Reverse Inequality Research Articles

Sharp Moment-Entropy Inequalities and Capacity Bounds for Symmetric Log-Concave Distributions

We show that the uniform distribution minimizes entropy among all one-dimensional symmetric log-concave distributions with fixed variance, as well as various generalizations of this fact to R\'enyi entropies of orders less than 1 and with moment constraints involving $p$-th absolute moments with $p\leq 2$. As consequences, we give new capacity bounds for additive noise channels with symmetric log-concave noises, as well as for timing channels involving positive signal and noise where the noise has a decreasing log-concave density. In particular, we show that the capacity of an additive noise channel with symmetric, log-concave noise under an average power constraint is at most 0.254 bits per channel use greater than the capacity of an additive Gaussian noise channel with the same noise power. Consequences for reverse entropy power inequalities and connections to the slicing problem in convex geometry are also discussed.

The reverse isoperimetric inequality for convex plane curves through a length-preserving flow

By a length-preserving flow, we provide a new proof of a conjecture on the reverse isoperimetric inequality composed by Pan et al. (Math Inequal Appl 13:329–338, 2010), which states that if $$\gamma $$ is a convex curve with length L and enclosed area A, then the best constant $$\varepsilon $$ in the inequality $$\begin{aligned} L^2\le 4\pi A+\varepsilon |{\tilde{A}}| \end{aligned}$$ is $$\pi $$ , where $${\tilde{A}}$$ denotes the oriented area of the locus of its curvature centers.

Diagnosis inspired by the thermodynamic geometry for different thermodynamic schemes of the charged BTZ black hole

Due to the asymptotic structure of the black hole solution, there are two different thermodynamic schemes for the charged Banados–Teitelboim–Zanelli (BTZ) black hole. In one scheme, the charged BTZ black hole is super-entropic, while in the other, it is not (the reverse isoperimetric inequality is saturated). In this paper, we investigate the thermodynamic curvature of the charged BTZ black hole in different coordinate spaces. We find that in both schemes, the thermodynamic curvature is always positive, which may be related to the information of repulsive interaction between black hole molecules for the charged BTZ black hole if we accept an empirical relationship between the thermodynamic curvature and interaction of a system. More importantly, we provide a diagnosis for the discrimination of the two schemes from the point of view of the thermodynamics geometry. For the charged BTZ black hole, when the reverse isoperimetric inequality is saturated, the thermodynamic curvature of an extreme black hole tends to be infinity, while when the reverse isoperimetric inequality is violated, the thermodynamic curvature of the extreme black hole goes to a finite value.

Social inequalities in overweight and obesity in 26 European countries

Abstract 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.

Numerical Radius Inequalities in 2-Inner Product Spaces

In this paper, we have obtained the analogue results on numerical radius inequalities from the classical inner product spaces to 2-inner product spaces. We have established several related reverse inequalities and some well known results in 2-inner product spaces

Dynamic Hardy-type inequalities with non-conjugate parameters

Within this paper, we derive entire series of new Hilbert and Hardy-Hilbert form integral inequalities with non-conjugate exponents on time scales. Firstly, we demonstrate and discuss two general equivalent inequalities of this type, as well as their corresponding reverse inequalities. We also extend Hilbert and Hardy-Hilbert inequalities in their most general form. The results obtained are then applied to a kernel of special Hardy form. Our results as special cases extend some of the dynamic inequalities achieved on time scales and also include some integral inequalities as special cases when T=R.

Some reverse inequalities of Hardy type on time scales

In this article, we obtain some new dynamic inequalities of Hardy type on time scales. The main results are derived using Fubini’s theorem and the chain rule on time scales. We apply the main results to the continuous calculus, discrete calculus, and q-calculus as special cases.

On a reverse isoperimetric inequality for relative outer parallel bodies

We show a reverse isoperimetric inequality within the class of relative outer parallel bodies, with respect to a general convex body E E , along with its equality condition. Based on the convexity of the sequence of quermassintegrals of Minkowski sums we also prove further inequalities.

A reverse Minkowski-type inequality

The famous Minkowski inequality provides a sharp lower bound for the mixed volume V ( K , M [ n − 1 ] ) V(K,M[n-1]) of two convex bodies K , M ⊂ R n K,M\subset \mathbb {R}^n in terms of powers of the volumes of the individual bodies K K and M M . The special case where K K is the unit ball yields the isoperimetric inequality. In the plane, Betke and Weil (1991) found a sharp upper bound for the mixed area of K K and M M in terms of the perimeters of K K and M M . We extend this result to general dimensions by proving a sharp upper bound for the mixed volume V ( K , M [ n − 1 ] ) V(K,M[n-1]) in terms of the mean width of K K and the surface area of M M . The equality case is completely characterized. In addition, we establish a stability improvement of this and related geometric inequalities of isoperimetric-type.

Generalized Minkowski-type Fractional Inequalities Involving Extended Mittag-leffler Function

In this paper the reverse fractional Minkowski integral inequality using extended Mittag-Leffler function with the corresponding fractional integral operator is proved, as well as several related Minkowskitype inequalities.

New generalized reverse Minkowski and related integral inequalities involving generalized fractional conformable integrals

This paper gives some novel generalizations by considering the generalized conformable fractional integrals operator for reverse Minkowski type and reverse Hölder type inequalities. Furthermore, novel consequences connected with this inequality, together with statements and confirmation of various variants for the advocated generalized conformable fractional integral operator, are elaborated. Moreover, our derived results are provided to show comparisons of convergence between old and modified operators towards a function under different parameters and conditions. The numerical approximations of our consequence have several utilities in applied sciences and fractional integro-differential equations.

Turán-Type Reverse Markov Inequalities for Polynomials with Restricted Zeros

Let $$\mathcal{P}_n^c$$ denote the set of all algebraic polynomials of degree at most n with complex coefficients. Let $$\begin{aligned} D^+ := \{z \in \mathbb {C}: |z| \le 1, \text { Im}(z) \ge 0\}\,. \end{aligned}$$For integers $$0 \le k \le n$$ let $$\mathcal{F}_{n,k}^c$$ be the set of all polynomials $$P \in \mathcal{P}_n^c$$ having at least $$n-k$$ zeros in $$D^+$$. Let $$\begin{aligned} \Vert f\Vert _A := \sup _{z \in A}{|f(z)|} \end{aligned}$$for complex-valued functions defined on $$A \subset \mathbb {C}$$. We prove that there are absolute constants $$c_1 > 0$$ and $$c_2 > 0$$ such that $$\begin{aligned} c_1 \left( \frac{n}{k+1}\right) ^{1/2} \le \inf _{P}{\frac{\Vert P^{\prime }\Vert _{[-1,1]}}{\Vert P\Vert _{[-1,1]}}} \le c_2 \left( \frac{n}{k+1}\right) ^{1/2} \end{aligned}$$for all integers $$0 \le k \le n$$, where the infimum is taken for all $$0 \not \equiv P \in \mathcal{F}_{n,k}^c$$ having at least one zero in $$[-1,1]$$. This is an essentially sharp reverse Markov-type inequality for the classes $$\mathcal{F}_{n,k}^c$$ extending earlier results of Turan and Komarov from the case $$k=0$$ to the cases $$0 \le k \le n$$.

Quantum Reverse Hypercontractivity: Its Tensorization and Application to Strong Converses

In this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log-Sobolev inequalities. Next we prove a generalization of the Stroock–Varopoulos inequality in the non-commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2-log-Sobolev and 1-log-Sobolev inequalities respectively. We then prove some tensorization-type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers. Finally as an application of these results, we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting. We prove strong converse bounds for the problems of quantum hypothesis testing and classical-quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive.

A series of refinements on the Young and reverse Young inequalities through a recursive algorithm

In this article, we present a recursive algorithm to obtain a series of refinements of the classical Young inequality. These inequalities approach to equalities when the number of the iteration in the recursive algorithm tends to infinity. Also these refinements applied to establish some improved reverse Young and matrix Young inequalities with Hilbert- Schmidt norm.

Distributing Responsibility

Distributing Responsibility

Improvements of operator reverse AM-GM inequality involving positive linear maps

In this paper, we shall present some reverse arithmetic-geometric mean operator inequalities for unital positive linear maps. These inequalities improve some corresponding results due to Xue (J. Inequal. Appl. 2017:283, 2017).

A reverse Sidorenko inequality

Let H be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph G without isolated vertices, the weighted number of graph homomorphisms \(\hom (G, H)\) satisfies the inequality $$\begin{aligned} \hom (G, H ) \le \prod _{uv \in E(G)} \hom (K_{d_u,d_v}, H )^{1/(d_ud_v)}, \end{aligned}$$where \(d_u\) denotes the degree of vertex u in G. In particular, one has $$\begin{aligned} \hom (G, H )^{1/|E(G)|} \le \hom (K_{d,d}, H )^{1/d^2} \end{aligned}$$for every d-regular triangle-free G. The triangle-free hypothesis on G is best possible. More generally, we prove a graphical Brascamp–Lieb type inequality, where every edge of G is assigned some two-variable function. These inequalities imply tight upper bounds on the partition function of various statistical models such as the Ising and Potts models, which includes independent sets and graph colorings. For graph colorings, corresponding to \(H = K_q\), we show that the triangle-free hypothesis on G may be dropped; this is also valid if some of the vertices of \(K_q\) are looped. A corollary is that among d-regular graphs, \(G = K_{d,d}\) maximizes the quantity \(c_q(G)^{1/|V(G)|}\) for every q and d, where \(c_q(G)\) counts proper q-colorings of G.Finally, we show that if the edge-weight matrix of H is positive semidefinite, then $$\begin{aligned} \hom (G, H) \le \prod _{v \in V(G)} \hom (K_{d_v+1}, H )^{1/(d_v+1)}. \end{aligned}$$This implies that among d-regular graphs, \(G = K_{d+1}\) maximizes \(\hom (G, H)^{1/|V(G)|}\). For 2-spin Ising models, our results give a complete characterization of extremal graphs: complete bipartite graphs maximize the partition function of 2-spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models. These results settle a number of conjectures by Galvin–Tetali, Galvin, and Cohen–Csikvári–Perkins–Tetali, and provide an alternate proof to a conjecture by Kahn.

Hypo-q-Norms on a Cartesian Product of Algebras of Operators on Banach Spaces

Abstract In this paper we consider the hypo-q-operator norm and hypo-q-numerical radius on a Cartesian product of algebras of bounded linear operators on Banach spaces. A representation of these norms in terms of semi-inner products, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy-Buniakowski-Schwarz inequality are also given.

Notes on the thermodynamics of superentropic AdS black holes

The super-entropic black hole, which possesses a noncompact horizon topology and violates the reverse isoperimetric inequality, has been found to satisfy both the thermodynamic first law and the Bekenstein-Smarr mass formula. In this paper, we first derive a new Christodoulou-Ruffini-like squared-mass formula for the four-dimensional Kerr-Newman-AdS super-entropic black hole, and then establish a set of very simple relations between thermodynamic quantities of the super-entropic Kerr-Newman-AdS$_4$ black hole and its usual counterparts. Using these relations, the thermodynamic quantities of the Kerr-Newman-AdS$_4$ super-entropic black hole can be obtained from those of the usual pro-type by taking the ultra-spinning limit properly. Then these relations are extended to the singly-rotating Kerr-AdS black holes in arbitrary dimensions and the double-rotating charged black hole in the five-dimensional minimal gauged supergravity. It can be inferred that the thermodynamic quantities of all super-entropic black holes obey similar limiting relations to those of their corresponding conventional rotating AdS black holes, and thus can be obtained by taking the ultra-spinning limit appropriately.

Instability of super-entropic black holes in extended thermodynamics

The charged black hole of Bañados, Teitelbiom and Zanelli is studied in extended gravitational thermodynamics where there is a dynamical pressure and volume. It is a simple example of a super-entropic black hole, violating the reverse isoperimetric inequality. It is proven that this property implies that its specific heat at constant volume is negative, signaling a new kind of fundamental instability for black holes. It is conjectured that this instability is present for other super-entropic black holes, and this is demonstrated numerically for a large family of known solutions.