It is well-known that every convex function admits an affine support at every interior point of a domain. Convex functions of higher order (precisely of an odd order) have a similar property: they are supported by the polynomials of degree no greater than the order of convexity. In this paper the attaching method is developed. It is applied to obtain the general result Theorem 2, from which the mentioned above support theorem and some related properties of convex functions of higher (both odd and even) order are derived. They are applied to obtain some known and new Hadamard-type inequalities between the quadrature operators and the integral approximated by them. It is also shown that the error bounds of quadrature rules follow by inequalities of this kind.
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, 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, 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 obtain some Hermite-Hadamard type inequalities for functions whose third derivatives in absolute value are preinvex and prequasiinvex.
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
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, we established some new Hadamard-type integral inequalities for functions whose derivatives of absolute values are m-convex and (α,m)-convex functions via Riemann-Liouville fractional integrals.
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 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 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 achieve some new Hadamard type inequalities using elementary well known inequalities for functions whose first derivatives absolute values are s-geometrically and geometrically convex. And also we get some applications for special means for positive numbers.
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 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 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 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.
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 the paper, we establish the Hermite-Hadamard type inequalities for the generalized s-convex functions in the second sense on real linear fractal set $\mathbb{R}^α(0<α<1).$
We prove new inequalities for the spectral radius, essential spectral radius, operator norm, measure of noncompactness and numerical radius of Hadamard weighted geometric means of positive kernel operators on Banach function and sequence spaces. Several inequalities appear to be new even in the finite dimensional case.
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 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.
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.
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.
In this paper, we not only give the extensions of the results given in [7] by Gill et al. for log-convex functions, but also obtain some new Hadamard type inequalities for log-convex, m-convex and (alpha,m)-convex functions.
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.
We prove new inequalities and equalities for the generalized and the joint spectral radius (and their essential versions) of Hadamard (Schur) geometric means of bounded sets of positive kernel operators on Banach function spaces. In the case of nonnegative matrices that define operators on Banach sequences we obtain additional results. Our results extend results of several authors that appeared relatively recently.
Relatively recently, K.M.R. Audenaert (2010), R.A. Horn and F. Zhang (2010), Z. Huang (2011), A.R. Schep (2011), A. Peperko (2012), D. Chen and Y. Zhang (2015) have proved inequalities on the spectral radius and the operator norm of Hadamard products and ordinary matrix products of finite and infinite non-negative matrices that define operators on sequence spaces. In the current paper we extend and refine several of these results and also prove some analogues for the numerical radius. Some inequalities seem to be new even in the case of $n\times n$ non-negative matrices.
In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) for radial derivations (i.e., the derivation along the geodesic curve) on Cartan--Hadamard manifolds. By Gauss lemma, our new Hardy inequality are stronger than the classical one. We also established the improvements of these inequalities in terms of sectional curvature of underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and…
We investigate the validity, as well as the failure, of Sobolev-type inequalities on Cartan-Hadamard manifolds under suitable bounds on the sectional and the Ricci curvatures. We prove that if the sectional curvatures are bounded from above by a negative power of the distance from a fixed pole (times a negative constant), then all the $ L^p $ inequalities that interpolate between Poincaré and Sobolev hold for radial functions provided the power lies in the interval $ (-2,0) $. The Poincaré inequality was established by H.P. McKean under a constant negative bound from above on the sectional cur…
As one of the least squares mean, we consider the Wasserstein mean of positive definite Hermitian matrices. We verify in this paper the inequalities of the Wasserstein mean related with a strictly positive and unital linear map, the identity of the Wasserstein mean for tensor product, and some inequalities of the Wasserstein mean for Hadamard product.
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 $Ψ$ and $Σ$ be bounded sets of positive kernel operators on a Banach function space $L$. We prove several refinements of the known inequalities $$ρ\left(Ψ^{\left( \frac{1}{2} \right)} \circ Σ^{\left( \frac{1}{2} \right)} \right) \le ρ(ΨΣ) ^{\frac{1}{2}} \;\; \mathrm{and}\;\; \hatρ \left(Ψ^{\left( \frac{1}{2} \right)} \circ Σ^{\left( \frac{1}{2} \right)} \right) \le \hatρ (ΨΣ) ^{\frac{1}{2}} $$ for the generalized spectral radius $ρ$ and the joint spectral radius $\hatρ$, where $Ψ^{\left( \frac{1}{2} \right)} \circ Σ^{\left( \frac{1}{2} \right)} $ denotes the Hadamard (Schur) geometric mean…
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 …
Recently, several authors have proved inequalities on the spectral radius $ρ$, operator norm $\|\cdot\|$ and numerical radius of Hadamard products and ordinary products of non-negative matrices that define operators on sequence spaces, or of Hadamard geometric mean and ordinary products of positive kernel operators on Banach function spaces. In the present article we generalize and refine several of these results. In particular, we show that for a Hadamard geometric mean $A ^{\left( \frac{1}{2} \right)} \circ B ^{\left( \frac{1}{2} \right)}$ of positive kernel operators $A$ and $B$ on a Banach…
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.
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 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.
We prove new explicit asymptotic formulae between (geometric) eigenvalues in max-algebra and classical distinguished eigenvalues of nonnegative matrices, which are useful tools for transferring results between both settings. We establish new inequalities for both types of eigenvalues of Hadamard products and Hadamard weighted geometric means of nonnegative matrices. Moreover, a version of the spectral mapping theorem for the distinguished spectrum is pointed out.
The purpose of this paper is to propose and analyze a multi-step iterative algorithm to solve a convex optimization problem and a fixed point problem posed on a Hadamard space. The convergence properties of the proposed algorithm are analyzed by employing suitable conditions on the control sequences of parameters and the structural properties of the under lying space. We aim to establish strong and del-convergence results of the proposed iterative algorithm and compute an optimal solution for a minimizer of proper convex lower semicontinuous function and a common fixed point of a finite family…
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 establish some new Hadamard type inequalities using elementary well known inequalities for functions whose inequalities absolute values are α-, m-, (α,m)-logarithmically convex.
We show that cyclic products of projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bačák. Namely, we construct an example of convex subsets in a Hadamard space such that the corresponding cyclic product of projections is not asymptotically regular.
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.
The Cartan-Hadamard conjecture states that, on every $n$-dimensional Cartan-Hadamard manifold $ \mathbb{M}^n $, the isoperimetric inequality holds with Euclidean optimal constant, and any set attaining equality is necessarily isometric to a Euclidean ball. This conjecture was settled, with positive answer, for $n \le 4$. It was also shown that its validity in dimension $n$ ensures that every $p$-Sobolev inequality ($ 1 < p < n $) holds on $ \mathbb{M}^n $ with Euclidean optimal constant. In this paper we address the problem of classifying all Cartan-Hadamard manifolds supporting an optimal fun…
We derive new comparison inequalities between weak and strong moments of norms of random vectors with optimal (up to an universal factor) constants. We discuss applications to the concentration of log-concave random vectors and bounds on $p$-summing norms of finite rank operators.
