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In this work, Levinson type inequalities involving two types of data points are proved using Green functions and the Hermite interpolating polynomial for the class of n-convex functions. In seek of applications to information theory some estimates for new functionals are obtained, based on $$\mathbf {f}$$ -divergence. Moreover, some inequalities involving Shannon entropies are deduced as well.

We started with the generalization of the Csisz ́ar’s f -divergence. We stated and proved Jensen’s type inequality for L-Lipschitzian functions. The results for commonly used examples of f-divergences, such as the Kullbach-Leibler divergence, the Hellinger divergence, the R ́enyi divergence and χ2 -distance are derived. Further, we examined two specific averaging functions, previously known in the literature. Finally, we obtained interesting results concerning the Zipf-Mandelbrot law.

We estimated the different entropies like Shannon entropy, Renyi divergences, Csiszar divergence by using the Jensen’s type functionals. The Zipf’s mandelbrot law and hybrid Zipf’s mandelbrot law are used to estimate the Shannon entropy. Further the Hermite interpolating polynomial is used to generalize the new inequalities for m-convex function.

The first issue of the National Security and The Future journal was published in 2000. In the first 20 years of journal, 134 authors published 257 papers. In this paper we provide an alphabetical overview of the authors and their biographical information. The authors' biographical information is published in its original form as it was published in the journal. The information about the authors is published in the language (English or Croatian) used for the publication of their paper in the journal. Biographical information is not updated and it refers to the year in which the papers were published. The author's information also includes information on the issue number or numbers of the journal in which the author's papers were published. Only the author's name and the issue number or numbers in which the author published their papers are given for the papers that do not include the biographical information of the author.

U radu je bibliometrijskim metodama analiziran časopis National Security and The Future. Obrađeno je razdoblje od prvog broja časopisa objavljenog 2000. godine do zadnjeg broja objavljenog u 2018. godini. Analizirana građa je prikupljena s portala HRČAK (Portal hrvatskih znanstvenih i stručnih časopisa) ili sa stranice samog časopisa. Ukupno je prikupljeno i analizirano objavljenih 257 članaka, koje potpisuje 134 različita autora tijekom 19 godina izlaženja časopisa. Analizom su obuhvaćeni različiti parametri od autora, čitanost časopisa do ključnih riječi i supojavnosti ključnih riječi točnije pojavnosti i supojavnosti obrađenih tema.

In this work, some new functional of Jensen-type inequalities are constructed using Shannon entropy, f-divergence, and Rényi divergence, and some estimates are obtained for these new functionals. Also using the Zipf–Mandelbrot law and hybrid Zipf–Mandelbrot law, we investigate some bounds for these new functionals. Furthermore, we generalize these new functionals for m-convex function using Lidstone polynomial.

Using Jensen's inequality and the converse Jensen's inequality for superquadratic functions we obtain new estimates for Shannon's entropy of the random variable X and derive new lower and upper bounds for the Shannon entropy in the terms of the Zipf and Zipf - Mandelbrot's law.

Jensen’s inequality plays pivotal role in attaining divergence between probability dis- tributions. Shannon, Relative and Zipf-Mandelbrot entropies have ample applications in many applied sciences, especially in information theory, biology, economics, etc. In the present paper, we have obtained new generalizations of cyclic refinements of Jensen’s inequality using differ- ent new Green functions by employing Lidstone’s polynomial. As an application of our obtained results we have given new entropic bounds. Also, we have established the connections between Shannon and Relative entropy with Zipf-Mandelbrot entropy

Zipf–Mandelbrot entropy has many applications in many applied sciences, for example, in information theory, ecology and economics, etc. The main purpose of this paper is to associate results for convex functions with Zipf–Mandelbrot entropy. First of all, we present some inequalities for convex functions and their applications for Csiszar divergence. Then we use two classes of convex functions and apply inequalities for convex functions and establish several results for Zipf–Mandelbrot entropy. Related results for Zipf–Mandelbrot entropies corresponding to different parameters have been proved.

There is a unified approach, maximization of Shannon entropy, that naturally follows the path of generalization from Zipf’s to hybrid Zipf’s law. Extending this idea we make transition from Zipf-Mandelbrot to hybrid Zipf-Mandelbrot law. It is interesting that examination of its densities provides some new insights of Lerch’s transcendent

In this work, we estimated the different entropies like Shannon entropy, Rényi divergences, Csiszár divergence by using Jensen’s type functionals. The Zipf’s–Mandelbrot law and hybrid Zipf’s–Mandelbrot law are used to estimate the Shannon entropy. The Abel–Gontscharoff Green functions and Fink’s Identity are used to construct new inequalities and generalized them for m-convex function.

In this paper, we consider the definition of “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law associated with the real utility distribution to give the results for majorizatioQn inequalities by using monotonic sequences. We obtain the equivalent statements between continuous convex functions and Green functions via majorization inequalities, “useful” Csiszár functional and “useful” Zipf-Mandelbrot law. By considering “useful” Csiszár divergence in the integral case, we give the results for integral majorization inequality. Towards the end, some applications are given.

In this paper we derive some Jensen and Edmundson-Lah-Ribarič type inequalities for positive linear functionals and 3-convex functions. Obtained results are then applied to generalized means and power means, as well as to the generalized f-divergence functional. Examples with Zipf-Mandelbrot law are given.

Shannon and Zipf-Mandelbrot entropies have many applications in many applied sciences, for example, in information theory, biology and economics, etc. In this paper, we consider two refinements of the well-know Jensen inequality and obtain different bounds for Shannon and Zipf-Mandelbrot entropies. First of all, we use some convex functions and manipulate the weights and domain of the functions and deduce results for Shannon entropy. We also discuss their particular cases. By using Zipf-Mandelbrot laws for different parameters in Shannon entropies results, we obtain bounds for Zipf-Mandelbrot entropy. The idea used in this paper for obtaining the results may stimulate further research in this area, particularly for Zipf-Mandelbrot entropy.

The Jensen’s inequality plays a crucial role to obtain inequalities for divergences between probability distributions. In this paper, we introduce a new functional, based on the f- divergence functional, and then, we obtain some estimates for the new functional, the f- divergence and the Rényi divergence by applying a cyclic refinement of the Jensen’s inequality. Some inequalities for Rényi and Shannon entropies are obtained too. Zipf–Mandelbrot law is used to illustrate the results.

In this paper we derive some Jensen and Edmundson-Lah-Ribaric type inequalities for positive linear functionals without the assumption about the convexity of the functions that are involved. General results are then applied to generalized f-divergence functional. Examples with Zipf's law and Zipf-Mandelbrot law are given.