Many of the most common inequalities in analysis have their origin in the notion of convexity. From the definition of the convex function, we obtain the function's continuation, induce Jensen's ā¦
Many of the most common inequalities in analysis have their origin in the notion of convexity. From the definition of the convex function, we obtain the function's continuation, induce Jensen's inequality, and obtain the familiar inequality between the arithmetic and geometric means of n positive numbers.
This chapter focuses on convex functions and their generalized function. Thus, we start this chapter by defining convex functions and some of their properties and discussing a simple geometric property. ā¦
This chapter focuses on convex functions and their generalized function. Thus, we start this chapter by defining convex functions and some of their properties and discussing a simple geometric property. Then, we generalize E-convex functions and establish some of their properties. Moreover, we give generalized s -convex functions in the second sense and present some new inequalities of generalized HermiteāHadamard's type for the class of functions whose second local fractional derivatives of order Ī± in absolute value at certain powers are generalized s -convex functions in the second sense. At the end, some examples are shown how these inequalities can be applied to some special means.
This study focuses on convex functions and their generalized. Thus, we start this study by giving the definition of convex functions and some of their properties and discussing a simple ā¦
This study focuses on convex functions and their generalized. Thus, we start this study by giving the definition of convex functions and some of their properties and discussing a simple geometric property. Then we generalize E-convex functions and establish some their properties. Moreover, we give generalized $ s $-convex functions in the second sense and present some new inequalities of generalized Hermite-Hadamard type for the class of functions whose second local fractional derivatives of order $ \alpha $ in absolute value at certain powers are generalized $ s $-convex functions in the second sense. At the end, some examples that these inequalities are able to be applied to some special means are showed.
Subject Physics Combinatorics and Graph Theory Collection: Oxford Scholarship Online
Subject Physics Combinatorics and Graph Theory Collection: Oxford Scholarship Online
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Abstract Here we discuss the theory and applications of convex functions both in IR1 and IRn. Section 5.6 deals with an important convex function associated with a non-empty bounded convex ā¦
Abstract Here we discuss the theory and applications of convex functions both in IR1 and IRn. Section 5.6 deals with an important convex function associated with a non-empty bounded convex set, its support function.
The paper focuses on the derivation of the integral variants of Jensen's inequality for convex functions of several variables. The work is based on the integral method, using convex combinations ā¦
The paper focuses on the derivation of the integral variants of Jensen's inequality for convex functions of several variables. The work is based on the integral method, using convex combinations as input, and set barycentres as output.
It is the purpose of this paper to propose a novel class of convex functions associated with strong Ī·-convexity. A relationship between the newly defined function and an earlier generalized ā¦
It is the purpose of this paper to propose a novel class of convex functions associated with strong Ī·-convexity. A relationship between the newly defined function and an earlier generalized class of convex functions is hereby established. To point out the importance of the new class of functions, some examples are presented. Additionally, the famous HermiteāHadamard inequality is derived for this generalized family of convex functions. Furthermore, some inequalities and results for strong Ī·-convex function are also derived. We anticipate that this new class of convex functions will open the research door to more investigations in this direction.
Choquet and Sugeno integrals have wide applications in several practical areas, especially as aggregation functions in decision theory. Universal integral is a generalization of Choquet and Sugeno integrals. The Jensen ā¦
Choquet and Sugeno integrals have wide applications in several practical areas, especially as aggregation functions in decision theory. Universal integral is a generalization of Choquet and Sugeno integrals. The Jensen type inequality related to the smallest universal integral which special cases are Sugeno, Shilkret, seminormed fuzzy integrals and pseudo-integral has been recently proposed. Based on this result, we have obtained the Jensen's inequality for the pseudo-integral and sufficient conditions under which that inequality holds.
Abstract In this work, the notion of a multiplicative harmonic convex function is examined, and HermiteāHadamard inequalities for this class of functions are established. Many inequalities of HermiteāHadamard type are ā¦
Abstract In this work, the notion of a multiplicative harmonic convex function is examined, and HermiteāHadamard inequalities for this class of functions are established. Many inequalities of HermiteāHadamard type are also taken into account for the product and quotient of multiplicative harmonic convex functions. In addition, new multiplicative integral-based inequalities are found for the quotient and product of multiplicative harmonic convex and harmonic convex functions. In addition, we provide certain upper limits for such classes of functions. The obtained results have been verified by providing examples with included graphs. The findings of this study may encourage more research in several scientific areas.
Abstract A novel generating mechanism for non-strict bivariate Archimedean copulas via the Lorenz curve of a non-negative random variable is proposed. Lorenz curves have been extensively studied in economics and ā¦
Abstract A novel generating mechanism for non-strict bivariate Archimedean copulas via the Lorenz curve of a non-negative random variable is proposed. Lorenz curves have been extensively studied in economics and statistics to characterize wealth inequality and tail risk. In this paper, these curves are seen as integral transforms generating increasing convex functions in the unit square. Many of the properties of these āLorenz copulasā, from tail dependence and stochastic ordering, to their Kendall distribution function and the size of the singular part, depend on simple features of the random variable associated to the generating Lorenz curve. For instance, by selecting random variables with a lower bound at zero it is possible to create copulas with asymptotic upper tail dependence. An āalchemyā of Lorenz curves that can be used as general framework to build multiparametric families of copulas is also discussed.
Abstract In this study, the authors establish and generalize some inequalities of Hadamard and Simpson type based on s -convexity in the second sense. Some applications are also given and ā¦
Abstract In this study, the authors establish and generalize some inequalities of Hadamard and Simpson type based on s -convexity in the second sense. Some applications are also given and generalized. Examples are given to show the results. The results generalize the integral inequalities in articles of Sarikaya and Xi.
Abstract In this paper, we investigate maps on sets of positive operators which are induced by the continuous functional calculus and transform a KuboāAndo mean $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ļ</mml:mi></mml:math> into ā¦
Abstract In this paper, we investigate maps on sets of positive operators which are induced by the continuous functional calculus and transform a KuboāAndo mean $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ļ</mml:mi></mml:math> into another $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ļ</mml:mi></mml:math> . We establish that under quite mild conditions, a mapping $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ļ</mml:mi></mml:math> can have this property only in the trivial case, i.e. when $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ļ</mml:mi></mml:math> and $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ļ</mml:mi></mml:math> are nontrivial weighted harmonic means and $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ļ</mml:mi></mml:math> stems from a function which is a constant multiple of the generating function of such a mean. In the setting where exactly one of $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ļ</mml:mi></mml:math> and $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ļ</mml:mi></mml:math> is a weighted arithmetic mean, we show that under fairly weak assumptions, the mentioned transformer property never holds. Finally, when both of $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ļ</mml:mi></mml:math> and $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ļ</mml:mi></mml:math> are such a mean, it turns out that the latter property is only satisfied in the trivial case, i.e. for maps induced by affine functions.
Abstract The current status concerning Hardy-type inequalities with sharp constants is presented and described in a unified convexity way. In particular, it is then natural to replace the Lebesgue measure ā¦
Abstract The current status concerning Hardy-type inequalities with sharp constants is presented and described in a unified convexity way. In particular, it is then natural to replace the Lebesgue measure dx with the Haar measure $dx/x$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>/</mml:mo> <mml:mi>x</mml:mi> </mml:math> . There are also derived some new two-sided Hardy-type inequalities for monotone functions, where not only the two constants are sharp but also the involved function spaces are (more) optimal. As applications, a number of both well-known and new Hardy-type inequalities are pointed out. And, in turn, these results are used to derive some new sharp information concerning sharpness in the relation between different quasi-norms in Lorentz spaces.
In this paper, we use the generalized version of convex functions, known as strongly convex functions, to derive improvements to the JensenāMercer inequality. We achieve these improvements through the newly ā¦
In this paper, we use the generalized version of convex functions, known as strongly convex functions, to derive improvements to the JensenāMercer inequality. We achieve these improvements through the newly discovered characterizations of strongly convex functions, along with some previously known results about strongly convex functions. We are also focused on important applications of the derived results in information theory, deducing estimates for Ļ-divergence, KullbackāLeibler divergence, Hellinger distance, Bhattacharya distance, Jeffreys distance, and JensenāShannon divergence. Additionally, we prove some applications to Mercer-type power means at the end.
<!-- *** Custom HTML *** --> Kiefer and Wolfowitz showed that if $F$ is a strictly curved concave distribution function (corresponding to a strictly monotone density $f$), then the Maximum ā¦
<!-- *** Custom HTML *** --> Kiefer and Wolfowitz showed that if $F$ is a strictly curved concave distribution function (corresponding to a strictly monotone density $f$), then the Maximum Likelihood Estimator $\widehat{F}_n$, which is, in fact, the least concave majorant of the empirical distribution function $\FF_n$, differs from the empirical distribution function in the uniform norm by no more than a constant times $(n^{-1} \log n)^{2/3}$ almost surely. We review their result and give an updated version of their proof. We prove a comparable theorem for the class of distribution functions $F$ with convex decreasing densities $f$, but with the maximum likelihood estimator $\widehat{F}_n$ of $F$ replaced by the least squares estimator $\widetilde{F}_n$: if $X_1 , \ldots , X_n$ are sampled from a distribution function $F$ with strictly convex density $f$, then the least squares estimator $\widetilde{F}_n$ of $F$ and the empirical distribution function $\FF_n$ differ in the uniform norm by no more than a constant times $(n^{-1} \log n )^{3/5}$ almost surely. The proofs rely on bounds on the interpolation error for complete spline interpolation due to Hall, Hall and Meyer. These results, which are crucial for the developments here, are all nicely summarized and exposited in de Boor.
<abstract><p>In this paper, we firstly give improvement of Hermite-Hadamard type and Fej$ \acute{e} $r type inequalities. Next, we extend Hermite-Hadamard type and Fej$ \acute{e} $r types inequalities to a new ā¦
<abstract><p>In this paper, we firstly give improvement of Hermite-Hadamard type and Fej$ \acute{e} $r type inequalities. Next, we extend Hermite-Hadamard type and Fej$ \acute{e} $r types inequalities to a new class of functions. Further, we give bounds for newly defined class of functions and finally presents refined estimates of some already proved results. Furthermore, we obtain some new discrete inequalities for univariate harmonic convex functions on linear spaces related to a variant most recently presented by Baloch <italic>et al.</italic> of Jensen-type result that was established by S. S. Dragomir.</p></abstract>
The main objective of this paper is to derive some new <i>k</i>-fractional refinements of HermiteHadamard like inequalities. We also discuss some new special cases of the main results. In the ā¦
The main objective of this paper is to derive some new <i>k</i>-fractional refinements of HermiteHadamard like inequalities. We also discuss some new special cases of the main results. In the last section, we discuss applications, which shows the significance of the obtained results.
In this paper, we obtain a new class of functions, which is developed via the HermiteāHadamard inequality for convex functions. The well-known one-one correspondence between the class of operator monotone ā¦
In this paper, we obtain a new class of functions, which is developed via the HermiteāHadamard inequality for convex functions. The well-known one-one correspondence between the class of operator monotone functions and operator connections declares that the obtained class represents the weighted logarithmic means. We shall also consider weighted identric mean and some relationships between various operator means. Among many things, we extended the weighted arithmeticāgeometric operator mean inequality as and involving the considered operator means.
In this work, several inequalities of Popoviciu type for h-MN-convex functions are proved, where M and N are denote to Arithmetic, Geometric and Harmonic means and h is a non-negative ā¦
In this work, several inequalities of Popoviciu type for h-MN-convex functions are proved, where M and N are denote to Arithmetic, Geometric and Harmonic means and h is a non-negative superadditive or subadditive function.
In this paper, we prove some Hermite-Hadamard type inequalities for operator geometrically convex functions for non-commutative operators.
In this paper, we prove some Hermite-Hadamard type inequalities for operator geometrically convex functions for non-commutative operators.
This paper introduces the monotone extended second order cone (MESOC), which is related to the monotone cone and the second order cone. Some properties of the MESOC are presented and ā¦
This paper introduces the monotone extended second order cone (MESOC), which is related to the monotone cone and the second order cone. Some properties of the MESOC are presented and its dual cone is computed. Projecting onto the MESOC is reduced to the pool-adjacent-violators algorithm (PAVA) of isotonic regression. An application of MESOC to portfolio optimisation is provided. Some broad descriptions of possible MESOC-regression models are also outlined.ble.
We give Fejer-Hadamard inequality for convex functions on coordinates in the rectangle from the plane. We define some mappings associated to it and discuss their properties.
We give Fejer-Hadamard inequality for convex functions on coordinates in the rectangle from the plane. We define some mappings associated to it and discuss their properties.
We prove that the Hamy symmetric function is Schur harmonic convex for . As its applications, some analytic inequalities including the well-known Weierstrass inequalities are obtained.
We prove that the Hamy symmetric function is Schur harmonic convex for . As its applications, some analytic inequalities including the well-known Weierstrass inequalities are obtained.
Abstract In this paper, firstly, we prove a JensenāMercer inequality for GA-convex functions. After that, we establish weighted HermiteāHadamardās inequalities for GA-convex functions using the new JensenāMercer inequality, and we ā¦
Abstract In this paper, firstly, we prove a JensenāMercer inequality for GA-convex functions. After that, we establish weighted HermiteāHadamardās inequalities for GA-convex functions using the new JensenāMercer inequality, and we establish some new inequalities connected with HermiteāHadamardāMercer type inequalities for differentiable mappings whose derivatives in absolute value are GA-convex.
New identities and inequalities are given for weighted majorization theorem for n-convex functions by using extension of the Montgomery identity and Green function. Various bounds for the reminders in new ā¦
New identities and inequalities are given for weighted majorization theorem for n-convex functions by using extension of the Montgomery identity and Green function. Various bounds for the reminders in new generalizations of weighted majorization formulae are provided using ÄebyÅ”ev type inequalities. Mean value theorems are also discussed for functional related to new results.
Evaluation of the Bellman functions is a difficult task. The exact Bellman functions of the dyadic Carleson Embedding Theorem 1.1 and the dyadic maximal operators are obtained in [3] and ā¦
Evaluation of the Bellman functions is a difficult task. The exact Bellman functions of the dyadic Carleson Embedding Theorem 1.1 and the dyadic maximal operators are obtained in [3] and [4]. Actually, the same Bellman functions also work for the tree-like structure. In this paper, we give a self-complete proof of the coincidence of the Bellman functions on the more general infinitely refining filtered probability space, see Definition 1.3. The proof depends on a remodeling of the Bellman function of the dyadic Carleson Embedding Theorem.
In this paper, we establish various inequalities for some mappings that are linked with the illustrious Hermite-Hadamard integral inequality for mappings whose absolute values belong to the class K?;s m;1 ā¦
In this paper, we establish various inequalities for some mappings that are linked with the illustrious Hermite-Hadamard integral inequality for mappings whose absolute values belong to the class K?;s m;1 and K?;s m;2.
In the paper, the authors find some new integral inequalities of Hermite-Hadamard type for functions whose derivatives of the n-th order are (Ī±,m)-convex and deduce some known results. As applications ā¦
In the paper, the authors find some new integral inequalities of Hermite-Hadamard type for functions whose derivatives of the n-th order are (Ī±,m)-convex and deduce some known results. As applications of the newly-established results, the authors also derive some inequalities involving special means of two positive real numbers.