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Gilda Pecaric

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Elizabeth FitzGerald (née Brown)

The Open University

University of California, Santa Barbara

Güzin Mazman Akar

Hacettepe University

Kansas State University

Walden University

National University of Malaysia

Hacettepe University

Canakkale Onsekiz Mart University

Katya (Ekaterina) Pechenkina

Swinburne University of Technology

Md. Saifuddin Khalid

Technical University of Denmark (DTU)

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Papers by Gilda Pecaric

Uniform Treatment of Jensen’s Inequality by Montgomery Identity

Journal of Mathematics, May 15, 2021

We generalize Jensen's integral inequality for real Stieltjes measure by using Montgomery identit... more 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.

Estimation of divergence measures on time scales via Taylor’s polynomial and Green’s function with applications in q-calculus

Advances in Difference Equations

Taylor’s polynomial and Green’s function are used to obtain new generalizations of an inequality ... more Taylor’s polynomial and Green’s function are used to obtain new generalizations of an inequality for higher order convex functions containing Csiszár divergence on time scales. Various new inequalities for some divergence measures in quantum calculus and h-discrete calculus are also established.

Estimation of divergences on time scales via the Green function and Fink’s identity

Advances in Difference Equations

The aim of the present paper is to obtain new generalizations of an inequality for n-convex funct... more The aim of the present paper is to obtain new generalizations of an inequality for n-convex functions involving Csiszár divergence on time scales using the Green function along with Fink’s identity. Some new results in h-discrete calculus and quantum calculus are also presented. Moreover, inequalities for some divergence measures are also deduced.

Uniform Treatment of Jensen’s Inequality by Montgomery Identity

Journal of Mathematics

We generalize Jensen’s integral inequality for real Stieltjes measure by using Montgomery identit... more 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.

Estimation of f-divergence and Shannon entropy by Bullen type inequalities via Fink’s identity

Filomat, 2022

By using Fink?s identity some new generalizations of Levinson type inequalities for n-convex func... more By using Fink?s identity some new generalizations of Levinson type inequalities for n-convex functions are obtained. In seek of applications of our results to information theory, new generalizations based on f-divergence estimates are also proven. Moreover, some inequalities for Shannon entropies are deduced as well.

Časopis National Security and the Future: bibliometrijska analiza časopisa u razdoblju 2000.-2008. godine

National security and the future, Mar 24, 2020

<jats:p>U radu je bibliometrijskim metodama analiziran časopis National Security and The Fu... more <jats:p>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.</jats:p>

New entropic bounds on time scales via Hermite interpolating polynomial

Journal of Inequalities and Applications, 2021

Hermite’s interpolation is utilized to establish a new generalization of an inequality for higher... more Hermite’s interpolation is utilized to establish a new generalization of an inequality for higher order convex functions containing Csiszár divergence on time scales. New entropic bounds in q-calculus and h-discrete calculus are also deduced. Some estimates for Zipf–Mandelbrot entropy are also given.

Pre-School Teachers' Informatics and Information Literacy

The life and activities of every man in the period of transition from the second into the third m... more The life and activities of every man in the period of transition from the second into the third millennium have been marked by epochal changes which appear as the consequence of scientific and technological revolution dominated by highly developed information and communication technology. Informatics and information education based on information and communication technologies is information society’s basic trait and premise for technological education of all persons involved in the educational process. The work deals with the necessity of transformation of the present pre-school teachers’ concept of informatics and information education subject to conditions of reformatory processes in higher education of the Republic of Croatia. Suggestions and possible solutions for the change of curriculum and pedagogical aspects of the realization of preschool teachers’ education are provided.

Converses of the Edmundson-Lah-Ribaric inequality for the generalized Csiszar divergence with applications to Zipf-Mandelbrot law

Annals of the University of Craiova - Mathematics and Computer Science Series, 2018

In this paper we obtain some estimates for the generalized f-divergence functional via converses ... more 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.

Generalized Csiszár's f-divergence for Lipschitzian functions

Mathematical Inequalities & Applications, 2021

We started with the generalization of the Csiszár's f-divergence. We stated and proved Jensen's t... more We started with the generalization of the Csiszár'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 Rényi divergence and χ 2-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.

Superquadratic functions in information theory

Journal of Mathematical Inequalities, 2020

Using Jensen's inequality and the converse Jensen's inequality for superquadratic functions we ob... more

Estimation of divergence measures via weighted Jensen inequality on time scales

Journal of Inequalities and Applications, 2021

The main purpose of the presented paper is to obtain some time scale inequalities for different d... more The main purpose of the presented paper is to obtain some time scale inequalities for different divergences and distances by using weighted time scales Jensen’s inequality. These results offer new inequalities in h-discrete calculus and quantum calculus and extend some known results in the literature. The lower bounds of some divergence measures are also presented. Moreover, the obtained discrete results are given in the light of the Zipf–Mandelbrot law and the Zipf law.

Some inequalities for Csiszár divergence via theory of time scales

Advances in Difference Equations, 2020

In this paper, we present some inequalities for Csiszárf-divergence between two probability measu... more In this paper, we present some inequalities for Csiszárf-divergence between two probability measures on time scale. These results extend some known results in the literature and offer new results inh-discrete calculus and quantum calculus. We also present several inequalities for divergence measures.

Generalizations of cyclic refinements of Jensen's inequality by Lidstone's polynomial with applications in information theory

Journal of Mathematical Inequalities, 2020

Jensen's inequality plays pivotal role in attaining divergence between probability distributions.... more Jensen's inequality plays pivotal role in attaining divergence between probability distributions. 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 different 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.

Shannon type inequalities via time scales theory

Advances in Difference Equations, 2020

The aim of present paper is to obtain Shannon type inequalities using the extended version of Jen... more The aim of present paper is to obtain Shannon type inequalities using the extended version of Jensen’s inequality in time scales settings. The concept of differential entropy of a continuous random variable on time scales is introduced, and its bounds for some particular distributions are also estimated.

Converse to the Sherman inequality with applications

Mathematical Inequalities & Applications, 2019

In this paper we proved a converse to Sherman's inequality. Using the concept of fdivergence we o... more

On a Jensen-type inequality for generalized f-divergences and Zipf-Mandelbrot law

Mathematical Inequalities & Applications, 2019

By means of one new Jensen-type inequality for signed measures which is characterized via several... more 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.

Levinson type inequalities for higher order convex functions via Abel–Gontscharoff interpolation

Advances in Difference Equations, 2019

In this paper, Levinson type inequalities are studied for the class of higher order convex functi... more

Several new cyclic Jensen type inequalities and their applications

Journal of Inequalities and Applications, 2019

We present some fundamental results and definitions regarding Jensen's inequality with the aim of... more We present some fundamental results and definitions regarding Jensen's inequality with the aim of obtaining new generalizations of cyclic refinements of Jensen's inequality from convex to higher order convex functions using Taylor's formula. We discuss the monotonicity of functionals for n-convex functions at a point. Applications of our work include new bounds for some important inequalities used in information theory.

On Shannon and Zipf–Mandelbrot entropies and related results

Journal of Inequalities and Applications, 2019

In this paper, we present some interesting results related to the bounds of the Shannon and the Z... more

Uniform Treatment of Jensen’s Inequality by Montgomery Identity

Journal of Mathematics, May 15, 2021

We generalize Jensen's integral inequality for real Stieltjes measure by using Montgomery identit... more 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.

Estimation of divergence measures on time scales via Taylor’s polynomial and Green’s function with applications in q-calculus

Advances in Difference Equations

Taylor’s polynomial and Green’s function are used to obtain new generalizations of an inequality ... more Taylor’s polynomial and Green’s function are used to obtain new generalizations of an inequality for higher order convex functions containing Csiszár divergence on time scales. Various new inequalities for some divergence measures in quantum calculus and h-discrete calculus are also established.

Estimation of divergences on time scales via the Green function and Fink’s identity

Advances in Difference Equations

The aim of the present paper is to obtain new generalizations of an inequality for n-convex funct... more The aim of the present paper is to obtain new generalizations of an inequality for n-convex functions involving Csiszár divergence on time scales using the Green function along with Fink’s identity. Some new results in h-discrete calculus and quantum calculus are also presented. Moreover, inequalities for some divergence measures are also deduced.

Uniform Treatment of Jensen’s Inequality by Montgomery Identity

Journal of Mathematics

We generalize Jensen’s integral inequality for real Stieltjes measure by using Montgomery identit... more 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.

Estimation of f-divergence and Shannon entropy by Bullen type inequalities via Fink’s identity

Filomat, 2022

By using Fink?s identity some new generalizations of Levinson type inequalities for n-convex func... more By using Fink?s identity some new generalizations of Levinson type inequalities for n-convex functions are obtained. In seek of applications of our results to information theory, new generalizations based on f-divergence estimates are also proven. Moreover, some inequalities for Shannon entropies are deduced as well.

Časopis National Security and the Future: bibliometrijska analiza časopisa u razdoblju 2000.-2008. godine

National security and the future, Mar 24, 2020

<jats:p>U radu je bibliometrijskim metodama analiziran časopis National Security and The Fu... more <jats:p>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.</jats:p>

New entropic bounds on time scales via Hermite interpolating polynomial

Journal of Inequalities and Applications, 2021

Hermite’s interpolation is utilized to establish a new generalization of an inequality for higher... more Hermite’s interpolation is utilized to establish a new generalization of an inequality for higher order convex functions containing Csiszár divergence on time scales. New entropic bounds in q-calculus and h-discrete calculus are also deduced. Some estimates for Zipf–Mandelbrot entropy are also given.

Pre-School Teachers' Informatics and Information Literacy

The life and activities of every man in the period of transition from the second into the third m... more The life and activities of every man in the period of transition from the second into the third millennium have been marked by epochal changes which appear as the consequence of scientific and technological revolution dominated by highly developed information and communication technology. Informatics and information education based on information and communication technologies is information society’s basic trait and premise for technological education of all persons involved in the educational process. The work deals with the necessity of transformation of the present pre-school teachers’ concept of informatics and information education subject to conditions of reformatory processes in higher education of the Republic of Croatia. Suggestions and possible solutions for the change of curriculum and pedagogical aspects of the realization of preschool teachers’ education are provided.

Converses of the Edmundson-Lah-Ribaric inequality for the generalized Csiszar divergence with applications to Zipf-Mandelbrot law

Annals of the University of Craiova - Mathematics and Computer Science Series, 2018

In this paper we obtain some estimates for the generalized f-divergence functional via converses ... more 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.

Generalized Csiszár's f-divergence for Lipschitzian functions

Mathematical Inequalities & Applications, 2021

We started with the generalization of the Csiszár's f-divergence. We stated and proved Jensen's t... more We started with the generalization of the Csiszár'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 Rényi divergence and χ 2-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.

Superquadratic functions in information theory

Journal of Mathematical Inequalities, 2020

Using Jensen's inequality and the converse Jensen's inequality for superquadratic functions we ob... more

Estimation of divergence measures via weighted Jensen inequality on time scales

Journal of Inequalities and Applications, 2021

The main purpose of the presented paper is to obtain some time scale inequalities for different d... more The main purpose of the presented paper is to obtain some time scale inequalities for different divergences and distances by using weighted time scales Jensen’s inequality. These results offer new inequalities in h-discrete calculus and quantum calculus and extend some known results in the literature. The lower bounds of some divergence measures are also presented. Moreover, the obtained discrete results are given in the light of the Zipf–Mandelbrot law and the Zipf law.

Some inequalities for Csiszár divergence via theory of time scales

Advances in Difference Equations, 2020

In this paper, we present some inequalities for Csiszárf-divergence between two probability measu... more In this paper, we present some inequalities for Csiszárf-divergence between two probability measures on time scale. These results extend some known results in the literature and offer new results inh-discrete calculus and quantum calculus. We also present several inequalities for divergence measures.

Generalizations of cyclic refinements of Jensen's inequality by Lidstone's polynomial with applications in information theory

Journal of Mathematical Inequalities, 2020

Jensen's inequality plays pivotal role in attaining divergence between probability distributions.... more Jensen's inequality plays pivotal role in attaining divergence between probability distributions. 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 different 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.

Shannon type inequalities via time scales theory

Advances in Difference Equations, 2020

The aim of present paper is to obtain Shannon type inequalities using the extended version of Jen... more The aim of present paper is to obtain Shannon type inequalities using the extended version of Jensen’s inequality in time scales settings. The concept of differential entropy of a continuous random variable on time scales is introduced, and its bounds for some particular distributions are also estimated.

Converse to the Sherman inequality with applications

Mathematical Inequalities & Applications, 2019

In this paper we proved a converse to Sherman's inequality. Using the concept of fdivergence we o... more

On a Jensen-type inequality for generalized f-divergences and Zipf-Mandelbrot law

Mathematical Inequalities & Applications, 2019

By means of one new Jensen-type inequality for signed measures which is characterized via several... more 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.

Levinson type inequalities for higher order convex functions via Abel–Gontscharoff interpolation

Advances in Difference Equations, 2019

In this paper, Levinson type inequalities are studied for the class of higher order convex functi... more

Several new cyclic Jensen type inequalities and their applications

Journal of Inequalities and Applications, 2019

We present some fundamental results and definitions regarding Jensen's inequality with the aim of... more We present some fundamental results and definitions regarding Jensen's inequality with the aim of obtaining new generalizations of cyclic refinements of Jensen's inequality from convex to higher order convex functions using Taylor's formula. We discuss the monotonicity of functionals for n-convex functions at a point. Applications of our work include new bounds for some important inequalities used in information theory.

On Shannon and Zipf–Mandelbrot entropies and related results

Journal of Inequalities and Applications, 2019

In this paper, we present some interesting results related to the bounds of the Shannon and the Z... more