On Jensen-McShane’s inequality

A sequence of inequalities which include McShane’s generalization of Jensen’s inequality for isotonic positive linear functionals and convex functions are proved and compared with results in [3]. As applications some results for the means are pointed out. Moreover, further inequalities of Hölder type are presented.     

Inequalities for the generalized trigonometric and hyperbolic functions

The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p$p$-Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinovi?–Adamovi?’s inequality, Lazarevi?’s inequality, Huygens-type inequalities, Wilker-type inequalities, and Cusa–Huygens-type inequalities, are generalized to the case of generalized functions.     

Generalizations of the Jensen-Steffensen and related inequalities

We present a couple of general inequalities related to the Jensen-Steffensen inequality in its discrete and integral form. The Jensen-Steffensen inequality, Slater’s inequality and a generalization of the counterpart to the Jensen-Steffensen inequality are deduced as special cases from these general inequalities.     

Some matrix inequalities related to Kwong functions

The aim of this note is to investigate some inequalities related to Kwong functions. Refinements of Kaur’s and Zhan’s inequality are presented by the Hadamard product. In addition, variations of the Heinz–Heron type means on Kwong functions are also given.     

三阶可微函数的若干不等式

通过建立与Bullen不等式有关的积分恒等式,利用Halder不等式、Grüss不等式、Chebychev不等式,给出三阶可微函数的一些不等式.     

The Hadamard determinant inequality — Extensions to operators on a Hilbert space

A generalization of classical determinant inequalities like Hadamard's inequality and Fischer's inequality is studied. For a version of the inequalities originally proved by Arveson for positive operators in von Neumann algebras with a tracial state, we give a different proof. We also improve and generalize to the setting of finite von Neumann algebras, some ‘Fischer-type’ inequalities by Matic for determinants of perturbed positive-definite matrices. In the process, a conceptual framework is established for viewing these inequalities as manifestations of Jensen's inequality in conjunction with the theory of operator monotone and operator convex functions on $[0,\infty )$. We place emphasis on documenting necessary and sufficient conditions for equality to hold.     

关于双曲函数的Huygens型不等式及应用

给出了关于双曲函数的Huygens型不等式,并得到了双曲函数的Wilker型不等式的加强.定义了一类新的Seiffert型平均,给出两个有关不等式.     

On general Bernstein and Nikol’ski type inequalities

The goal of this paper is to establish the relations between general Bernstein and Nikol’ski type inequalities under some weak conditions. From these relations some known classical inequalities are implied. Also, a family of functions equipped with Bernstein type inequality which satisfies Nikol’ski type inequality is found.     

Hardy-Hilbert型积分不等式的一种新改进

研究了带积分核函数和参数λ（λ〉1）的Hardy-Hilbert型不等式,并利用加强的Hlder’s不等式对Hardy-Hilbert不等式作了改进,建立了一些新的不等式.     

On some generalizations of the Hilbert–Hardy type discrete inequalities

New Hilbert-type discrete inequalities are presented by using new techniques in proof. By specializing the weight coefficient functions in the hypothesis and the parameters, we obtain many special cases which include, in particular, the discrete inequality derived by Hilbert and Hardy. Many improvements and generalizations of known results are given in this paper.     

Refinements of the Cauchy–Bunyakovsky–Schwarz inequality for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous functions of selfadjoint operators in Hilbert spaces that improve the Cauchy–Bunyakovsky–Schwarz inequality, are given.     

A half-discrete Hilbert-type inequality in the whole plane related to the Riemann zeta function

By the use of Hermite–Hadamard’s inequality and weight functions, a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic cotangent function and multi-parameters is given. The constant factor related to the Riemann zeta function is proved to be the best possible. The equivalent forms, two kinds of particular inequalities, the operator expressions and some equivalent reverses are considered.     

An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces

We generalize the Beckner’s type Poincaré inequality (Beckner, W. Proc. Amer. Math. Soc. (1989) 105:397–400) to a large class of probability measures on an abstract Wiener space of the form μ?ν, where μ is the reference Gaussian measure and ν is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and integration with respect to those convolution measures. In addition, we prove that in the finite-dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.     

Some trace inequalities of Čebyšev type for functions of operators in hilbert spaces

Some trace operator inequalities for synchronous functions that are related to the ?eby?ev inequality for sequences of real numbers are given.     

Generalization of Popoviciu-Type Inequalities Via Fink’s Identity

We obtained useful identities via Fink’s identity, by which the inequality of Popoviciu for convex functions is generalized for higher order convex functions. We investigate the bounds for the identities related to the generalization of the Popoviciu inequality using inequalities for the ?eby?ev functional. Some results relating to the Grüss- and Ostrowski-type inequalities are constructed. Further, we also construct new families of exponentially convex functions and Cauchy-type means by looking at linear functional associated with the obtained inequalities.     

A refinement of von Neumann—Heinz theorem and related inequalities

The paper gives a quantitative refinement of von Neumann’s theorem with some relevant inequalities permitting to estimate from below the distance between the origin of the coordinate system and the so-called normalized numerical range of an operator acting in the Hilbert space.     

General Hardy’s inequalities for functions nonzero on the boundary

Consider Hardy’s inequalities with general weight ϕ for functions nonzero on the boundary. By an integral identity in C ¹( ), define Hilbert spaces H _k ¹ (Ω, ϕ) called Sobolev-Hardy spaces with weight ϕ. As a corollary of this identity, Hardy’s inequalities with weight ϕ in C ¹ ( ) follow. At last, by Hardy’s inequalities with weight ϕ = 1, discuss the eigenvalue problem of the Laplace-Hardy operator with critical parameter (N − 2)²/4 in H ¹ ₁ (Ω).      

Radon不等式的新推广

利用Ho。lder不等式、Young不等式、Chebyshev不等式、幂平均不等式建立Radon不等式的指数推广形式,得到一个具有广泛应用价值的不等式.指出文[7]中给出的关于Radon不等式的推广结果是错误的,并在本文中作了修正.     

与GA-凸函数有关的两个函数

引入两个与GA-凸函数有关的函数,研究了它们的凸性、单调性及相互关系,获得了已有文献关于GA-凸函数的Hadamard型不等式的加细,也得到一些新的不等式.     

A new approach for the derivation of bounds for the Jensen difference

There are many useful applications of Jensen's inequality in several fields of science, and due to this reason, a lot of results are devoted to this inequality in the literature. The main theme of this article is to present a new method of finding estimates of the Jensen difference for differentiable functions. By applying definition of convex function, and integral Jensen's inequality for concave function in the identity pertaining the Jensen difference, we derive bounds for the Jensen difference. We present integral version of the bounds in Riemann sense as well. The sharpness of the proposed bounds through examples are discussed, and we conclude that the proposed bounds are better than some existing bounds even with weaker conditions. Also, we present some new variants of the Hermite–Hadamard and Hölder inequalities and some new inequalities for geometric, quasi-arithmetic, and power means. Finally, we give some applications in information theory.