A. G. Ghazanfari, M. Shakoori, A. Barani, S. S. Dragomir
In this paper we establish a Hermite- Hadamard type inequality for operator preinvex functions and an estimate of the right hand side of a Hermite- Hadamard type inequality in which some operator preinvex functions of selfadjoint operators in Hilbert spaces are involved.
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In this paper, we extend some estimates of the left hand side of a Hermite- Hadamard type inequality for nonconvex functions whose derivatives absolute values are preinvex and log-preinvex.
In this paper we introduce operator preinvex functions and es- tablish a Hermite-Hadamard type inequality for such functions. We give an estimate of the right hand side of a Hermite-Hadamard type inequality in which some operator preinvex functions of selfadjoint operators in Hilbert spaces are involved. Also some Hermite-Hadamard type inequalities for the product of two operator preinvex functions are given.
In this paper, we obtain some Hermite-Hadamard type inequalities for functions whose third derivatives in absolute value are preinvex and prequasiinvex.
In this paper we establish some new Hermite-Hadamard type inequalities for two operator convex functions of selfadjoint operators in Hilbert spaces.
We present several matrix and operator inequalities of Hermite-Hadamard type. We first establish a majorization version for monotone convex functions on matrices. We then utilize the Mond-Pecaric method to get an operator version for convex functions. We also present some applications. Finally we obtain an Hermite-Hadamard inequality for operator convex functions, positive linear maps and operators acting on Hilbert spaces.
A new variant of weighted Hermite-Hadamard inequalities and applications
该研究借助合适的概率测度构建算子凸映射的新型加权Hermite-Hadamard型不等式，并将其用于改进知名算子不等式、定义新加权算子平均，拓展了相关算子不等式理论与应用。
The aim of this paper is to establish Hermite-Hadamard, Hermite-Hadamard-Fejér, Dragomir-Agarwal and Pachpatte type inequalities for new fractional integral operators with exponential kernel. These results allow us to obtain a new class of functional inequalities which generalizes known inequalities involving convex functions. Furthermore, the obtained results may act as a useful source of inspiration for future research in convex analysis and related optimization fields.
In this paper, we prove some Hermite-Hadamard type inequalities for operator geometrically convex functions for non-commutative operators. Keywords: Operator geometrically convex function, Hermite-Hadamard inequality.
This research aimed to explore some new Hermite-Hadamard inequalities for strongly harmonic convex set-valued functions with modulus c > 0 introduced by G. Santana
In this paper, obtained some new class of Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities via fractional integrals for the p-hyperbolic convex functions. It is shown that such inequalities are simple consequences of Hermite-Hadamard-Fejer inequality for the p-hyperbolic convex function.
In this paper, we introduce the notion of log-φ-convex functions and present some properties and representation of such functions. We obtain some results of the Hermite Hadamard inequalities for product log-φ-convex functions.
We study an elementary inequality supporting the classical Hermite-Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such new Schatten p-norm estimates and new majorization
The main target of this paper is to discuss operator Hermite--Hadamard inequality for convex functions, without appealing to operator convexity. Several forms of this inequality will be presented and some applications including norm and mean inequalities will be shown too.
In this paper, we derive a new proof on some sharp double integral inequalities of the Hermite-Hadamard type. Our approach is mainly based on well-known Taylor's theorem with the integral remainder.
In this paper we introduce operator s-convex func- tions and establish some Hermite-Hadamard type inequalities in which some operator s-convex functions of positive operators in Hilbert spaces are involved.
New identity for fractional integrals have been defined. By using of this identity, some new Hermite-Hadamard type inequalities for Riemann-Liouville fractional integral have been developed. Our results have some relationships with the result of Avci et al., proved in AKO.
In this paper, we extend some estimates of the right hand side of a Hermite- Hadamard type inequality for nonconvex functions whose second derivatives absolute values are ϕ-convex, log-ϕ-convex, and quasi-ϕ-convex.
In a recent paper [9], Ozdemir, Tunc and Akdemir defined two new classes of convex functions with which they proved some Hermite-Hadamard type inequalities. As an Open problem, they asked for conditions under which the composition of two functions belong to their newly defined class of convex functions and if Hermite-Hadamard type inequalities can be obtained. In this paper, we respond to the Open problems and prove some new Hermite-Hadamard inequalities for twice differentiable composition whose second derivative is $((h-s)_{2}, I)$-convex. Our results are applied to some special means of rea…
In this paper, we introduce the concept of operator geometrically convex functions for positive linear operators and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for operators which give some refinements of previous results.
In this paper, we introduce the notion of strongly φ-convex functions with respect to c>0 and present some properties and representation of such functions. We obtain a characterization of inner product spaces involving the notion of strongly φ-convex functions. Finaly, a version of Hermite Hadamard-type inequalities for strongly φ-convex functions are established.
In this paper, we establish various inequalities for some differentiable mappings that are linked with the illustrious Hermite- Hadamard integral inequality for mappings whose derivatives are (h -($α$?;m))-convex.The generalized integral inequalities contribute some better estimates than some already presented. The inequalities are then applied to numerical integration and some special means.
In this paper, some Hermite-Hadamard type inequalities are established for harmonically $(α,m)$-convex functions via fractional integrals and some Hermite-Hadamard type inequalities are obtained for these classes of functions.
On a weighted Hermite-Hadamard inequality involving convex functional arguments
本文研究含凸泛函的加权Hermite-Hadamard型不等式，提出的方法可改进逆转已有不等式，还构建了新加权凸泛函均值及相关性质。
In this paper, we establish various inequalities for some differentiable mappings that are linked with the illustrious Hermite-Hadamard integral inequality for mappings whose derivatives are $s$-$(α,m)$-convex.The generalised integral inequalities contribute some better estimates than some already presented. The inequalities are then applied to numerical integration and some special means.
We give a slight extension of the Hermite-Hadamard inequality on simplices and we use it to establish error bounds of the operators connected with the approximate integration.
In this article, we define a new class of convexity called generalized $(h-m)$-convexity, which generalizes $h$-convexity and $m$-convexity on fractal sets $\mathbb{R}^α$ $(0<α\leq 1)$. Some properties of this new class are discussed. Using local fractional integrals and generalized $(h-m)$-convexity, we generalized Hermite-Hadamard (H-H) and Fejér-Hermite-Hadamard (Fejér-H-H) types inequalities. We also obtained a new result of the Fejér-H-H type for the function whose derivative in absolute value is the generalized $(h-m)$-convexity on fractal sets. Some applications to random variables and …
In the paper, we establish the Hermite-Hadamard type inequalities for the generalized s-convex functions in the second sense on real linear fractal set $\mathbb{R}^α(0<α<1).$
In this paper, we have established some generalized integral inequalities of Hermite-Hadamard-Fejér type for generalized fractional integrals. The results presented here would provide generalizations of those given in earlier works.
On the weighted hermite-hadamard inequality in multiple variables, application for weighted multivariate means
该研究针对单变量Hermite-Hadamard不等式的拓展需求，提出多变量加权形式的不等式变体，并将其应用于构造推广现有双变量均值的加权多元均值。
It is well-known that the left term of the classical Hermite-Hadamard inequality is closer to the integral mean value than the right one. We show that in the multivariate case it is not true. Moreover, we introduce some related inequality comparing the methods of the approximate integration, which is optimal. We also present its counterpart of Fejer type.
Inequalities play important roles not only in mathematics, but also in other fields, such as economics and engineering. Even though many results are published on Hermite-Hadamard (H-H) type inequalities, new researcher to this fields often found it difficult to understand them. Thus, some important discoverers, such as the formulations of H-H type inequalities via various classes of convexity, through differentiable mappings and for fractional integrals, are presented. Some well-known examples from previous literature are used as illustrations.
Fractional Integral Estimates of Hermite-Hadamard type in Global Nonpositive Curvature Spaces
该研究在整体非正曲率空间中引入测地$h$-凸函数以拓展函数凸性概念，通过Katugampola分数阶积分证明Hermite-Hadamard型估计，得到涉及平方距离映射的本质精确推论，丰富了弯曲几何空间的分析理论。
In this paper, we establish Hermite-Hadamard inequality for interval-valued convex function on the co-ordinates on the rectangle from the plane. We also present Hermite-Hadamard inequality for the product of interval-valued convex functions on co-ordinates. Some examples are also given to clarify our new results
Let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a subharmonic function, $Δf \geq 0$, which satisfies $f \geq 0$ on the boundary $\partial Ω$. Then $$ \int_Ω{f ~dx} \leq |Ω|^{\frac{1}{n}} \int_{\partial Ω}{f ~dσ}.$$ Our proof is based on a new gradient estimate for the torsion function, $Δu = -1$ with Dirichlet boundary conditions, which is of independent interest.
Generalised Local Fractional Hermite-Hadamard Type Inequalities on Fractal Sets
本文提出广义凸性概念，定义广义$\phi_{h-s}$凸函数，证明分形集上的Hermite-Hadamard型不等式，并应用于概率论。
The Hermite-Hadamard inequality states that the average value of a convex function on an interval is bounded from above by the average value of the function at the endpoints of the interval. We provide a generalization to higher dimensions: let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a convex function satisfying $f \big|_{\partial Ω} \geq 0$, then $$ \frac{1}{|Ω|} \int_Ω{f ~d \mathcal{H}^n} \leq \frac{2 π^{-1/2} n^{n+1}}{|\partial Ω|} \int_{\partial Ω}{f~d \mathcal{H}^{n-1}}.$$ The constant $2 π^{-1/2} n^{n+1}$ is presumably far from optimal, however, …
In this paper we show the following result: if C is an n-dimensional 0-symmetric convex compact set, $f:C\rightarrow[0,1)$ is concave, and $g:[0,1)\rightarrow[0,1)$ is not identically zero, convex, with g(0)=0, then \[ \frac{1}{|C|}\int_C g(f(x))dx \leq \frac12 \int_{-1}^1g(f(0)(1+t))dt, \] where |C| denotes the volume of C. If g? is strictly convex, equality holds if and only if f is affine, C is a generalized symmetric cylinder and f becomes 0 at one of the basis of C. We exploit this inequality to answer a question of Francisco Santos on estimating the volume of a convex set by means of the…
Let $Ω\subset \mathbb{R}^d$, $d \geq 2$, be a bounded convex domain and $f\colon Ω\to \mathbb{R}$ be a non-negative subharmonic function. In this paper we prove the inequality \[ \frac{1}{|Ω|}\int_Ωf(x)\,dx \leq \frac{d}{|\partialΩ|}\int_{\partialΩ} f(x)\,dσ(x)\,. \] Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $Ω\subset \mathbb{R}^d$ is a bounded convex domain and $u$ is the solution of $-Δu =1$ with homogeneous Dirichlet boundary conditions, then \[ \|\nabla u\|_{L^\infty(Ω)} < d\frac{|Ω|}{|\partialΩ|}\,. \] Moreove…
Let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $Δf \geq 0$). Then $$ \frac{1}{|Ω|} \int_Ω{f dx} \leq \frac{c_n}{ |\partial Ω| } \int_{\partial Ω}{ f dσ},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ Ω_2 \subset Ω_1 \subset \mathbb{R}^n$: $$ \frac{|\partial Ω_1|}{|Ω_1|}…
In this paper, we have derived certain classical inequalities, namely, Young's, Hölder's, Minkowski's and Hermite-Hadamard inequalities for pseudo-integral (also known as $g$-integral). For Young's, Hölder's, Minkowski's inequalities, both the cases $p>1$ as well as $p<1,\,p\ne 0$ have been covered. Moreover, in the case of Hermite-Hadamard inequality, a refinement has also been proved and as a special case, $g$-analogue of geometric-logarithmic-arithmetic inequality has been deduced.
This monograph is associated with the renowned Hermite-Hadamard's integral inequality of $2$-variables on the co-ordinates. In this article we established several inequalities of the type of Hadamard's for the mappings whose absolute values of second order partial derivatives are $s-(α,m)$-convex mappings.
In this paper, a general form of integral inequalities of Hermite-Hadamard's type through differentiability for s-Convex function in second sense and whose all derivatives are absolutely continuous are established. The generalized integral inequalities contributes some better estimates than some already presented. The inequalities are then applied to numerical integration and some special means.
This article improves the triangle inequality for complex numbers, using the Hermite-Hadamard inequality for convex functions. Then, applications of the obtained refinement are presented to include some operator inequalities. The operator applications include numerical radius inequalities and operator mean inequalities.
In this paper, we obtain some new inequalities of Hermite-Hadamard type and Simpson type for functions whose third derivatives belong to Godunova-Levin class.
In this paper, we first provide a better estimate of the second inequality in Hermite-Hadamard inequality. Next, we study the reverse of the celebrated Davis-Choi-Jensen's inequality. Our results are employed to establish a new bound for the operator Kantorovich inequality.
In this paper, we provide some inequalities for $P$-class functions and self-adjoint operators on a Hilbert space including an operator version of the Jensen's inequality and the Hermite-Hadamard's type inequality. We improve the Hölder-MacCarthy inequality by providing an upper bound. Some refinements of the Jensen type inequality for $P$-class functions will be of interest.
In this paper, we introduce the concept of operator arithmetic-geometrically convex functions for positive linear operators and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for operators which give some refinements of previous results. Moreover, some unitarily invariant norm inequalities are established.
Some Generalizations of Mercer inequality and its operator extensions
本文研究了超二次函数的Mercer不等式及其算子推广，通过用正算子替换常数得到更一般的形式，并导出了Jensen算子不等式及Hermite-Hadamard型不等式。
In this paper, we prove an operator version of the Jensen's inequality and its converse for $h$-convex functions. We provide a refinement of the Jensen type inequality for $h$-convex functions. Moreover, we prove the Hermite-Hadamard's type inequality and a multiple operator version of the Jensen's inequality for $h$-convex functions. In particular, a result for convex, $P$-class, $s$-convex, Godunova-Levin, and $s$-Godunova-Levin functions can be deduced.
In this paper, we obtain a new class of functions, which is developed via the Hermite--Hadamard 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 arithmetic--geometric operator mean inequality as $A\#_{t}B\leq A\ell_t B\leq \frac{1}{2}(A\#_{t}B + A\nabla_{t} B)\le A\nabla_tB$ and $A\…
On new types of convex functions and their properties
本文针对 $-1<n-α\leq 1$ 提出两个新凸函数及引理，探讨其与 Caputo 左侧导数的相关性，并基于新定义重写 Hermite Hadamard 积分不等式及给出初等不等式。
New algebraic-analytic properties of a previously studied Banach algebra $\mathcal{A}({\bf{p}})$ of entire functions are established. For a given fixed sequence $(\bf{p}(n))_{n\geq 0}$ of positive real numbers, such that $\lim_{n\rightarrow \infty} {\bf{p}}(n)^{\frac{1}{n}}=\infty$, the Banach algebra $\mathcal{A}({\bf{p}})$ is the set of all entire functions $f$ such that $f(z)=\sum_{n=0}^\infty \hat{f}(n) z^n $ ($z\in \mathbb{C}$), where the sequence $(\hat{f}(n))_{n\geq 0}$ of Taylor coefficients of $f$ satisfies $\hat{f}(n)=O({\bf{p}}(n)^{-1})$ for $n\rightarrow \infty$, with pointwise add…
Let $A$ be a positive definite operator on a Hilbert space $H$, and $|||.|||$ be a unitarily invariant norm on $B(H)$. We show that if $f$ is an operator monotone function on $(0,\infty)$ and $n\in \mathbb{N}$, then $|||D^n f(A)|||\leq\|f^{(n)}(A)\|$ and $\|f^{(n)}(\cdot)\|$ is a quasi-convex function on the set of all positive definite operators in $B(H)$. We establish some estimates of the right hand side of some Hermite-Hadamard type inequalities in which differentiable functions are involved, and norms of the maps induced by them on the set of self adjoint operators are convex, quasi-conve…
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