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 numerical integrations are studied.
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 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.
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 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 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.
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 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, 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.
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
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 $\mathbb{R}^α(0<α<1).$
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.
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.
The purpose of this paper is to study a generalization of strongly $η$-convex functions using the fractal calculus developed by Yang \cite{Yang}, namely generalized strongly $η$-convex function. Among other results, we obtain some Hermite-Hadamard and Fejér type inequalities for this class of functions.
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.
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|}…
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, 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.
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 $ α$ in absolute value at certain powers are generalized $ s $-convex functions in the second sen…
