Sharp Inequalities Research Articles

Comparing three possible hypoelliptic Laplacians on the 5-dimensional Cartan group via div-curl type estimates

Abstract On general Carnot groups, the definition of a possible hypoelliptic Hodge-Laplacian on forms using the Rumin complex has been considered in (M. Rumin, “Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups,” C. R. Acad. Sci., Paris Sér. I Math., vol. 329, no. 11, pp. 985–990, 1999, M. Rumin, “Sub-Riemannian limit of the differential form spectrum of contact manifolds,” Geom. Funct. Anal., vol. 10, no. 2, pp. 407–452, 2000), where the author introduced a 0-order pseudodifferential operator on forms. However, for questions regarding regularity for example, where one needs sharp estimates, this 0-order operator is not suitable. Up to now, there have only been very few attempts to define hypoelliptic Hodge-Laplacians on forms that would allow for such sharp estimates. Indeed, this question is rather difficult to address in full generality, the main issue being that the Rumin exterior differential d c is not homogeneous on arbitrary Carnot groups. In this note, we consider the specific example of the free Carnot group of step 3 with 2 generators, and we introduce three possible definitions of hypoelliptic Hodge-Laplacians. We compare how these three possible Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups.

Conformally covariant boundary operators and sharp higher order CR Sobolev trace inequalities on the Siegel domain and complex ball

Conformally covariant boundary operators and sharp higher order CR Sobolev trace inequalities on the Siegel domain and complex ball

The sharp quantitative barycentric isoperimetric inequality for bounded sets

The sharp quantitative barycentric isoperimetric inequality for bounded sets

A sharp inequality relating Yamabe invariants on asymptotically Poincaré–Einstein manifolds with a Ricci curvature lower bound

Let [Formula: see text] be a conformally compact manifold with [Formula: see text]. If [Formula: see text] is asymptotically Poincaré–Einstein, we establish a sharp inequality relating the type II Yamabe invariant of [Formula: see text] and the Yamabe invariant of its conformal infinity.

On the Harmonic Characterization Of The Spheres: A Sharp Stability Inequality

On the Harmonic Characterization Of The Spheres: A Sharp Stability Inequality

Talenti’s comparison theorem on Finsler manifolds with nonnegative Ricci curvature

We establish a Talenti-type comparison theorem for the Dirichlet problem associated with Poisson’s equation on complete noncompact Finsler manifolds having nonnegative Ricci curvature and Euclidean volume growth. The proof relies on anisotropic symmetrization arguments and leverages the sharp isoperimetric inequality recently established by Manini [Preprint, arXiv:2212.05130, 2022 ]. In addition, we characterize the rigidity of the comparison principle under the additional assumption that the reversibility constant of the Finsler manifold is finite. As application, we prove a Faber-Krahn inequality for the first Dirichlet eigenvalue of the Finsler-Laplacian.

Type problem, the first eigenvalue and Hardy inequalities

In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues \lambda_{1}(B_{r}) of “balls” B_{r}:=\{\rho<r\} on a complete Riemannian manifold M as r\rightarrow +\infty , where \rho is a Lipschitz continuous exhaustion function with |\nabla\rho|\leq1 a.e. on M . We obtain several sharp results. First, if r^{2} \lambda_{1}(B_{r})\ge \gamma>0 for all r>r_{0} , we obtain a sharp estimate of the volume growth: |B_{r}|\ge cr^{\mu(\gamma)}. Moreover, when \gamma>j_{0}^{2}\approx 5.784 , where j_{0} denotes the first positive zero of the Bessel function J_{0} , then M is non-parabolic and we have a Hardy-type inequality. In the case where r_{0}=0 , a sharp Hardy-type inequality holds. These spectral conditions are satisfied if one assumes that \Delta\rho^{2}\geq2\mu(\gamma)>0 . In particular, when \inf_{M}\Delta\rho^{2}>4 , M is non-parabolic and we get a sharp Hardy-type inequality. Related results for finite volume case are also studied.

On Certain Analytic Functions Associated with Nephroid Function

The normalized analytic function ΦN(z)=1+z−z33, which connects the open unit disk onto a bounded domain within the right half of a nephroid-shaped region, is associated with the bounded turning of functions denoted by Rn. It calculates the sharp coefficient inequalities, which include the upper bound of the third Hankel determinant and Logarithmic coefficients related to the functions of the ΦN(z) class. This research mainly focuses on identifying solutions to specific coefficient-related problems for analytic functions within the domain of nephroid functions.

A sharp double inequality for sums of powers revisited

The proof of the monotonicity of the sequence n↦nΔ(n), presented in the 2011 article “A Sharp double inequality for sums of powers” by V. Lampret, is corrected. Namely, it is demonstrated that, for S(n):=∑k=1n(kn)n=∑j=0n(1−jn)n, the sequence n↦n(ee−1−S(n)) is strictly increasing.

On some sharp Chernoff type inequalities

On some sharp Chernoff type inequalities

A stability inequality for planar lens partition

Abstract Recently it has been shown that the unique local perimeter minimizing partitioning of the plane into three regions, where one region has finite area and the other two have infinite measure, is given by the so-called standard lens partition. Here we prove a sharp stability inequality for the standard lens, hence strengthening the local minimality of the lens partition in a quantitative form. As an application of this stability result we consider a nonlocal perturbation of an isoperimetric problem.

Sharp Hardy-type inequalities for non-compact harmonic manifolds and Damek–Ricci spaces

Sharp Hardy-type inequalities for non-compact harmonic manifolds and Damek–Ricci spaces

Bohr's phenomenon for certain classes of analytic functions

This paper aims to explore some sharp Bohr-type inequalities for various classes of analytic functions. We begin by examining the Bohr phenomenon in terms of convex combinations and we extend our study to various other Bohr-type inequalities for the class of analytic univalent functions. Furthermore, we investigate similar inequalities for the Ma-Minda class of convex as well as starlike functions. Also, we estimate the same for the class of concave univalent functions.

Cantelli’s Bounds for Generalized Tail Inequalities

Let X be a centered random vector in a finite-dimensional real inner product space E. For a subset C of the ambient vector space V of E and x,y∈V, write x⪯Cy if y−x∈C. If C is a closed convex cone in E, then ⪯C is a preorder on V, whereas if C is a proper cone in E, then ⪯C is actually a partial order on V. In this paper, we give sharp Cantelli-type inequalities for generalized tail probabilities such as PrX⪰Cb for b∈V. These inequalities are obtained by “scalarizing” X⪰Cb via cone duality and then by minimizing the classical univariate Cantelli’s bound over the scalarized inequalities. Three diverse applications to random matrices, tails of linear images of random vectors, and network homophily are also given.

Sharp inequalities for Zalcman functional of logarithmic coefficients of inverse functions in certain classes of analytic functions

Sharp inequalities for Zalcman functional of logarithmic coefficients of inverse functions in certain classes of analytic functions

Sharp bounds for Toader mean by arithmetic and centroidal means in exponential type with applications

We establish the optimal lower and upper bounds for the Toader mean by arithmetic and centroidal means in exponential type. That is, for any fixed p ? R, we find the best possible parameters ? = ?(p) and ? = ?(p) such that ?p(a, b; ?) ? T(a, b) ? ?p(a, b; ?) are valid for all a, b > 0 with a ?= b, where T(a, b) is the Toader mean of a and b, and ?p(a, b; ?) is the exponential type mean of a and b with ? ? (0, 1). As applications, several new sharp inequalities for the complete elliptic integral of the second kind will be derived.

Midpoint-type integral inequalities for (s, m)-convex functions in the third sense involving Caputo fractional derivatives and Caputo–Fabrizio integrals

In this study, midpoint-type integral inequalities for ( s , m ) -convex function in the third sense, involving Caputo fractional derivatives and Caputo–Fabrizio integral operators, are demonstrated. Inequalities, containing Caputo–Fabrizio integral operators for functions, whose derivatives are ( s , m ) -convex function in the third sense, are also established. This article utilized ( s , m ) -convex function in the third sense to generalized midpoint-type integral inequalities which give more sharp inequalities.

A Minkowski type inequality in warped cylinders

We prove a Minkowski type inequality for weakly mean convex and star-shaped hypersurfaces in warped cylinders which are asymptotically flat or hyperbolic. In particular, we show that this sharp inequality holds for outward minimizing hypersurfaces in the Schwarzschild manifold or the hyperbolic space using the weak solution of the inverse mean curvature flow.

Several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals

Abstract In the paper, in view of two monotonicity rules for the ratios of two functions and of two Maclaurin power series expansions, the authors establish several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals.

Best approximations for the weighted combination of the Cauchy-Szegö kernel and its derivative in the mean

In this paper, we study an extremal problem involving best approximation in the Hardy space $H^1$ on the unit disk $\mathbb D$. Specifically, we consider weighted combinations of the Cauchy-Szegö kernel and its derivative, parameterized by an inner funtion $\varphi$ and a complex number $\lambda$, and provide explicit formulas for the best approximation $e_{\varphi,z}(\lambda)$ by the subspace $H^1_0$. We also describe the extremal functions associated with this approximation. Our main result gives the form of $e_{\varphi,z}(\lambda)$ as a function of $\lambda$ and shows that, for a sufficiently large module of $\lambda$, the extremal function is linear in $\lambda$ and unique. We apply this result to establish a sharp inequality for holomorphic functions in the unit disk, leading to a new version of the Schwarz-Pick inequality.