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New bounds for Shannon, Relative and Mandelbrot entropies via Hermite interpolating polynomial

Mehmood N., Butt S. I., Pečarić D., Pečarić J.

Demonstratio Mathematica

2018

Vol. 51, nr 1

112--130

To procure inequalities for divergences between probability distributions, Jensen’s inequality is the key to success. Shannon, Relative and Zipf-Mandelbrot entropies have many applications in many applied sciences, such as, in information theory, biology and economics, etc. We consider discrete and continuous cyclic refinements of Jensen’s inequality and extend them from convex function to higher order convex function by means of different new Green functions by employing Hermite interpolating polynomial whose error term is approximated by Peano’s kernal. As an application of our obtained results, we give new bounds for Shannon, Relative and Zipf-Mandelbrot entropies.

Corrigendum to “Generalizations of Opial-type inequalities in several independent variables” published in Demonstratio Math. 4(47) (2014), 324–335

Andrić M., Barbir A., Pečarić J., Roqia G.

Demonstratio Mathematica

2016

Vol. 49, nr 2

149--154

The purpose of this corrigendum is to correct an error in the earlier paper by the authors: Generalizations of Opial-type inequalities in several independent variables, Demonstratio Math.

Generalizations of Opial-type inequalities in several independent variables

Andric M., Barbir A., Pečarić J., Roqia G.

Demonstratio Mathematica

2014

Vol. 47, nr 4

839--847

In this paper, we consider Willett’s and Rozanova’s generalizations of Opial’s inequality and extend them to inequalities in several independent variables. Also, we present some new Opial-type inequalities in several independent variables.

On a refinement of the Majorisation type inequality

Adil Khan M., Niezgoda M., Pečarić J.

Demonstratio Mathematica

2011

Vol. 44, nr 1

49-57

In this note, two mean value theorems are proved by using some recent results by Barnett et al. [N. S. Barnett, P. Cerone, S. S. Dragomir, Majorisation inequalities for Stieltjes integrals, Appl. Math. Lett. 22 (2009), 416-421]. A new class of Cauchy type means for two functions is studied. Logarithmic convexity for differences of power means is proved. Monotonicity of Cauchy means is shown.