1}^nf\left({x}_i\right)-f\left(\frac{1}{n}\sum \limits_{i=1}^n{x}_i\right). $$$$ The lower bound in () is the refinement, while the upper one is the reverse of the Jensen inequality.…”

“…$$ The lower bound in () is the refinement, while the upper one is the reverse of the Jensen inequality. Based on this property, numerous inequalities such as the Young inequality, the Hölder inequality, power mean inequalities, and so forth have been refined (see, e.g., [6–8] and the references cited therein). For a detailed overview of the classical and new results in connection to the Jensen inequality, the reader is referred to monographs [2, 4, 9], as well as to the references cited therein.…”