Slavko Simic
In this article we give some improvements and generalizations of the famous Jensen's and Jensen-Mercer inequalities for twice differentiable functions, where convexity property of the target function is not assumed in advance. They represents a refinement of these inequalities in the case of convex/concave functions with numerous applications in Theory of Means and Probability and Statistics.
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A Converse of the Jensen Inequality for Convex Mappings of Several Variables and Applications
本文给出了著名的多元可微凸映象Jensen不等式的一个逆结果，并将其应用于相应的著名解析不等式。应用香农和Renyi的熵映射也给出了。
In this paper, we give the refinement of an extension of Jensen's inequality to affine combinations. Furthermore, we present the functional form of Jensen's inequality for continuous 3-convex functions of one variable at a point.
Diamond-$α$ Jensen's Inequality on Time Scales
时间尺度上的动态导数理论及其应用近年来受到广泛关注。本文的主要目的是给出时间尺度上的Delta和Nabla动力导数的线性组合Diamond- $α$导数的基本性质。利用Diamond- $α$积分证明了时间尺度上Jensen不等式的一个推广形式，并给出了一些推论，包括Hölder和Minkowski的Diamond- $α$积分不等式.
A measure-theoretic version of the Dragomir-Jensen inequality
我们将Dragomir对Jensen不等式的精化从dicrete推广到一般情况，确定了等式条件。
The present paper is devoted to the study of Jensen-Mercer-type inequalities. Our results generalize and improve some earlier results in the literature.
Jensen convex functions and doubly stochastic matrices
本文研究双随机矩阵与 Jensen 凸性的关系，证明在适当线性代数条件下，相关矩阵型函数不等式可推出 f 为 Jensen 凸，并说明凸函数对任意阶双随机矩阵均满足该不等式。
Jensen's inequality for partial traces in von Neumann algebras
本文在半有限冯·诺依曼代数中证明了部分追踪的詹森不等式，并扩展至一般冯·诺依曼代数框架。
Inequalities play an important role in pure and applied mathematics. In particular, Jensen's inequality, one of the most famous inequalities, plays a main role in the study of the existence and uniqueness of initial and boundary value problems for differential equations. In this work we prove some new Jensen-type inequalities for $m$-convex functions, and we apply them to generalized Riemann-Liouville-type integral operators. It is remarkable that, if we consider $m=1$, we obtain new inequalities for convex functions.
It is well-known that if a real valued function acting on a convex set satisfies the $n$-variable Jensen inequality, for some natural number $n\geq 2$, then, for all $k\in\{1,\dots, n\}$, it fulfills the $k$-variable Jensen inequality as well. In other words, the arithmetic mean and the Jensen inequality (as a convexity property) are both reducible. Motivated by this phenomenon, we investigate this property concerning more general means and convexity notions. We introduce a wide class of means which generalize the well-known means for arbitrary linear spaces and enjoy a so-called reducibility …
A considerable amount of literature in the theory of inequality is devoted to the study of Jensen's and Young's inequality. This article presents a number of new inequalities involving the log-convex functions and the geometrically convex functions. As their consequences, we derive the refinements for Young's inequality and Jensen's inequality. In addition, the operator Jensen's type inequality is also developed for conditioned two functions. Utilizing these new inequalities, we investigate the operator inequalities related to the relative operator entropy.
We present both necessary and sufficient conditions to the convex closed shape $X$ such that the inequality $$ \frac{1}{|X|} \int_X f(x)\:dx \le \frac{1}{|\partial X|} \int_{\partial X} f(x)\:dx$$ is valid for every convex function $f \colon X \to \mathbb{R}$ ($\partial X$ stands for the boundary of $X$). It is proved that this inequality holds if $X$ is (i) an $n$-dimensional parallelotope, (ii) an $n$-dimensional ball, (iii) a convex polytope having an inscribed sphere (tangent to all its facets) with center in the center of mass of $\partial X$.
In this paper, by using Jensen's inequality and Chebyshev integral inequality, some generalizations and new refined Hardy type integral inequalities are obtained. In addition, the corresponding reverse relation are also proved.
Generalized Jensen's inequality motivated from thermodynamics
本文受热力学定律启发提出并证明广义 Jensen 不等式，统一导出加权 AM-GM-HM 等不等式，同时证明多物体热平衡温度的唯一性与过程的自发性。
Estimates for approximately Jensen convex functions
本文研究带误差函数 φ 的近似 Jensen 凸函数，建立其对任意有理凸组合的显式凸性误差估计，并证明相应无穷级数收敛且在有理参数下可化为有限和。
Generalized Convex Functions and Some Inequalities on Fractal Sets
在实数分形集上引入了广义凸函数，并研究了广义凸函数的性质。基于这些性质，我们建立了广义Jensen不等式和广义Hermite-Hadamard不等式.并给出了一些应用实例。
In this paper we investigate continuity properties of functions $f:\mathbb{R}_+\to\mathbb{R}_+$ that satisfy the $(p,q)$-Jensen convexity inequality $$ f\big(H_p(x,y)\big)\leq H_q(f(x),f(y)) \qquad(x,y>0), $$ where $H_p$ stands for the $p$th power (or Hölder) mean. One of the main results shows that there exist discontinuous multiplicative functions that are $(p,p)$-Jensen convex for all positive rational number $p$. A counterpart of this result states that if $f$ is $(p,p)$-Jensen convex for all $p\in P\subseteq\mathbb{R}_+$, where $P$ is a set of positive Lebesgue measure, then $f$ must be c…
Considering the weighted concept of majorization, Sherman obtained generalization of majorization inequality for convex functions known as Sherman's inequality. We extend Sherman's result to the class of n-strongly convex functions using extended idea of convexity to the class of strongly convex functions. We also obtaine upper bound for Sherman's inequality, so called the converse Sherman inequality, and as easy consequences we get Jensen's as well as majorization inequality and their conversions for strongly convex functions. Obtained results are stronger versions for analogous results for c…
Some mathematical inequalities among various weighted means are studied. Inequalities on weighted logarithmic mean are given. Besides, the gap in Jensen's inequality is studied as a convex function approach. Consequently, some non-trivial inequalities on means are given. Some operator inequalities are also shown.
We have recently established some integral inequalities for convex functions via the Hermite-Hadamard's inequalities. In continuation here, we also establish some interesting new integral inequalities for convex functions via the Hermite--Hadamard's inequalities and Jensen's integral inequality. Useful applications involving special means are also included.
We discuss a rather general condition under which the inequality of Jensen works for certain convex combinations of points not all in the domain of convexity of the function under attention. Based on this fact, an extension of the Hardy-Littlewood-Pólya theorem of majorization is proved and new insight is given into the problem of risk aversion in mathematical finance.
We present a tight parametrical Hermite-Hadamard type inequality with probability measure, which yields a considerably closer upper bound for the mean value of convex function than the classical one. Our inequality becomes equality not only with affine functions, but also with a family of V-shaped curves determined by the parameter. The residual (error) of this inequality is strictly smaller than in the classical Hermite-Hadamard inequality under any probability measure and with all non-affine convex functions. In the framework of Karamata's theorem on the inequalities with convex functions, w…
Jacob's ladders and some generalizations of certain Ramachandra's inequality
本文得到了一类Ramachandra不等式的一些基本推广，即e.对临界线上的黎曼ζ函数产生的复杂信号的能量给出了新的下界估计。
A generalization of Hilbert's inequality
在Hardy，Littlewood和Pólya对经典Hilbert不等式的推广中，只要相应矩阵的母函数满足一定的单调性条件，就可以确定不等式的最佳常数.本文给出了一类不等式在不再满足单调性条件时的最佳常数。
In this work, we introduce the class of $h$-${\rm{MN}}$-convex functions by generalizing the concept of ${\rm{MN}}$-convexity and combining it with $h$-convexity. Namely, Let $I,J$ be two intervals subset of $\left(0,\infty\right)$ such that $\left(0,1\right)\subseteq J$ and $\left[a,b\right]\subseteq I$. Consider a non-negative function $h: (0,\infty)\to \left(0,\infty\right)$ and let ${\rm{M}}:\left[0,1\right]\to \left[a,b\right] $ $(0<a<b)$ be a Mean function given by ${\rm{\rm{M}}}\left(t\right)={\rm{\rm{M}}}\left( {h(t);a,b} \right)$; where by ${\rm{\rm{M}}}\left( {h(t);a,b} \right)$ we m…
In literature the Hermite-Hadamard inequality was eligible for many reasons, one of the most surprising and interesting that the Hermite-Hadamard inequality combine the midpoint and trapezoid formulae in an inequality. In this work, a Hermite-Hadamard like inequality that combines the composite trapezoid and composite midpoint formulae is proved. So that, the classical Hermite-Hadamard inequality becomes a special case of the presented result. Some Ostrowski's type inequalities for convex functions are proved as well.
Sharpening and generalizations of Shafer-Fink's double inequality for the arc sine function
本文对反正弦函数的Shafer-Fink二重不等式进行了改进和推广。
In this paper we give some sharper refinements and generalizations of inequalities related to Shafer's inequality for the arctangent function, stated in Theorems 1, 2 and 4 in [1], by C. Mortici and H.M. Srivastava.
A convexity-type functional inequality with infinite convex combinations
本文证明了在 $d$ 维欧几里得空间非空凸子集上，满足无穷凸组合凸性型泛函不等式的下有界函数必为凸函数，并利用 Jensen 不等式的概率版本对已有结果给出了替代证明。
Sharpening and generalizations of Carlson's inequality for the arc cosine function
本文对反余弦函数的Carlson二重不等式进行了改进和推广。
In the paper, the authors review several refinements of Young's integral inequality via several mean value theorems, such as Lagrange's and Taylor's mean value theorems of Lagrange's and Cauchy's type remainders, and via several fundamental inequalities, such as Čebyšev's integral inequality, Hermite--Hadamard's type integral inequalities, Hölder's integral inequality, and Jensen's discrete and integral inequalities, in terms of higher order derivatives and their norms, survey several applications of several refinements of Young's integral inequality, and further refine Young's integral inequa…
Gruss Type Discrete Inequalities in Inner Product Spaces, Revisited
本文重新探讨内积空间中的 Gruss 型离散不等式，得到尖锐结果并应用于凸函数的 Jensen 不等式。
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