Abstract
The main aim of this paper is to give an improvement of the recent result on the sharpness of the Jensen inequality. The results given here are obtained using different Green functions and considering the case of the real Stieltjes measure, not necessarily positive. Finally, some applications involving various types of f-divergences and Zipf–Mandelbrot law are presented.
Highlights
The Jensen inequality is one of the most famous and most important inequalities in mathematical analysis.In [2], some estimates about the sharpness of the Jensen inequality are given
The authors in [2] expanded φ(f (x)) around any given value of f (x), say around c = f (x0), which can be arbitrarily chosen in the domain I of φ, such that c = f (x0) is in the interior of I, and as their first result, they get the following inequalities: 0 ≤ φ f (x) dx – φ f (x) dx
The main aim of our paper is to give an improvement of that result using various Green functions and considering the case of the real Stieltjes measure, not necessarily positive
Summary
The Jensen inequality is one of the most famous and most important inequalities in mathematical analysis. In [2], some estimates about the sharpness of the Jensen inequality are given. The difference φ f (x) dx – φ f (x) dx is estimated, where φ is a convex function of class C2. The authors in [2] expanded φ(f (x)) around any given value of f (x), say around c = f (x0), which can be arbitrarily chosen in the domain I of φ, such that c = f (x0) is in the interior of I, and as their first result, they get the following inequalities:. The main aim of our paper is to give an improvement of that result using various Green functions and considering the case of the real Stieltjes measure, not necessarily positive
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