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The more intense and stricter demands of scientific policy is that scientific production in the Humanities and Social Sciences be evaluated with bibliometrics methods and indicators (by impact factor, publishing in most cited journals, citations and visibility in „prestigious“ data bases, etc.). Theoreticians and advocates of bibliometrics and scientometrics methods are very skeptical regarding the application of these indicators in Humanities and Social Sciences. Social sciences and humanities do not have only scientific function. Papers are published on national languages and are not primarily intended for scientific communication but for much larger audience. That means that papers, essays and books in Humanities and Social Sciences have also cultural, educational, social and political purposes. In spite of actual philosophy of globalization and theory that science is (or must be) international - it is a fact that Humanities and Social Sciences are primarily oriented to the research and promotion of cultural and national identity. Their subjects bind them to the national domain, although this does not exclude a multinational research, international comparisons and exchange of experiences. Our stand is that papers in Humanities and Social Sciences cannot be measured by strict standards of scientific communication taken from the natural, medical and engineering sciences (in terms of communication forms). Because the data do suggest that this approach eliminates as non-scientific work 2/3 of production of workers in the Humanities and Social Sciences. It is obvious that cultural, educational, political and pedagogical functions are an integral part of the social and scientific responsibility of the author in the Humanities and Social Sciences. Therefore, their activity can be measured rather by influence factor and not by impact factor on which literal application of scientometric’s methods insists.

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 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.

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.

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

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 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.

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