This article analyses the asymptotic behaviour of solutions of linear Volterra difference equations. Some sufficient conditions are presented under which the solutions to a general linear equation converge to limits, …
This article analyses the asymptotic behaviour of solutions of linear Volterra difference equations. Some sufficient conditions are presented under which the solutions to a general linear equation converge to limits, which are given by a limit formula. This result is then used to obtain the exact asymptotic representation of the solutions of a class of convolution scalar difference equations, which have real characteristic roots. We give examples showing the accuracy of our results.
It is found that every solution of a system of linear delay difference equations has finite limit at infinity, if some conditions are satisfied. These are much weaker than the …
It is found that every solution of a system of linear delay difference equations has finite limit at infinity, if some conditions are satisfied. These are much weaker than the known sufficient conditions for asymptotic constancy of the solutions. When we impose some positivity assumptions on the coefficient matrices, our conditions are also necessary. The novelty of our results is illustrated by examples.
We give a refinement of the discrete Jensen's inequality in the convex and mid-convex cases.For mid-convex functions our result is a common generalization of known inequalities.We illustrate the scope of …
We give a refinement of the discrete Jensen's inequality in the convex and mid-convex cases.For mid-convex functions our result is a common generalization of known inequalities.We illustrate the scope of the results by applying them to some special situations.
Abstract In this paper we introduce new refinements of both the discrete and the classical Jensen’s inequality. First, we give the weighted version of a recent cyclic refinement. By using …
Abstract In this paper we introduce new refinements of both the discrete and the classical Jensen’s inequality. First, we give the weighted version of a recent cyclic refinement. By using this result, we obtain new refinements of the classical Jensen’s inequality. We investigate
In this paper some integral inequalities are proved in probability spaces, which go back to some discrete variants of the Jensen's inequality.Especially, we refine the classical Jensen's inequality.Convergence results corresponding …
In this paper some integral inequalities are proved in probability spaces, which go back to some discrete variants of the Jensen's inequality.Especially, we refine the classical Jensen's inequality.Convergence results corresponding to the inequalities are also studied.
In this paper, a new parameter-dependent refinement of the discrete Jensen's inequality is given for convex and mid-convex functions. The convergence of the introduced sequences is also studied. One of …
In this paper, a new parameter-dependent refinement of the discrete Jensen's inequality is given for convex and mid-convex functions. The convergence of the introduced sequences is also studied. One of the proofs requires an interesting convergence theorem with probability theoretical background. We apply the results to define some new quasi-arithmetic and mixed symmetric means and study their monotonicity and convergence.
The present paper develops a framework for a Halanay type nonautonomous delay differential inequality with maxima, and establishes necessary and/or sufficient conditions for the global attractivity of the zero solution. …
The present paper develops a framework for a Halanay type nonautonomous delay differential inequality with maxima, and establishes necessary and/or sufficient conditions for the global attractivity of the zero solution. The emphasis is put on the rate of convergence based on the theory of the generalized characteristic equation. The applicability and the sharpness of the results are illustrated by examples. This work aspires to serve as a remarkable step towards a unified theory of the nonautonomous Halanay inequality.
There are a lot of refinements of the discrete Jensen's inequality, and this problem has been studied by many authors. It is also a natural problem to give analogous results …
There are a lot of refinements of the discrete Jensen's inequality, and this problem has been studied by many authors. It is also a natural problem to give analogous results for the classical Jensen's inequality. In spite of this, few papers have been published dealing with this problem. The purpose of this paper is to give some refinements of the classical Jensen's inequality. The results give a new approach of this topic. Moreover, new discrete inequalities can be derived, and the integral analogous of discrete inequalities can be obtained. We also have new refinements of the left-hand side of the Hermite-Hadamard inequality. MSC:26D07, 26A51.
Refinements of the operator Jensen's inequality for convex and operator convex functions are given by using cyclic refinements of the discrete Jensen's inequality. Similar refinements are fairly rare in the …
Refinements of the operator Jensen's inequality for convex and operator convex functions are given by using cyclic refinements of the discrete Jensen's inequality. Similar refinements are fairly rare in the literature. Some applications of the results to norm inequalities, the Holder McCarthy inequality and generalized weighted power means for operators are presented.
Abstract In this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We …
Abstract In this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We essentially generalize and extend these results by using permutations of finite sets and bijections of the set of positive numbers. We get refinements of the discrete Jensen’s inequality for infinite convex combinations in Banach spaces. Similar results are rare. Finally, some applications are given on different topics.
Various attempts have been made to give an upper bound for the solutions of the delayed version of the Gronwall–Bellman integral inequality, but the obtained estimations are not sharp. In …
Various attempts have been made to give an upper bound for the solutions of the delayed version of the Gronwall–Bellman integral inequality, but the obtained estimations are not sharp. In this paper a new approach is presented to get sharp estimations for the nonnegative solutions of the considered delayed inequalities. The results are based on the idea of the generalized characteristic inequality. Our method gives sharp estimation, and therefore the results are more exact than the earlier ones.
Recently, Xiao, Srivastava and Zhang (see [10]) have introduced a new refinement of the discrete Jensen's inequality for mid-convex functions.We give and discuss the weighted form of their results.This leads …
Recently, Xiao, Srivastava and Zhang (see [10]) have introduced a new refinement of the discrete Jensen's inequality for mid-convex functions.We give and discuss the weighted form of their results.This leads to some new inequlities and limit formulas.We illustrate the scope of the results by applying them to introduce and study some new quasi-arithmetic means.
We generalize cyclic refinements of Jensen’s inequality from a convex function to a higher-order convex function by means of Lagrange–Green’s function and Fink’s identity. We formulate the monotonicity of the …
We generalize cyclic refinements of Jensen’s inequality from a convex function to a higher-order convex function by means of Lagrange–Green’s function and Fink’s identity. We formulate the monotonicity of the linear functionals obtained from these identities utilizing the theory of inequalities for n-convex functions at a point. New Grüss- and Ostrowski-type bounds are found for identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity and mean value theorems.
Abstract In this paper we derive majorization type integral inequalities using measure spaces with signed measures. We obtain necessary and sufficient conditions for the studied integral inequalities to be satisfied. …
Abstract In this paper we derive majorization type integral inequalities using measure spaces with signed measures. We obtain necessary and sufficient conditions for the studied integral inequalities to be satisfied. To apply our results, we first generalize Hardy–Littlewood–Pólya and Fuchs inequalities. Then we deal with the nonnegativity of some integrals with nonnegative convex functions. As a consequence, the known characterization of Steffensen–Popoviciu measures on compact intervals is extended to arbitrary intervals. Finally, we give necessary and sufficient conditions for the satisfaction of the integral Jensen inequality and the integral Lah–Ribarič inequality for signed measures. All the considered problems are also studied for special classes of convex functions. To prove the main assertions some approximation results for nonnegative convex functions are also developed.
In this paper, we give a refinement of discrete Jensen’s inequality for the operator convex functions. We launch the corresponding mixed symmetric means for positive self-adjoint operators defined on Hilbert …
In this paper, we give a refinement of discrete Jensen’s inequality for the operator convex functions. We launch the corresponding mixed symmetric means for positive self-adjoint operators defined on Hilbert space and also establish the refinement of inequality between power means of strictly positive operators.
In this paper we consider Bihari type integral inequalities in measure spaces.We give explicit bounds for the solutions under very weak conditions.The studied inequality essentially contains all inequalities of similar …
In this paper we consider Bihari type integral inequalities in measure spaces.We give explicit bounds for the solutions under very weak conditions.The studied inequality essentially contains all inequalities of similar forms that was considered previously, the results and the proofs give a unified approach of the problem.The results are applied to establish the existence of a solution to the integral equation corresponding to the integral inequality.
Abstract There are a lot of papers dealing with applications of the so-called cyclic refinement of the discrete Jensen’s inequality. A significant generalization of the cyclic refinement, based on combinatorial …
Abstract There are a lot of papers dealing with applications of the so-called cyclic refinement of the discrete Jensen’s inequality. A significant generalization of the cyclic refinement, based on combinatorial considerations, has recently been discovered by the author. In the present paper we give the integral versions of these results. On the one hand, a new method to refine the integral Jensen’s inequality is developed. On the other hand, the result contains some recent refinements of the integral Jensen’s inequality as elementary cases. Finally some applications to the Fejér inequality (especially the Hermite–Hadamard inequality), quasi-arithmetic means, and f -divergences are presented.
Recently, Horvath introduced a new method to refine the well known discrete Jensen’s inequality (see [2]). He also gave a parameter dependant refinement of the discrete Jensen’s inequality (see [3]). …
Recently, Horvath introduced a new method to refine the well known discrete Jensen’s inequality (see [2]). He also gave a parameter dependant refinement of the discrete Jensen’s inequality (see [3]). We apply the new exponential convexity method as illustrated in [7], to the functionals obtained from the refinement results of [2] and [3]. In this way we are able to generalize the results given in [4] as well as given in [1].
In this paper we give a new refinement of discrete Jensen’s inequality, which generalizes a former result. The introduced sequences depend on parameters. The strict monotonicity and the convergence are …
In this paper we give a new refinement of discrete Jensen’s inequality, which generalizes a former result. The introduced sequences depend on parameters. The strict monotonicity and the convergence are investigated. We also study the behavior of the sequences when the parameters vary. One of the proofs requires an interesting convergence theorem with probability theoretical background. This result is an extension of a former result, but its proof is simpler. The results are applied to define and study some new quasi-arithmetic means. MSC:26D07, 26A51.
A recent refinement of the classical discrete Jensen inequality is given by Horváth and Pečarić. In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined. We …
A recent refinement of the classical discrete Jensen inequality is given by Horváth and Pečarić. In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined. We investigate the exponential convexity of some functions, study mean value theorems, and prove the monotonicity of the introduced means.
In this paper, we present a general framework that provides a comprehensive and uniform treatment of integral majorization inequalities for convex functions and finite signed measures. Along with new results, …
In this paper, we present a general framework that provides a comprehensive and uniform treatment of integral majorization inequalities for convex functions and finite signed measures. Along with new results, we present unified and simple proofs of classical statements. To apply our results, we deal with Hermite-Hadamard-Fejér-type inequalities and their refinements. We present a general method to refine both sides of Hermite-Hadamard-Fejér-type inequalities. The results of many papers on the refinement of the Hermite-Hadamard inequality, whose proofs are based on different ideas, can be treated in a uniform way by this method. Finally, we establish a necessary and sufficient condition for when a fundamental inequality of f-divergences can be refined by another f-divergence.
Asymptotic behavior of a convolution of a function with a measure is investigated. Our results give conditions which ensure that the exact rate of the convolution function can be determined …
Asymptotic behavior of a convolution of a function with a measure is investigated. Our results give conditions which ensure that the exact rate of the convolution function can be determined using a positive weight function related to the given function and measure. Many earlier related results are included and generalized. Our new limit formulas are applicable to subexponential functions, to tail equivalent distributions, and to polynomial-type convolutions, among others.
In this paper we study some integral inequalities in measure spaces which are natural generalizations of special Bihari type integral inequalities.Explicit upper bounds for the solutions are given.The classical arguments …
In this paper we study some integral inequalities in measure spaces which are natural generalizations of special Bihari type integral inequalities.Explicit upper bounds for the solutions are given.The classical arguments can not be extended to this more general situation, we develop new methods.The results are applied to establish the existence of a solution to the corresponding integral equations.
In this paper we obtain refinements of the discrete Hölder's and Minkowski's inequalities for finite and infinite sequences by using cyclic refinements of the discrete Jensen's inequality.
In this paper we obtain refinements of the discrete Hölder's and Minkowski's inequalities for finite and infinite sequences by using cyclic refinements of the discrete Jensen's inequality.
Abstract The main purpose of this work is to present essential extensions of results in [7] and [8], and apply them to some special situations. Of particular interest is the …
Abstract The main purpose of this work is to present essential extensions of results in [7] and [8], and apply them to some special situations. Of particular interest is the refinement of the integral Jensen inequality for vector valued integrable functions. The applications related to four topics, namely f -divergences in information theory (an interesting refinement of the weighted geometric mean–arithmetic mean inequality is obtained as a consequence), norm inequalities, quasi-arithmetic means, Hölder’s and Minkowski’s inequalities.
In this paper we study inequalities corresponding to Jensen-Mercer's inequality.Some new extensions of Niezgoda's inequality and the integral version of Jensen-Mercer's inequality are given.The obtained inequalities do not only generalize …
In this paper we study inequalities corresponding to Jensen-Mercer's inequality.Some new extensions of Niezgoda's inequality and the integral version of Jensen-Mercer's inequality are given.The obtained inequalities do not only generalize the former ones, but our proofs are natural and simple.They clearly show the structure of such inequalities: they consist of two parts, a discrete or integral Jensen's inequality and then a majorization type inequality.Another purpose of the paper is to provide a deeper understanding of the methods used to refine Jensen-Mercer's and the corresponding inequalities.Moreover, some new refinements of these inequalities are obtained.Finally, some applications related to Fejér's and Hermite-Hadamard inequalities are given.
In this paper, we consider the class of self-adjoint operators defined on a Hilbert space, whose spectra are contained in an interval. We give parameter dependent renement of the well …
In this paper, we consider the class of self-adjoint operators defined on a Hilbert space, whose spectra are contained in an interval. We give parameter dependent renement of the well known discrete Jensen's inequality in this class. The parameter dependent mixed symmetric means are defined for a subclass of positive self-adjoint operators which insure the refinements of inequality between power means of strictly positive operators.
In this paper we consider a class of integral equations in measure spaces.Remarkable and important special integral equations are contained among them, which have been extensively investigated nowadays.The main results …
In this paper we consider a class of integral equations in measure spaces.Remarkable and important special integral equations are contained among them, which have been extensively investigated nowadays.The main results of this paper are existence theorems for the studied integral equations under the condition that the operator defined by the equation is increasing.Moreover, there are some auxiliary results which are interesting in their own rights.We shall see that some of the problems formulated for the classical integral equations can be solved in a very satisfactory way in this essentially more general case, and the results give unified approaches of the problems.Finally, some applications are given.
f-divergences play important role in probability theory, especially in information theory and in mathematical statistics. Remarkable divergences can be found among them. Inequalities for f-divergences are very useful and applicable …
f-divergences play important role in probability theory, especially in information theory and in mathematical statistics. Remarkable divergences can be found among them. Inequalities for f-divergences are very useful and applicable in information theory. In this paper we give a precise equality condition and a refinement for one of the basic inequalities of f-divergences. The results are illustrated by some applications.
In this paper new refinements of classical Jensen’s inequality are obtained by using some refinements of discrete Jensen’s inequality. To apply our refinements, new quasi-arithmetic means are introduced, the properties …
In this paper new refinements of classical Jensen’s inequality are obtained by using some refinements of discrete Jensen’s inequality. To apply our refinements, new quasi-arithmetic means are introduced, the properties of these means are studied, and refinements of the left hand side of the Hermite-Hadamard inequality are given.
We study some special nonlinear integral inequalities and the corresponding integral equations in measure spaces. They are significant generalizations of Bihari type integral inequalities and Volterra and Fredholm type integral …
We study some special nonlinear integral inequalities and the corresponding integral equations in measure spaces. They are significant generalizations of Bihari type integral inequalities and Volterra and Fredholm type integral equations. The kernels of the integral operators are determined by concave functions. Explicit upper bounds are given for the solutions of the integral inequalities. The integral equations are investigated with regard to the existence of a minimal and a maximal solution, extension of the solutions, and the generation of the solutions by successive approximations.
In this paper we study the periodicity of higher order nonlinear equations. They are defined by a recursion which is generated by a mapping , where X is a state …
In this paper we study the periodicity of higher order nonlinear equations. They are defined by a recursion which is generated by a mapping , where X is a state set. Our main objective is to prove sharp conditions for the global periodicity of our equations assuming the weakest possible assumptions on the state set X. As an application of our general algebraic-like conditions we prove a new linearized global periodicity theorem assuming that X is a normed space. We needed a new proof-technique since in the infinite dimensional case the Jacobian does not exist. We give new necessary and/or sufficient conditions as well as new examples for global periodicity, for instance whenever the state set X is a group.
In this paper we give very general refinements of the discrete Jensen's inequality for convex and mid-convex functions defined by recursion.Conditions are given for strict inequality which is rare in …
In this paper we give very general refinements of the discrete Jensen's inequality for convex and mid-convex functions defined by recursion.Conditions are given for strict inequality which is rare in this topic.In some cases explicit formulas are obtained.The results contain and generalize earlier statements.As an application we define some new quasi-arithmetic means and study their (strict) monotonicity.
The variation-of-constants formula is one of the principal tools of the theory of differential equations. There are many papers dealing with different versions of the variation-of-constants formula and its applications. …
The variation-of-constants formula is one of the principal tools of the theory of differential equations. There are many papers dealing with different versions of the variation-of-constants formula and its applications. Our main purpose in this paper is to give a variation-of-constants formula for inhomogeneous linear functional differential systems determined by general Volterra type operators with delay. Our treatment of the delay in the considered systems is completely different from the usual methods. We deal with the representation of the studied Volterra type operators. Some existence and uniqueness theorems are obtained for the studied linear functional differential and integral systems. Finally, some applications are given.
In this paper, starting from abstract versions of a result of Bennett given by Niculescu, we derive new majorization-type integral inequalities for convex functions using finite signed measures. The proof …
In this paper, starting from abstract versions of a result of Bennett given by Niculescu, we derive new majorization-type integral inequalities for convex functions using finite signed measures. The proof of the main result is based on a generalization of a recently discovered majorization-type integral inequality. As applications of the results, we give simple proofs of the integral Jensen and Lah–Ribarič inequalities for finite signed measures, generalize and extend known results, and obtain an interesting new refinement of the Hermite–Hadamard–Fejér inequality.
In this paper, we give necessary and sufficient conditions for the integral Jensen–Mercer inequality and closely related inequalities to be satisfied for finite signed measures. As applications, we obtain new …
In this paper, we give necessary and sufficient conditions for the integral Jensen–Mercer inequality and closely related inequalities to be satisfied for finite signed measures. As applications, we obtain new special inequalities that are related to the integral Jensen–Steffensen inequality. We also provide refinements of the majorization-type inequality associated with the Jensen–Mercer inequality for finite signed measures. Using the result obtained, we extend a known refinement. The majorization-type inequalities needed for the proofs are interesting in themselves.
Abstract Among inequalities that use the concept of convexity, Jensen-type inequalities and majorization-type inequalities are significant and fundamental. An important and widely researched area in the study of Jensen-type inequalities …
Abstract Among inequalities that use the concept of convexity, Jensen-type inequalities and majorization-type inequalities are significant and fundamental. An important and widely researched area in the study of Jensen-type inequalities is the refinement of such inequalities. In this paper, we provide a general method for refining the integral Jensen inequality for finite signed measures using integral majorization inequalities. Under the conditions considered in the paper, the results are unique, and even for measures, they give a new approach. We also provide interesting specific refinements, some of which relate to Jensen–Steffensen’s inequality. A new and extended version of this inequality is also obtained in the paper.
Abstract In this paper we give a necessary and sufficient condition for the discrete Jensen inequality to be satisfied for real (not necessarily nonnegative) weights. The result generalizes and completes …
Abstract In this paper we give a necessary and sufficient condition for the discrete Jensen inequality to be satisfied for real (not necessarily nonnegative) weights. The result generalizes and completes the classical Jensen–Steffensen inequality. The validity of the strict inequality is studied. As applications, we first give the form of our result for strongly convex functions, then we study discrete quasi-arithmetic means with real (not necessarily nonnegative) weights, and finally we refine the inequality obtained.
In this paper, using the results of a recent paper by the author, we give a new method for proving refinements of inequalities related to convex functions on intervals.In many …
In this paper, using the results of a recent paper by the author, we give a new method for proving refinements of inequalities related to convex functions on intervals.In many cases, the proof is simpler and more transparent than using the usual techniques, and the essence of the refinement is clearer.This is illustrated by two refinements of the Jensen's inequality and one refinement of the Lah-Ribarič inequality.As an application we generalize a recent result for strongly convex functions.
Abstract Motivated by a paper of Dragomir, we give new refinements for both discrete and integral Jensen inequalities using the Jensen’s gap. As applications, we give refinements of various inequalities …
Abstract Motivated by a paper of Dragomir, we give new refinements for both discrete and integral Jensen inequalities using the Jensen’s gap. As applications, we give refinements of various inequalities verifiable by Jensen’s inequality. Topics covered: norms, quasi-arithmetic means, Hölder’s inequality and f -divergences in information theory.
In this paper, we examine the relationship between a recent new discrete majorization-type inequality and classical majorization-type inequalities. The multiplicative analog of the studied new inequality is also given, which …
In this paper, we examine the relationship between a recent new discrete majorization-type inequality and classical majorization-type inequalities. The multiplicative analog of the studied new inequality is also given, which is a wide generalization of Weyl’s inequality. As an application, we give a parametric refinement of Popoviciu’s version of the Petrović inequality.
In this paper, we present a general framework that provides a comprehensive and uniform treatment of integral majorization inequalities for convex functions and finite signed measures. Along with new results, …
In this paper, we present a general framework that provides a comprehensive and uniform treatment of integral majorization inequalities for convex functions and finite signed measures. Along with new results, we present unified and simple proofs of classical statements. To apply our results, we deal with Hermite-Hadamard-Fejér-type inequalities and their refinements. We present a general method to refine both sides of Hermite-Hadamard-Fejér-type inequalities. The results of many papers on the refinement of the Hermite-Hadamard inequality, whose proofs are based on different ideas, can be treated in a uniform way by this method. Finally, we establish a necessary and sufficient condition for when a fundamental inequality of f-divergences can be refined by another f-divergence.
Abstract In this paper we derive majorization type integral inequalities using measure spaces with signed measures. We obtain necessary and sufficient conditions for the studied integral inequalities to be satisfied. …
Abstract In this paper we derive majorization type integral inequalities using measure spaces with signed measures. We obtain necessary and sufficient conditions for the studied integral inequalities to be satisfied. To apply our results, we first generalize Hardy–Littlewood–Pólya and Fuchs inequalities. Then we deal with the nonnegativity of some integrals with nonnegative convex functions. As a consequence, the known characterization of Steffensen–Popoviciu measures on compact intervals is extended to arbitrary intervals. Finally, we give necessary and sufficient conditions for the satisfaction of the integral Jensen inequality and the integral Lah–Ribarič inequality for signed measures. All the considered problems are also studied for special classes of convex functions. To prove the main assertions some approximation results for nonnegative convex functions are also developed.
Abstract The main purpose of this work is to present essential extensions of results in [7] and [8], and apply them to some special situations. Of particular interest is the …
Abstract The main purpose of this work is to present essential extensions of results in [7] and [8], and apply them to some special situations. Of particular interest is the refinement of the integral Jensen inequality for vector valued integrable functions. The applications related to four topics, namely f -divergences in information theory (an interesting refinement of the weighted geometric mean–arithmetic mean inequality is obtained as a consequence), norm inequalities, quasi-arithmetic means, Hölder’s and Minkowski’s inequalities.
Abstract There are a lot of papers dealing with applications of the so-called cyclic refinement of the discrete Jensen’s inequality. A significant generalization of the cyclic refinement, based on combinatorial …
Abstract There are a lot of papers dealing with applications of the so-called cyclic refinement of the discrete Jensen’s inequality. A significant generalization of the cyclic refinement, based on combinatorial considerations, has recently been discovered by the author. In the present paper we give the integral versions of these results. On the one hand, a new method to refine the integral Jensen’s inequality is developed. On the other hand, the result contains some recent refinements of the integral Jensen’s inequality as elementary cases. Finally some applications to the Fejér inequality (especially the Hermite–Hadamard inequality), quasi-arithmetic means, and f -divergences are presented.
In this paper we study inequalities corresponding to Jensen-Mercer's inequality.Some new extensions of Niezgoda's inequality and the integral version of Jensen-Mercer's inequality are given.The obtained inequalities do not only generalize …
In this paper we study inequalities corresponding to Jensen-Mercer's inequality.Some new extensions of Niezgoda's inequality and the integral version of Jensen-Mercer's inequality are given.The obtained inequalities do not only generalize the former ones, but our proofs are natural and simple.They clearly show the structure of such inequalities: they consist of two parts, a discrete or integral Jensen's inequality and then a majorization type inequality.Another purpose of the paper is to provide a deeper understanding of the methods used to refine Jensen-Mercer's and the corresponding inequalities.Moreover, some new refinements of these inequalities are obtained.Finally, some applications related to Fejér's and Hermite-Hadamard inequalities are given.
In this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative Volterra type operators. In a recent paper of us …
In this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative Volterra type operators. In a recent paper of us a variation of constants formula has been given for such systems, and this is particularly suited to handling the studied problems. First, we investigate the boundedness of the solutions. Then some estimates are given for the upper and lower limits of the solutions. By using this, we study the existence of the limit of the solutions, and obtain a formula for the limit. The applicability of our results is illustrated by studying the dependence of limits of the solutions on the choice of inhomogeneities, synchronisation, and Lotka-Volterra type delay functional differential systems.
In this paper we obtain refinements of the discrete Hölder's and Minkowski's inequalities for finite and infinite sequences by using cyclic refinements of the discrete Jensen's inequality.
In this paper we obtain refinements of the discrete Hölder's and Minkowski's inequalities for finite and infinite sequences by using cyclic refinements of the discrete Jensen's inequality.
The variation-of-constants formula is one of the principal tools of the theory of differential equations. There are many papers dealing with different versions of the variation-of-constants formula and its applications. …
The variation-of-constants formula is one of the principal tools of the theory of differential equations. There are many papers dealing with different versions of the variation-of-constants formula and its applications. Our main purpose in this paper is to give a variation-of-constants formula for inhomogeneous linear functional differential systems determined by general Volterra type operators with delay. Our treatment of the delay in the considered systems is completely different from the usual methods. We deal with the representation of the studied Volterra type operators. Some existence and uniqueness theorems are obtained for the studied linear functional differential and integral systems. Finally, some applications are given.
Abstract In this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We …
Abstract In this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We essentially generalize and extend these results by using permutations of finite sets and bijections of the set of positive numbers. We get refinements of the discrete Jensen’s inequality for infinite convex combinations in Banach spaces. Similar results are rare. Finally, some applications are given on different topics.
We generalize cyclic refinements of Jensen’s inequality from a convex function to a higher-order convex function by means of Lagrange–Green’s function and Fink’s identity. We formulate the monotonicity of the …
We generalize cyclic refinements of Jensen’s inequality from a convex function to a higher-order convex function by means of Lagrange–Green’s function and Fink’s identity. We formulate the monotonicity of the linear functionals obtained from these identities utilizing the theory of inequalities for n-convex functions at a point. New Grüss- and Ostrowski-type bounds are found for identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity and mean value theorems.
f-divergences play important role in probability theory, especially in information theory and in mathematical statistics. Remarkable divergences can be found among them. Inequalities for f-divergences are very useful and applicable …
f-divergences play important role in probability theory, especially in information theory and in mathematical statistics. Remarkable divergences can be found among them. Inequalities for f-divergences are very useful and applicable in information theory. In this paper we give a precise equality condition and a refinement for one of the basic inequalities of f-divergences. The results are illustrated by some applications.
The present paper develops a framework for a Halanay type nonautonomous delay differential inequality with maxima, and establishes necessary and/or sufficient conditions for the global attractivity of the zero solution. …
The present paper develops a framework for a Halanay type nonautonomous delay differential inequality with maxima, and establishes necessary and/or sufficient conditions for the global attractivity of the zero solution. The emphasis is put on the rate of convergence based on the theory of the generalized characteristic equation. The applicability and the sharpness of the results are illustrated by examples. This work aspires to serve as a remarkable step towards a unified theory of the nonautonomous Halanay inequality.
Refinements of the operator Jensen's inequality for convex and operator convex functions are given by using cyclic refinements of the discrete Jensen's inequality. Similar refinements are fairly rare in the …
Refinements of the operator Jensen's inequality for convex and operator convex functions are given by using cyclic refinements of the discrete Jensen's inequality. Similar refinements are fairly rare in the literature. Some applications of the results to norm inequalities, the Holder McCarthy inequality and generalized weighted power means for operators are presented.
Abstract In this paper we introduce new refinements of both the discrete and the classical Jensen’s inequality. First, we give the weighted version of a recent cyclic refinement. By using …
Abstract In this paper we introduce new refinements of both the discrete and the classical Jensen’s inequality. First, we give the weighted version of a recent cyclic refinement. By using this result, we obtain new refinements of the classical Jensen’s inequality. We investigate
In this paper we establish infinite chains of integral inequalities related to the classical Jensen's inequality by using special refinements of the discrete Jensen's inequality.As applications, we introduce and study …
In this paper we establish infinite chains of integral inequalities related to the classical Jensen's inequality by using special refinements of the discrete Jensen's inequality.As applications, we introduce and study new integral means (generalized quasi-arithmetic means), and give refinements of the left hand side of Hermite-Hadamard inequality.
Various attempts have been made to give an upper bound for the solutions of the delayed version of the Gronwall–Bellman integral inequality, but the obtained estimations are not sharp. In …
Various attempts have been made to give an upper bound for the solutions of the delayed version of the Gronwall–Bellman integral inequality, but the obtained estimations are not sharp. In this paper a new approach is presented to get sharp estimations for the nonnegative solutions of the considered delayed inequalities. The results are based on the idea of the generalized characteristic inequality. Our method gives sharp estimation, and therefore the results are more exact than the earlier ones.
In this paper new refinements of classical Jensen’s inequality are obtained by using some refinements of discrete Jensen’s inequality. To apply our refinements, new quasi-arithmetic means are introduced, the properties …
In this paper new refinements of classical Jensen’s inequality are obtained by using some refinements of discrete Jensen’s inequality. To apply our refinements, new quasi-arithmetic means are introduced, the properties of these means are studied, and refinements of the left hand side of the Hermite-Hadamard inequality are given.
In this paper we give very general refinements of the discrete Jensen's inequality for convex and mid-convex functions defined by recursion.Conditions are given for strict inequality which is rare in …
In this paper we give very general refinements of the discrete Jensen's inequality for convex and mid-convex functions defined by recursion.Conditions are given for strict inequality which is rare in this topic.In some cases explicit formulas are obtained.The results contain and generalize earlier statements.As an application we define some new quasi-arithmetic means and study their (strict) monotonicity.
In this paper, we give a refinement of discrete Jensen’s inequality for the operator convex functions. We launch the corresponding mixed symmetric means for positive self-adjoint operators defined on Hilbert …
In this paper, we give a refinement of discrete Jensen’s inequality for the operator convex functions. We launch the corresponding mixed symmetric means for positive self-adjoint operators defined on Hilbert space and also establish the refinement of inequality between power means of strictly positive operators.
Recently, Xiao, Srivastava and Zhang (see [10]) have introduced a new refinement of the discrete Jensen's inequality for mid-convex functions.We give and discuss the weighted form of their results.This leads …
Recently, Xiao, Srivastava and Zhang (see [10]) have introduced a new refinement of the discrete Jensen's inequality for mid-convex functions.We give and discuss the weighted form of their results.This leads to some new inequlities and limit formulas.We illustrate the scope of the results by applying them to introduce and study some new quasi-arithmetic means.
In this paper we give a new refinement of discrete Jensen’s inequality, which generalizes a former result. The introduced sequences depend on parameters. The strict monotonicity and the convergence are …
In this paper we give a new refinement of discrete Jensen’s inequality, which generalizes a former result. The introduced sequences depend on parameters. The strict monotonicity and the convergence are investigated. We also study the behavior of the sequences when the parameters vary. One of the proofs requires an interesting convergence theorem with probability theoretical background. This result is an extension of a former result, but its proof is simpler. The results are applied to define and study some new quasi-arithmetic means. MSC:26D07, 26A51.
In this paper we study the periodicity of higher order nonlinear equations. They are defined by a recursion which is generated by a mapping , where X is a state …
In this paper we study the periodicity of higher order nonlinear equations. They are defined by a recursion which is generated by a mapping , where X is a state set. Our main objective is to prove sharp conditions for the global periodicity of our equations assuming the weakest possible assumptions on the state set X. As an application of our general algebraic-like conditions we prove a new linearized global periodicity theorem assuming that X is a normed space. We needed a new proof-technique since in the infinite dimensional case the Jacobian does not exist. We give new necessary and/or sufficient conditions as well as new examples for global periodicity, for instance whenever the state set X is a group.
It is proved that any first-order globally periodic linear inhomogeneous autonomous difference equation defined by a linear operator with closed range in a Banach space has an equilibrium. This result …
It is proved that any first-order globally periodic linear inhomogeneous autonomous difference equation defined by a linear operator with closed range in a Banach space has an equilibrium. This result is extended for higher order linear inhomogeneous system in a real or complex Euclidean space. The work was highly motivated by the early works of Smith (1934, 1941) and the papers of Kister (1961) and Bas (2011).
Recently, Horvath introduced a new method to refine the well known discrete Jensen’s inequality (see [2]). He also gave a parameter dependant refinement of the discrete Jensen’s inequality (see [3]). …
Recently, Horvath introduced a new method to refine the well known discrete Jensen’s inequality (see [2]). He also gave a parameter dependant refinement of the discrete Jensen’s inequality (see [3]). We apply the new exponential convexity method as illustrated in [7], to the functionals obtained from the refinement results of [2] and [3]. In this way we are able to generalize the results given in [4] as well as given in [1].
There are a lot of refinements of the discrete Jensen's inequality, and this problem has been studied by many authors. It is also a natural problem to give analogous results …
There are a lot of refinements of the discrete Jensen's inequality, and this problem has been studied by many authors. It is also a natural problem to give analogous results for the classical Jensen's inequality. In spite of this, few papers have been published dealing with this problem. The purpose of this paper is to give some refinements of the classical Jensen's inequality. The results give a new approach of this topic. Moreover, new discrete inequalities can be derived, and the integral analogous of discrete inequalities can be obtained. We also have new refinements of the left-hand side of the Hermite-Hadamard inequality. MSC:26D07, 26A51.
In this paper, we consider the class of self-adjoint operators defined on a Hilbert space, whose spectra are contained in an interval. We give parameter dependent renement of the well …
In this paper, we consider the class of self-adjoint operators defined on a Hilbert space, whose spectra are contained in an interval. We give parameter dependent renement of the well known discrete Jensen's inequality in this class. The parameter dependent mixed symmetric means are defined for a subclass of positive self-adjoint operators which insure the refinements of inequality between power means of strictly positive operators.
In this paper, a new parameter-dependent refinement of the discrete Jensen's inequality is given for convex and mid-convex functions. The convergence of the introduced sequences is also studied. One of …
In this paper, a new parameter-dependent refinement of the discrete Jensen's inequality is given for convex and mid-convex functions. The convergence of the introduced sequences is also studied. One of the proofs requires an interesting convergence theorem with probability theoretical background. We apply the results to define some new quasi-arithmetic and mixed symmetric means and study their monotonicity and convergence.
A recent refinement of the classical discrete Jensen inequality is given by Horváth and Pečarić. In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined. We …
A recent refinement of the classical discrete Jensen inequality is given by Horváth and Pečarić. In this paper, the corresponding weighted mixed symmetric means and Cauchy-type means are defined. We investigate the exponential convexity of some functions, study mean value theorems, and prove the monotonicity of the introduced means.
We give a refinement of the discrete Jensen's inequality in the convex and mid-convex cases.For mid-convex functions our result is a common generalization of known inequalities.We illustrate the scope of …
We give a refinement of the discrete Jensen's inequality in the convex and mid-convex cases.For mid-convex functions our result is a common generalization of known inequalities.We illustrate the scope of the results by applying them to some special situations.
It is found that every solution of a system of linear delay difference equations has finite limit at infinity, if some conditions are satisfied. These are much weaker than the …
It is found that every solution of a system of linear delay difference equations has finite limit at infinity, if some conditions are satisfied. These are much weaker than the known sufficient conditions for asymptotic constancy of the solutions. When we impose some positivity assumptions on the coefficient matrices, our conditions are also necessary. The novelty of our results is illustrated by examples.
We give a refinement of the discrete Jensen's inequality in the convex and mid-convex cases.For mid-convex functions our result is a common generalization of known inequalities.We illustrate the scope of …
We give a refinement of the discrete Jensen's inequality in the convex and mid-convex cases.For mid-convex functions our result is a common generalization of known inequalities.We illustrate the scope of the results by applying them to some special situations.
Generalized refinement of Jensen's inequality is given. Appplications are done for cyclic mixed symmetric means and Cauchy means.
Generalized refinement of Jensen's inequality is given. Appplications are done for cyclic mixed symmetric means and Cauchy means.
In this paper some integral inequalities are proved in probability spaces, which go back to some discrete variants of the Jensen's inequality.Especially, we refine the classical Jensen's inequality.Convergence results corresponding …
In this paper some integral inequalities are proved in probability spaces, which go back to some discrete variants of the Jensen's inequality.Especially, we refine the classical Jensen's inequality.Convergence results corresponding to the inequalities are also studied.
In this paper, a new parameter-dependent refinement of the discrete Jensen's inequality is given for convex and mid-convex functions. The convergence of the introduced sequences is also studied. One of …
In this paper, a new parameter-dependent refinement of the discrete Jensen's inequality is given for convex and mid-convex functions. The convergence of the introduced sequences is also studied. One of the proofs requires an interesting convergence theorem with probability theoretical background. We apply the results to define some new quasi-arithmetic and mixed symmetric means and study their monotonicity and convergence.
Abstract In this paper we introduce new refinements of both the discrete and the classical Jensen’s inequality. First, we give the weighted version of a recent cyclic refinement. By using …
Abstract In this paper we introduce new refinements of both the discrete and the classical Jensen’s inequality. First, we give the weighted version of a recent cyclic refinement. By using this result, we obtain new refinements of the classical Jensen’s inequality. We investigate
Abstract In this paper we derive majorization type integral inequalities using measure spaces with signed measures. We obtain necessary and sufficient conditions for the studied integral inequalities to be satisfied. …
Abstract In this paper we derive majorization type integral inequalities using measure spaces with signed measures. We obtain necessary and sufficient conditions for the studied integral inequalities to be satisfied. To apply our results, we first generalize Hardy–Littlewood–Pólya and Fuchs inequalities. Then we deal with the nonnegativity of some integrals with nonnegative convex functions. As a consequence, the known characterization of Steffensen–Popoviciu measures on compact intervals is extended to arbitrary intervals. Finally, we give necessary and sufficient conditions for the satisfaction of the integral Jensen inequality and the integral Lah–Ribarič inequality for signed measures. All the considered problems are also studied for special classes of convex functions. To prove the main assertions some approximation results for nonnegative convex functions are also developed.
There are a lot of refinements of the discrete Jensen's inequality, and this problem has been studied by many authors. It is also a natural problem to give analogous results …
There are a lot of refinements of the discrete Jensen's inequality, and this problem has been studied by many authors. It is also a natural problem to give analogous results for the classical Jensen's inequality. In spite of this, few papers have been published dealing with this problem. The purpose of this paper is to give some refinements of the classical Jensen's inequality. The results give a new approach of this topic. Moreover, new discrete inequalities can be derived, and the integral analogous of discrete inequalities can be obtained. We also have new refinements of the left-hand side of the Hermite-Hadamard inequality. MSC:26D07, 26A51.
In this note we introduce a notion of M -dominated k -sample mean, where M is fixed multiset, which unifies several refinements of the Jensen's inequality for any mid-convex function.The …
In this note we introduce a notion of M -dominated k -sample mean, where M is fixed multiset, which unifies several refinements of the Jensen's inequality for any mid-convex function.The approach treats on equal basis sample means over samples with of without repetitions.
We consider continuous time and discrete time Halanay-type inequalities for nonautonomous scalar systems with discrete and distributed delays. The results obtained generalise the existing results of Halanay and improve certain …
We consider continuous time and discrete time Halanay-type inequalities for nonautonomous scalar systems with discrete and distributed delays. The results obtained generalise the existing results of Halanay and improve certain results of Baker and Tang. Furthermore, it is shown that the discrete time inequalities which are analogues of continuous time inequalities preserve the stability conditions corresponding to the continuous time Halanay-type inequalities.
Improvements of the Giaccardi and the Petrovi´c inequality are given. The notion of nexponentially convex functions is introduced. An elegant method of producing nexponentially convex and exponentially convex functions is …
Improvements of the Giaccardi and the Petrovi´c inequality are given. The notion of nexponentially convex functions is introduced. An elegant method of producing nexponentially convex and exponentially convex functions is applied using the Giaccardi and the Petrovi´c differences. Cauchy mean value theorems are proved and shown to be useful in studying Stolarsky type means defined by using the Giaccardi and the Petrovi´c differences.
The paper deals with the f-divergences of Csiszar generalizing the discrimination information of Kullback, the total variation distance, the Hellinger divergence, and the Pearson divergence. All basic properties of f-divergences …
The paper deals with the f-divergences of Csiszar generalizing the discrimination information of Kullback, the total variation distance, the Hellinger divergence, and the Pearson divergence. All basic properties of f-divergences including relations to the decision errors are proved in a new manner replacing the classical Jensen inequality by a new generalized Taylor expansion of convex functions. Some new properties are proved too, e.g., relations to the statistical sufficiency and deficiency. The generalized Taylor expansion also shows very easily that all f-divergences are average statistical informations (differences between prior and posterior Bayes errors) mutually differing only in the weights imposed on various prior distributions. The statistical information introduced by De Groot and the classical information of Shannon are shown to be extremal cases corresponding to alpha=0 and alpha=1 in the class of the so-called Arimoto alpha-informations introduced in this paper for 0<alpha<1 by means of the Arimoto alpha-entropies. Some new examples of f-divergences are introduced as well, namely, the Shannon divergences and the Arimoto alpha-divergences leading for alphauarr1 to the Shannon divergences. Square roots of all these divergences are shown to be metrics satisfying the triangle inequality. The last section introduces statistical tests and estimators based on the minimal f-divergence with the empirical distribution achieved in the families of hypothetic distributions. For the Kullback divergence this leads to the classical likelihood ratio test and estimator
Abstract In this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We …
Abstract In this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We essentially generalize and extend these results by using permutations of finite sets and bijections of the set of positive numbers. We get refinements of the discrete Jensen’s inequality for infinite convex combinations in Banach spaces. Similar results are rare. Finally, some applications are given on different topics.
Various attempts have been made to give an upper bound for the solutions of the delayed version of the Gronwall–Bellman integral inequality, but the obtained estimations are not sharp. In …
Various attempts have been made to give an upper bound for the solutions of the delayed version of the Gronwall–Bellman integral inequality, but the obtained estimations are not sharp. In this paper a new approach is presented to get sharp estimations for the nonnegative solutions of the considered delayed inequalities. The results are based on the idea of the generalized characteristic inequality. Our method gives sharp estimation, and therefore the results are more exact than the earlier ones.
The present paper develops a framework for a Halanay type nonautonomous delay differential inequality with maxima, and establishes necessary and/or sufficient conditions for the global attractivity of the zero solution. …
The present paper develops a framework for a Halanay type nonautonomous delay differential inequality with maxima, and establishes necessary and/or sufficient conditions for the global attractivity of the zero solution. The emphasis is put on the rate of convergence based on the theory of the generalized characteristic equation. The applicability and the sharpness of the results are illustrated by examples. This work aspires to serve as a remarkable step towards a unified theory of the nonautonomous Halanay inequality.
A refinement of the discrete Jensen's inequality for convex functions defined on a convex subset in linear spaces is given. Application for $f$-divergence measures including the Kullback-Leibler and Jeffreys divergences …
A refinement of the discrete Jensen's inequality for convex functions defined on a convex subset in linear spaces is given. Application for $f$-divergence measures including the Kullback-Leibler and Jeffreys divergences are provided as well.
We present an extension of the well-known $3/2$-stability criterion by Yorke for two term functional differential equations. We prove the exact nature of the obtained stability region which coincides with …
We present an extension of the well-known $3/2$-stability criterion by Yorke for two term functional differential equations. We prove the exact nature of the obtained stability region which coincides with the Yorke result in the special case when the decay term is absent. Moreover, we reveal some interesting links existing between the Yorke conditions, Halanay inequalities and differential equations with maxima, all of them essentially involving the maximum functionals.
Recently, Xiao, Srivastava and Zhang (see [10]) have introduced a new refinement of the discrete Jensen's inequality for mid-convex functions.We give and discuss the weighted form of their results.This leads …
Recently, Xiao, Srivastava and Zhang (see [10]) have introduced a new refinement of the discrete Jensen's inequality for mid-convex functions.We give and discuss the weighted form of their results.This leads to some new inequlities and limit formulas.We illustrate the scope of the results by applying them to introduce and study some new quasi-arithmetic means.
We derive new nonlinear discrete analogue of the continuous Halanay-type inequality. These inequalities can be used as basic tools in the study of the global asymptotic stability of the equilibrium …
We derive new nonlinear discrete analogue of the continuous Halanay-type inequality. These inequalities can be used as basic tools in the study of the global asymptotic stability of the equilibrium of certain generalized difference equations.
A refinement of the discrete Jensen's inequality for convex functions defined on a convex subset in linear spaces is given. Application for $f$% -divergence measures including the Kullback-Leibler and Jeffreys …
A refinement of the discrete Jensen's inequality for convex functions defined on a convex subset in linear spaces is given. Application for $f$% -divergence measures including the Kullback-Leibler and Jeffreys divergences are provided as well.
Some improvements of classical Jensen's inequality are used to define the weighted mixed symmetric means. Exponential convexity and mean value theorems are proved for the differences of these improved inequalities. …
Some improvements of classical Jensen's inequality are used to define the weighted mixed symmetric means. Exponential convexity and mean value theorems are proved for the differences of these improved inequalities. Related Cauchy means are also defined, and their monotonicity is established as an application.
In this paper we give a new refinement of discrete Jensen’s inequality, which generalizes a former result. The introduced sequences depend on parameters. The strict monotonicity and the convergence are …
In this paper we give a new refinement of discrete Jensen’s inequality, which generalizes a former result. The introduced sequences depend on parameters. The strict monotonicity and the convergence are investigated. We also study the behavior of the sequences when the parameters vary. One of the proofs requires an interesting convergence theorem with probability theoretical background. This result is an extension of a former result, but its proof is simpler. The results are applied to define and study some new quasi-arithmetic means. MSC:26D07, 26A51.
We present a probabilistic approach which proves blow-up of solutions of the Fujita equation $\partial w/\partial t = -(-\Delta )^{\alpha /2}w + w^{1+\beta }$ in the critical dimension $d=\alpha /\beta$. …
We present a probabilistic approach which proves blow-up of solutions of the Fujita equation $\partial w/\partial t = -(-\Delta )^{\alpha /2}w + w^{1+\beta }$ in the critical dimension $d=\alpha /\beta$. By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as $t\to \infty$. In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Escobedo and Levine (1995) to the case of $\alpha$-Laplacians with possibly different parameters $\alpha$.