Hermite-Hadamard’s type inequalities for operator convex functions

Some Hermite-Hadamard’s type inequalities for operator convex functions of selfadjoint operators in Hilbert spaces are given. Applications for particular cases of interest are also provided.     

Opial type inequalities for cosine and sine operator functions

Various L _p form Opial type inequalities are given for cosine and sine operator functions with applications.     

rP—凸函数与琴生型不等式

给出 r P—凸函数的定义以及判定 r P—凸函数的方法 ,建立关于 r P—凸函数的琴生型不等式 ,最后给出它的应用 ,包括改进一些已知不等式和建立一些新不等式 .     

Some extensions of the operator entropy type inequalities

In this paper, we define the generalised relative operator entropy and investigate some of its properties such as subadditivity and homogeneity. As application of our result, we obtain the information inequality. In continuation, we establish some reverses of the operator entropy inequalities under certain conditions by using the Mond–Pe?ari? method.     

New sharp inequalities for operator means

New sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator Pólya-Szegö inequality to arbitrary operator means. Furthermore, we obtain some new lower and upper bounds for the Tsallis relative operator entropy, operator monotone functions and positive linear maps.     

Some integral inequalities for logarithmically convex functions

The main aim of the present note is to establish new Hadamard like integral inequalities involving log-convex function. We also prove some Hadamard-type inequalities, and applications to the special means are given.     

Weighted Hermite–Hadamard–Mercer type inequalities for convex functions

In this paper, first, we prove the weighted Hermite–Hadamard–Mercer inequalities for convex functions, after we establish some new weighted inequalities connected with the right‐sides of weighted Hermite–Hadamard–Mercer type inequalities for differentiable functions whose derivatives in absolute value at certain powers are convex. The results presented here would provide extensions of those given in earlier works.     

Hadamard type local fractional integral inequalities for generalized harmonically convex functions and applications

In this paper, we establish a local fractional integral identity with a parameter $\lambda$ on Yang's fractal sets. Using this identity, by generalized power mean inequality and generalized Hölder inequality, two Hermite-Hadamard type local fractional integral inequalities for generalized harmonically convex functions are established. By giving some special values to the parameter, some inequalities with specific form can be obtained. Some applications to generalized special means are given.     

几何凸函数与琴生型不等式

给出几何凸函数的定义以及判定几何凸函数的方法 ,建立关于几何凸函数的琴生型不等式 ,最后给出它的应用 ,包括改进一些已知不等式和建立一些新不等式 .     

Simpson and Newton type inequalities for convex functions via newly defined quantum integrals

We first establish two new identities, based on the kernel functions with either two section or three sections, involving quantum integrals by using new definition of quantum derivative. Then, some new inequalities related to Simpson's 1/3 formula for convex mappings are provided. In addition, Newton type inequalities, for functions whose quantum derivatives in modulus or their powers are convex, are deduced. We also mention that the results in this work generalize inequalities given in earlier study.     

Integral inequalities for coordinated Harmonically convex functions

In this paper, we introduce the notion of coordinated harmonically convex functions. We derive some new integral inequalities of Hermite–Hadamard type for coordinated Harmonically convex functions. The interested readers are encouraged to find the applications of harmonically convex functions in pure and applied sciences.     

Refinements of the Heinz operator inequalities

In this article, we study the Heinz inequalities for two positive operators. We refine the ordering relations among the Heinz means with different parameters and obtain some improvements of the Heinz operator inequalities.     

Poincaré like inequalities for semigroups, cosine and sine operator functions

Here we present Poincaré type general L _p inequalities regarding semigroups, cosine and sine operator functions.     

Rellich type inequalities related to Grushin type operator and Greiner type operator

We prove some sharp Hardy inequality associated with the gradient ? _?? = (? _x ,|x| ^?? ? _y ) by a direct and simple approach. Moreover, similar method is applied to obtain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L ^p . We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.     

Converses of the Jensen inequality derived from the Green functions with applications in information theory

Jensen integral inequality has got much importance regarding their applications in different fields of mathematics. In this paper, two converses of Jensen integral inequality for convex function are obtained. The results are applied to establish converses of Hölder and Hermite-Hadamard inequalities as well. At the end, some useful applications in information theory of the obtained results are given. The idea used in this paper may inculcate further research.     

Some new inequalities involving the Hardy operator

In this paper we derive some new inequalities involving the Hardy operator, using some estimates of the Jensen functional, continuous form generalization of the Bellman inequality and a Banach space variant of it. Some results are generalized to the case of Banach lattices on $(0,b],0<b\le \infty$.     

Quadratic operator inequalities and linear-fractional relations

We study properties of solution sets of inequalities of the form

$X^* AX + B^* X + X^* B + C \leqslant 0,$

, where A, B, and C are bounded Hilbert space operators and A and C are self-adjoint. The following properties are considered: closedness and inferior points in Standard operator topologies, convexity, and nonemptiness.