Mond-Pečarić Method in Operator Inequalities

Inequalities for bounded selfadjoint operators on a Hilbert space

Takayuki Furuta, Jadranka Mićić Hot, Josip Pečarić, Yuki Seo

The authors first state the fundamental properties of bounded linear operators on a Hilbert space. Then they describe how Jensen's type inequalities develop in the field of operator inequalities. The Mond-Pečarić method for convex functions has outlined a more complete picture of converses of Jensen's inequality in the field of operator inequalities. The better part of the book is devoted to presenting results by using said method.

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Euler integral identity, quadrature formulae and error estimations

From the point of view of inequality theory

Iva Franjić, Josip Pečarić, Ivan Perić, Ana Vukelić

The book contains generalizations of many classical quadrature formulas such as Simpson, dual Simpson, Maclaurin, Gauss, Lobatto and Radau. The specific feature of this book is that a unique method is used. This method is based on the Euler integral identities expressing expansion of a function in Bernoulli polynomials, proved by V. I. Krylov as a generalization of the first and the second Euler-Maclaurin sum formula. Mostly as error estimates, the book presents some aspects of generalizations, refinements and variants of four famous inequalities: the Hermite-Hadamard, the Ostrowski, the Bullen and the Iyengar inequality, which is an exception. Although related to the Hermite-Hadamard inequality in the same way as the Ostrowski inequality, Iyengar type inequalities are, it seems, beyond the reach of the method presented in this book. This is the reason why generalizations of the Iyengar inequality are given in the Addendum.

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Recent Advances in Hilbert-type Inequalities

A unified treatment of Hilbert-type inequalities

Mario Krnić, Josip Pečarić, Ivan Perić, Predrag Vuković

Recent Advances in Hilbert-type inequalities is a crown of decennial research of several authors in the theory of Hilbert-type inequalities. The book is based on some thirty significant papers dealing with this problem area, published in the course of the last ten years. The present book provides a unified approach to Hilbert-type inequalities, the main results are established in general sigma finite measure spaces, with general kernels and weight functions. A special emphasis has been dedicated to establishing diverse methods for improving original Hilbert-type inequalities. Diverse generalizations, refinements and applications of Hilbert-type inequalities are presented.

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Recent Developments of Mond-Pečarić Method in Operator Inequalities

Inequalities for bounded selfadjoint operators on a Hilbert space, II

Masatoshi Fujii, Jadranka Mićić Hot, Josip Pečarić, Yuki Seo

The authors first state the fundamental properties of bounded linear operators on a Hilbert space. Then they tell the history of the Kantorovich inequality, and describe how the Kantorovich inequality develops in the field of operator inequalities. The Mond- Pečarić method for convex functions has outlined a more complete picture of that inequality in the field of operator inequalities. The better part of the book is devoted to presenting many new results by using said method. In such context, the generalization and improvement of many famous inequalities are presented.

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General Integral Identities and Related Inequalities

Arising from Weighted Montgomery Identity

Andrea Aglić Aljinović, Ambroz Čivljak, Sanja Kovač, Josip Pečarić, Mihaela Ribičić Penava

The main objective of this book is to expose some recent research in the theory of weighted integral and discrete inequalities of Ostrowski type, with emphasis on its various useful applications such as generalizations of classical numerical quadrature rules. Obtained results generalize, refine, unify or sharpen some well-known or previously obtained inequalities. General identities from which the related inequalities are deduced in this book are various kinds of different generalizations of Montgomery identity and Euler integral formula which express expansion of a function in terms of Bernoulli polynomials. This book is intended for researchers in the field of mathematical inequalities, but it can also be used by graduate students and advanced undergraduates.

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Inequalities of Hardy and Jensen

New Hardy type inequalities with general kernels

Kristina Krulić Himmelreich, Josip Pečarić, Dora Pokaz

In this book some general aspects of generalizations, refinements, and variants of famous Hardy's inequality are presented. An integral operator with general non- negative kernel on measure spaces with positive σ-finite measure is considered and some new weighted Hardy type inequalities for convex functions and refinements of weighted Hardy type inequalities for superquadratic functions are obtained. Moreover, some refinements of weighted Hardy type inequalities for convex functions and some new refinements of discrete Hardy type inequalities are given. Furthermore, improvements and reverses of new weighted Hardy type inequalities with integral operators are stated and proved. By using the concept of the subdifferential of a convex function, the general Boas-type inequality is given and some new inequalities for superquadratic and subquadratic functions as well as for functions which can be bounded by non-negative convex or superquadratic function are obtained. The Boas functional and related inequality allow us to adjust Lagrange and Cauchy mean value theorems to the context and in that way define a new class of two-parametric means of the Cauchy-type. We also give some interesting, one-dimensional and multidimensional, examples related to balls and cones in Rⁿ.

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Steffensen's and Related Inequalities

A Comprehensive Survey and Recent Advances

Josip Pečarić, Ksenija Smoljak Kalamir, Sanja Varošanec

Since its appearance in 1918, Steffensen’s inequality is the subject of investigation by numerous mathematicians. The book is devoted to generalizations and refinements of Steffensen’s inequality and its connection to other inequalities, such as Gauss’, Jensen-Steffensen’s, Hölder’s and Iyengar’s inequality. A brief historical overview and a survey of weaker conditions for inverse Steffensen’s inequality is given. The book also contains Lp generalizations, generalizations for convex functions, refinements and sharpened versions, multidimensional generalizations and measure spaces generalizations of Steffensen’s inequality.

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Combinatorial Improvements of Jensen's Inequality

Classical and New Refinements of Jensen's Inequality with Applications

László Horváth, Khuram Ali Khan, Josip Pečarić

The present book aims to collect results about refinements of Jensen’s inequality; to provide methods of constructing refinements of Jensen’s inequality, with emphasis on the combinatorial improvements; conformation of old results from new points of view and insights; to define quasi-arithmetic and mixed symmetric means corresponding to the introduced refinements; to generate Cauchy means by using the refinements and the notion of exponential convexity; to study the monotonicity all of these means. The presented overview yields numerous possibilities for further exploration.

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Jensen Inequalities on Time Scales

Theory and Applications

Josipa Barić, Rabia Bibi, Martin Bohner, Ammara Nosheen, Josip Pečarić

This book summarizes very recent research related to Jensen-type inequalities on time scales. Many of the present results are proved via the theory of isotonic linear functionals. The book combines three areas of classical and active current research: classical inequalities in analysis, dynamic equations on time scales and isotonic linear functionals.

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Inequalities of Opial and Jensen

Improvements of Opial-type inequalities with applications to fractional calculus

Maja Andrić, Josip Pečarić Ivan Perić

Improvements of Opial-type inequalities with applications to fractional calculus.

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Superadditivity and monotonicity of the Jensen-type functionals

New methods for improving the Jensen-type inequalities in real and in operator cases

Mario Krnić, Neda Lovrinčević, Josip Pečarić Jurica Perić,

New methods for improving the Jensen-type inequalities in real and in operator cases.

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General Linear Inequalities and Positivity

Higher order convexity

Asif R. Khan, Josip Pečarić, Marjan Praljak, Sanja Varošanec

Higher order convexity.

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Further Development of Hilbert-type Inequalities

Selected Topics in Hilbert-type Inequalities

Tserendorj Batbold, Mario Krnić, Josip Pečarić, Predrag Vuković

This book is a result of five-year research of authors in Hilbert-type inequalities. The book is based on some twenty significant papers published in the course of the last five years. Roughly speaking, authors give some new generalizations, interpretations, refinements and applications of Hilbert-type inequalities.

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Weighted Energy Estimates for Convex Functions, Convex Vectors and Subsolution of Partial Differential Equation

Selected topics in reverse Poincare type inequalities

Julije Jakšetić, Josip Pečarić, Muhammad Shoaib Saleem

The regularity theory for solutions of certain parabolic partial differential equations is a well developed topic, but when it comes to subsolutions and supersolutions a lot remains to be done. The reverse Poincare type inequalities represent an important tool in the study of qualitative properties of solution of elliptic as well as parabolic partial differential equations.

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Inequalities and Zipf-Mandelbrot Law

Selected topics in information theory

Edited by Josip Pečarić, Đilda Pečarić

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 many others, so this book can be useful to different scientific communities.

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Majorization Inequality and Information Theory

Selected topics of majorization and applications

Muhammad Adil Khan, Slavica Ivelić Bradanović, Naveed Latif, Đilda Pečarić, Josip Pečarić

In this book, new methods are introduced that allow a link to be established between the concept of majorization with class of convex functions and theirs natural generalization, the class of convex functions of higher order, as well as classes of exponential and logarithmically convex functions.

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Weighted Steffensen’s Inequality

Recent Advances in Generalizations of Steffensen’s Inequality

Julije Jakšetić, Josip Pečarić, Anamarija Perušić Pribanić, Ksenija Smoljak Kalamir

Since its invention in 1918, Steffensen’s inequality is generalized in numerous directions under various settings. This book collects its most recent advances in generalizations. Under the vast diversities, in this book, Steffensen’s inequality is connected with the following notions: convex functions, higher order convexity, exponential convexity, h−convex functions, interpolating polynomials, measure theoretic aspects, weighted Bellman-Steffensen type inequalities, Gauss type inequalities, Hölder type inequalities. . .

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Cyclic Improvements of Jensen's Inequalities

Cyclic Inequalities in Information Theory

Saad Ihsan Butt, László Horváth, Đilda Pečarić, Josip Pečarić

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.

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Some New Improvements of Jensen’s Inequality

Jensen’s Type Inequalities in Information Theory

Muhammad Adil Khan, Khuram Ali Khan, Đilda Pečarić, Josip Pečarić

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.

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Analytical Inequalities for Fractional Calculus Operators and the Mittag-Leffler Function

Applications of integral operators containing an extended generalized Mittag-Leffler function in the kernel

Maja Andrić, Ghulam Farid, Josip Pečarić

Inequalities which involve integrals of functions and their derivatives occupy a central place in mathematical analysis and its applications. Fractional differentiation inequalities have applications to fractional differential equations; the most important ones are in establishing uniqueness of the solution of initial problems and giving upper bounds to their solutions. These applications have motivated many researchers in the field of integral inequalities to investigate certain extensions and generalizations using different fractional differential and integral operators. As a solution of fractional order differential or integral equations, the Mittag-Leffler function with its generalizations appears.

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Edmundson-Lah-Ribarič Type Inequalities

Reverses of the Edmundson-Lah-Ribarič Inequality with Applications to Classical Inequalities

Mario Krnić, Rozarija Mikić, Josip Pečarić

This book is based on several recent research papers on the subject of Jensen’s inequality, its converses and their variants, with special emphasis on the Edmundson-Lah Ribarič inequality in different settings and under various conditions. In the first chapter we will show some difference type converses of the mentioned inequalities for positive linear functionals, together with their refinements, improvements and applications to many famous classical inequalities. In the second chapter different classes of inequalities of vii the Jensen and Edmundson-Lah-Ribarič type for functions with bounded second order divided differences, Lipschitzian functions and 3-convex functions are derived. Also, several representations of the left side in the Edmundson-Lah-Ribarič inequality via Hermite’s interpolating polynomial in terms of divided differences are given and used for obtaining inequalities for the class of n-convex functions. Third chapter is dedicated to estimates for the Csisz´ar f -divergence functional and generalization of the f-divergence functional for different classes of functions via results from the previous two chapters. Application to Zipf and Zipf-Mandelbrot law is also given. In the fourth chapter we give difference type converses of the Jensen and Edmundson-Lah-Ribarič operator inequality for a unital field of positive linear mappings between C*-algebras of operators in compact Hausdorff space and their further refinements and improvements. Several mutual bounds for the operator version of the Jensen and Edmundson-Lah-Ribarič inequality which hold for the classes of bounded real-valued functions, Lipschitzian functions and n-convex functions are also given. In the fifth chapter we show some converses of Ando’s and Davis-Choi’s inequality of different types, as well as the Edmundson-Lah-Ribarič inequality and its difference type converse for positive linear mappings. Some results are extended to the class of n-convex functions. Difference type converse for solidarities and connections are also given. In the sixth chapter, some converses of the Jensen and Edmundson-Lah-Ribarič inequality in terms of time scale calculus are proved together with new refinements of those converse relations with respect to the multiple Lebesgue delta integral for convex functions. In the last chapter we give a short historical comment on the connection between the Edmundson-Madansky and the Lah-Ribariˇc inequality, and an overview of some already known results. Also we give a Levinson’s type generalization of the Edmundson-Lah-Ribarič inequality for a class of functions which contains the class of 3-convex functions, and analogous inequalities for the operator inequality in the Hilbert space and the scalar product of Hilbert space operators.

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New Developments for Jensen and Lah-Ribarič Inequalities

Current Trends in Convex Analysis

Maja Andrić, Vera Čuljak, Đilda Pečarić, Josip Pečarić, Jurica Perić

This book is divided in three chapters. The first chapter gives refinements of the Jensen and the Lah-Ribarič inequality, for their discrete and integral forms. The technique used in proving these refinements can be found in the proof of the Jensen-Boas inequality. Using this results, a refinement of the integral Hölder and discrete Hölder inequality, and refinements of some inequalities for power means and quasi arithmetic means are obtained. Also from the refinement of the integral forms we get a refinement of the famous Hermite-Hadamard inequality. As applications of these refinements, in the last part of the chapter we give some interesting estimates for the Csisz´ar divergence (discrete and integral case), and for the discrete case we also consider the Zipf-Mandelbrot law. In Chapter 2, we consider a generalization of the Jensen-McShane inequality for normalized positive isotonic linear functional and convex (concave) functions defined on a rectangle. We present the sequences of inequalities involving McShane generalization of Jensen’s inequality. As applications of these inequalities, for various choices of the functional F, we present extensions of known inequalities: the Diaz-Metcalf type inequalities for bounded random variables, the Fey´er and the Lupa¸s type inequalities for a function of two variables and inequalities of the Petrovi´c type for two non-negative real n-tuples. A conversion of the Jensen-McShane inequality is obtained by the two-variables function. Under special conditions, the Gheorghiu-type inequality is proved. The last chapter is dedicated to the recently introduced class of (h,g;m)-convex functions, which unifies a certain range of convexity, thus allowing the generalizations of known results. For this class, we present several types of inequalities such as Hermite-Hadamard, Fejér, Lah-Ribarič and Jensen, which generalize and extend corresponding inequalities. From Lah-Ribarič type inequalities for (h,g;m)-convex functions we obtain inequalities of Giaccardi, Popoviciu and Petrović. We also point out some special refined results. At the end, we use fractional calculus to obtain fractional version of the Hermite-Hadamard inequality, involving Riemann-Liouville fractional integral operators, which contain extended generalized Mittag-Leffler functions as their kernel. As an application, the upper bounds of fractional integral operators for (h,g;m)-convex functions are given.

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Extension, Dimension and Shape

Ivan Ivanšić Leonard R. Rubin

The following is a collection of notes that emanated from a topology seminar conducted at the University of Zagreb during the fall of 1999 and the spring of 2000. This work contains a compilation of a portion of the research that has been done in the areas of extension theory, dimension theory, and connections between these two and shape theory. The main emphasis will be on the former two, but we shall include the third because of the way it has been intertwined with the others. When we found it possible, we made steps to improve the efficiency and clarity of what had been published previously in the literature relative to these subjects. So the reader might even find some new results in the following even though it was not the authors’ intent to produce a research article. This writing is not meant to be a historical presentation, but to set the tone, we have provided, in the ensuing paragraphs, at least a rudimentary scan of the events leading to our survey.

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An Adventure in the Realm of Topology

Mohammad Sal Moslehian

Topology, also known as rubber sheet geometry, studies the shape of objects up to continuous deformations that can be continuously undone. Supported by his experience, not only in university teaching, but also in training teachers, professor Moslehian offers us a textbook, with quite advanced topics and many activities, that will certainly engage students, and create an interest in new concepts. The book contains many activities, nicely illustrated, and these will develop the natural tendency in teenagers to be explorers. For instance, in one activity the reader is asked to use a marker to draw a circle, a triangle, and a square on an uninflated balloon and then inflate the balloon. Then the reader is asked to explain why the resulting shapes are topologically equivalent to the original ones. The book also contains many challenging topics: four-dimensional spaces, knot theory, and fractals, to mention a few. The formal study of these topics requires many prerequisites. However, with his ample experience, Professor Moslehian manages to communicate them without technicalities, engaging the curious students in a pleasant ludic journey. The book will appeal to anyone interested in geometry and can be regarded as a great contribution to the popularization of mathematics.

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