The aim of this paper is to show that Jensen’s Inequality and an extension of Chebyshev’s Inequality complement one another, so that they both can be formulated in a pairing form, including a second inequality, that provides an estimate for the classical one. 1. Introduction The well known fact that the derivative and the integral are inverse each other has a lot of interesting consequences, on...

Keywords: Hermite–Hadamard inequalities Jensen's inequality Xiao–Srivastava–Zhang–Pečarić–Svrtan–Jensen type inequalities Refinements and extensions Convex functions a b s t r a c t The main object of this paper is to give several refinements and extensions of the Hermite–Hadamard and Jensen inequalities in n variables. Relevant connections of the results presented here and the various inequali...

Income inequality undermines societies: The more inequality, the more health problems, social tensions, and the lower social mobility, trust, life expectancy. Given people's tendency to legitimate existing social arrangements, the stereotype content model (SCM) argues that ambivalence-perceiving many groups as either warm or competent, but not both-may help maintain socio-economic disparities. ...

In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for Gromov hyperbolicity in terms of the isoperimetric function. We prove similar results for the linear filling radius inequality. Our theorems strengthen and gene...

A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santalo inequality and the affine isoperimetric inequality of affine differential geometry.

We give a refinement of the quantitative isoperimetric inequality. We prove that the isoperimetric gap controls not only the Fraenkel asymmetry but also the oscillation of the boundary.

Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality δ(l) < Clα for α > 2 then it satisfies the averaged isoperimetric inequality δ(l) < C...

Generalized refinement of Jensen’s inequality is given. Appplications are done for cyclic mixed symmetric means and Cauchy means.

This chapter examines the role of social inequality in crime and deviance by specifying a social psychological theory of the causal mechanisms by which inequality is associated with crime. We begin by noting that the powerful have more input into the content of criminal law, a point illustrated by the relatively soft penalties for white collar and corporate crimes compared to the harsh penaltie...