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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.

In this paper we give a new refinement of the Lah–Ribarič inequality and, using the same technique, we give a refinement of the Jensen inequality. Using these results, a refinement of the discrete Hölder inequality and a refinement of some inequalities for discrete weighted power means and discrete weighted quasi-arithmetic means are obtained. We also give applications in the information theory; namely, we give some interesting estimations for the discrete Csiszár divergence and for its important special cases.

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

We generalize Jensen's integral inequality for real Stieltjes measure by using Montgomery identity under the effect of n-convex functions; also, we give different versions of Jensen's discrete inequality along with its converses for real weights. As an application, we give generalized variants of Hermite-Hadamard inequality. Montgomery identity has a great importance as many inequalities can be obtained from Montgomery identity in q-calculus and fractional integrals. Also, we give applications in information theory for our obtained results, especially for Zipf and Hybrid Zipf-Mandelbrot entropies., H.E.C. Pakistan under NRPU Project [7906]; RUDN University Strategic Academic Leadership Program, This research is supported by H.E.C. Pakistan under NRPU Project 7906 and RUDN University Strategic Academic Leadership Program.

In this paper, we prove several Jensen-Grüss type inequalities under various conditions. Some applications in information theory are also given.

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.

The classical Jensen inequality is a famous tool to construct new results in the theory of inequalities. It has numerous applications in abstract and applied sciences. In this monograph, some recent developments in theory of inequalities with respect to Jensen's inequality are collected and presented. Applications are given in information theory by evaluating the estimates for different entropies and divergences via some recent and existing refinements of Jensen's inequality. The results for convex functions in this context are also generalized for higher order convex functions by means of interpolating polynomials.

We establish new refinements and improvements of Popoviciu’s inequality for n-convex functions using Abel-Gontscharoff interpolating polynomial along with the aid of new Green functions. We construct new inequalities for n-convex functions and compute new upper bounds for Ostrowski and Grüss type inequalities. As an application of our work in information theory, we give new estimations for Shannon, Relative and Zipf-Mandelbrot entropies using generalized Popoviciu’s inequality.

The most important inequality for convex functions is the Jensen's inequality. To give refinements of Jensen's inequality is an extensively investigated theme with numerous methods, results and applications. This book is mainly devoted to cyclic refinements (cyclic permutations are used to define the refining terms), and their applications in information theory. It contains the most recent research results of this promising topic.

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.

The Jensen functional in its discrete form is brought in relation to the Csiszar divergence functional, this time via its monotonicity property. This approach presents a generalization of the previously obtained results that made use of interpolating Jensen-type inequalities. Thus the monotonicity property is integrated with the Zipf-Mandelbrot law and applied to f-divergences for probability distributions that originate from the Csiszar divergence functional: Kullback-Leibler divergence, Hellinger distance, Bhattacharyya distance, chi-square divergence, total variation distance. The Zipf-Mandelbrot and the Zipf law are widely used in various scientific fields and interdisciplinary and here the focus is on the aspect of the mathematical inequalities.

Inequalities make important part of mathematics. Majorization are used for finding some nice and applicable inaqualities. Applicable to information Theory

‎‎‎‎‎In this paper‎, ‎we present some interesting results related to the bounds of Zipf-Mandelbrot entropy and the $3$-convexity of the function‎. ‎Further‎, ‎we define linear functionals as the nonnegative differences of the obtained inequalities and we present mean value theorems for the linear functionals‎. ‎Finally‎, ‎we discuss the $n$-exponential convexity and the log-convexity of the functions‎ ‎associated with the linear functionals‎.

By means of one new Jensen-type inequality for signed measures which is characterized via several different Green functions, in this paper we derive new inequalities for generalized f-divergences. The applications on the Zipf-Mandelbrot law, as one specific kind of probability distributions, are also given.

In this paper we proved a converse to Sherman’s inequality. Using the concept of f-divergence we obtained some inequalities for the well-known entropies. We also introduced a new entropy by applying the Zipf-Mandelbrot law and derived some related inequalities

In this paper we show how the Zipf-Mandelbrot law is connected to the theory of majorization. Firstly we consider the Csiszar $f$-divergence for the Zipf-Mandelbrot law and then develop important majorization inequalities for these divergences. We also discuss some special cases for our generalized results by using the Zipf-Mandelbrot law. As applications, we present the majorization inequalities for various distances obtaining by some special convex functions in the Csiszar $f$-divergence for Z-M law like the Renyi $\alpha$-order entropy for Z-M law, variational distance for Z-M law, the Hellinger distance for Z-M law, $\chi^{2}$-distance for Z-M law and triangular discrimination for Z-M law. At the end, we give important applications of the Zipf's law in linguistics and obtain the bounds for the Kullback-Leibler divergence of the distributions associated to the English and the Russian languages.

The Publisher has retracted this article [1]. Owing to an administrative error it was published without the amendments made by the authors at the proof stage. A second version of the article with the authors’ amendments that should remain in the literature is [2]. The Publisher apologises to the authors and to readers for this error. All authors agree with this retraction.

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.

In this paper we obtain some estimates for the generalized f-divergence functional via converses of the Jensen and Edmundson-Lah- Ribaric inequalities for convex functions, and then we obtain some estimates for the Kullback- Leibler divergence. All of the obtained results are applied to Zipf-Mandelbrot law and Zipf law

In this paper we deal with analytical properties of the Zipf-Mandelbrot law. If total mass of this law is spread all over positive integers we come to Hurwitz ζ -function. As we show, it is very natural first to examine properties of Hurwitz ζ -function to derive properties of Zipf-Mandelbrot law. Using some well-known inequalities such as Chebyshev’s and Lyapunov’s inequality we are able to deduce a whole variety of theoretical characterizations that include, among others, log-convexity, log-subadditivity, exponential convexity.

In this paper we obtain some estimates for the generalized f-divergence functional via converses of the Jensen and Edmundson-Lah- Ribarič inequalities for convex functions, and then we obtain some estimates for the Kullback- Leibler divergence. All of the obtained results are applied to Zipf-Mandelbrot law and Zipf law.

For the analysis of doctoral dissertations in information science in the Republic of Croatia (from 1978 to 2007), keywords are used in order to get an insight into the development of information science. By the method of co-word analysis of keywords with which doctoral dissertations are indexed, a network of clusters that match following scientific disciplines is obtained: archival and documentation science, librarianship, communicology, museology, information science, information systems and lexicography. By cluster and data visualization and the overview of keywords frequency, the development of subjects and the correlation of clusters in information science, during the period of thirty years in which doctoral dissertation are made, is shown. The results of the co-word analysis about the development of information science in the Republic of Croatia are shown according to time periods, but also according to affiliation to certain disciplines inside the information science.

Inequalities and Zipf-Mandelbrot law is devoted to recent advances in variety of inequalities in Information theory especially for Zipf law and Zipf-Mandelbrot law. Since, Zipf-Mandelbrot law has been applied in various scientific disciplines and different kind of natural or social phenomena: from text mining, information retrieval animal communication, to gene expression and meny others, we hope that this book will be useful to different scientific communities