Reversed Stein–Weiss Inequalities with Poisson-Type Kernel and Qualitative Analysis of Extremal Functions

Through conformal map, isoperimetric inequalities are equivalent to the Hardy–Littlewood–Sobolev (HLS) inequalities involved with the Poisson-type kernel on the upper half space. From the analytical point of view, we want to know whether there exists a reverse analogue for the Poisson-type kernel. In this work, we give an affirmative answer to this question. We first establish the reverse Stein–Weiss inequality with the Poisson-type kernel, finding that the range of index 𝑝,q^{\prime} appearing in the reverse inequality lies in the interval (0,1), which is perfectly consistent with the feature of the index for the classical reverse HLS and Stein–Weiss inequalities. Then we give the existence and asymptotic behaviors of the extremal functions of this inequality. Furthermore, for the reverse HLS inequalities involving the Poisson-type kernel, we establish the regularity for the positive solutions to the corresponding Euler–Lagrange system and give the sufficient and necessary conditions of the existence of their solutions. Finally, in the conformal invariant index, we classify the extremal functions of the latter reverse inequality and compute the sharp constant. Our methods are based on the reversed version of the Hardy inequality in high dimension, Riesz rearrangement inequality and moving spheres.

Social inequalities in overweight and obesity in 26 European countries

Background Previous studies have shown inequalities in overweight and obesity in disfavor of the socially disadvantaged groups. This study examines the extent of these inequalities in 26 European countries. Methods Data from the 2017 EU Statistics on Income and living Conditions (EU-SILC) were used (18 years and older, n = 482,595). A body mass index of 25.0 to 29.9 kg/m2 was classified as overweight and 30.0 and more as obese. Educational level (EL) was used as socioeconomic indicator. Generalized linear models were fitted to compute low-versus high absolute (RD) and relative (RR) inequality. Absolute inequality amplitude (RDA) was calculated as RD/Prevalence. Results Among men, average EU inequalities for overweight were slightly in disfavor of the low educated (RR = 1.05, RDA=5%). A mixed inequality pattern was observed across countries, as the risk of overweight was higher among high educated men in most Eastern countries, in contrast to other parts of Europe (RR from 0.74 to 1.19, RDA from -27% to 20%). Male obesity showed more pronounced inequalities (RR = 1.22, RDA=18%), and a consistent pattern of higher risk among the low educated and wide variation across countries (RR from 1.20 to 2.18, RDA from 16% to 49%). Among women, significant inequalities in overweight were observed (RR = 1.23, RDA=21%), with a consistent pattern of higher risk among the lowest EL, and substantial variation across countries (RR from 1.06 to 1.53, RDA from 7% to 36%). Inequalities were even larger for female obesity, with average RR and RDA reaching 1.49 and 35%, and wider variation (RR from 1.35 to 2.77, RDA from 12% to 88%). Conclusions Social inequalities in weight status are widespread in Europe, but vary substantially between countries. Inequalities are larger among women. For male overweight, a reverse inequality is observed in most Eastern countries. This study allows countries to benchmark the inequalities observed nationally to the situation in other EU countries. Key messages Social inequalities in weight status are widespread in Europe. The pattern of social inequalities in overweight and obesity varies substantially by country and gender.

s.VII Anxieties, Ch.29 Inequality of Arms Reversed?: Defendants in the Battle for Political Legitimacy

This chapter discusses a communicative advantage for ‘defiant defendants’, otherwise known as the ‘inequality of arms reversed’. A common critique of international criminal justice is that international criminal trials, when faced with high-profile and charismatic defendants, are basically doomed: either they silence the defendant’s political rhetoric and become show trials, or they let the defendant speak of the bias and inconsistencies in their institutional set-up, thus equally imperilling their legitimacy. This chapter argues that international criminal courts are not doomed by the reverse inequality: the communicative outcomes of international criminal trials remain contingent. For instance, prosecutors can make arguments that are politically and culturally attuned to local audiences. Moreover, the procedure of the trial can influence the defendant’s attitude. This chapter contends that it is possible for prosecutors and judges to acknowledge the political dimension of international criminal processes without turning them into show trials. Indeed, it is desirable for judges and prosecutors to confront the politics of the defendant head on.

Generalized Logarithmic Hardy–Littlewood–Sobolev Inequality

This paper is devoted to logarithmic Hardy–Littlewood–Sobolev inequalities in the 2D Euclidean space, in the presence of an external potential with logarithmic growth. The coupling with the potential introduces a new parameter, with two regimes. The attractive regime reflects the standard logarithmic Hardy–Littlewood–Sobolev inequality. The 2nd regime corresponds to a reverse inequality, with the opposite sign in the convolution term, which allows us to bound the free energy of a drift–diffusion–Poisson system from below. Our method is based on an extension of an entropy method proposed by E. Carlen, J. Carrillo, and M. Loss, and on a nonlinear diffusion equation.

Hard and Soft Adjusting of a Parameter With Its Known Boundaries by the Value Based on the Experts’ Estimations Limited to the Parameter

Adjustment of an unknown parameter of the multistage expert procedure is considered. The lower and upper boundaries of the parameter are counted to be known. A key condition showing that experts’ estimations are satisfactory in the current procedure is an inequality, in which the value based on the estimations is not greater than the parameter. The algorithms of hard and soft adjusting are developed. If the inequality is true and its both terms are too close for a long sequence of expert procedures, the adjusting can be early stopped. The algorithms are reversible, implying inversion to the reverse inequality and sliding up off the lower boundary.

ADDITIVITY PROPERTIES OF SOFIC ENTROPY AND MEASURES ON MODEL SPACES

Sofic entropy is an invariant for probability-preserving actions of sofic groups. It was introduced a few years ago by Lewis Bowen, and shown to extend the classical Kolmogorov–Sinai entropy from the setting of amenable groups. Some parts of Kolmogorov–Sinai entropy theory generalize to sofic entropy, but in other respects this new invariant behaves less regularly. This paper explores conditions under which sofic entropy is additive for Cartesian products of systems. It is always subadditive, but the reverse inequality can fail. We define a new entropy notion in terms of probability distributions on the spaces of good models of an action. Using this, we prove a general lower bound for the sofic entropy of a Cartesian product in terms of separate quantities for the two factor systems involved. We also prove that this lower bound is optimal in a certain sense, and use it to derive some sufficient conditions for the strict additivity of sofic entropy itself. Various other properties of this new entropy notion are also developed.

Hölder’s reverse inequality and its applications

We establish a new reverse Holder integral inequality and its discrete version. As applications, we prove Radon?s, Jensen?s reverse and weighted power mean inequalities and their discrete versions.

A SHARP VERSION OF BONSALL’S INEQUALITY

The best possible constant in a classical inequality due to Bonsall is established by relating that inequality to Young’s. Further, this extends the range of Bonsall’s inequality and yields a reverse inequality. It also provides a better constant in an inequality of Hardy, Littlewood and Pólya.

A Reverse Analytic Inequality for the Elementary Symmetric Function with Applications

We give a reverse inequality involving the elementary symmetric function by use of the Schur harmonic convexity theory. As applications, several new analytic inequalities for then-dimensional simplex are established.