Sharp Inequalities Research Articles

Coefficient functionals for a class of bounded turning functions connected to three leaf function

In this article, we define the class of bounded turning functions connected with three leaf function to investigate results for the estimates of four initial coefficients, Fekete-Szegö functional, the second-order Hankel determinant and Zalcman conjecture and these results are shown to be sharp. Furthermore, we estimate the bounds of the third-order Hankel determinants for this class and for its 2-fold and 3-fold symmetric functions. Finally we evaluate the sharp Krushkal inequality for the functions in this class.

Logarithm Sobolev and Shannon’s Inequalities Associated with the Deformed Fourier Transform and Applications

By using the symmetry of the Dunkl Laplacian operator, we prove a sharp Shannon-type inequality and a logarithmic Sobolev inequality for the Dunkl transform. Combining these inequalities, we obtain a new, short proof for Heisenberg-type uncertainty principles in the Dunkl setting. Moreover, by combining Nash’s inequality, Carlson’s inequality and Sobolev’s embedding theorems for the Dunkl transform, we prove new uncertainty inequalities involving the L∞-norm. Finally, we obtain a logarithmic Sobolev inequality in Lp-spaces, from which we derive an Lp-Heisenberg-type uncertainty inequality and an Lp-Nash-type inequality for the Dunkl transform.

Sharp concentration inequalities for the centred relative entropy

Abstract We study the relative entropy between the empirical estimate of a discrete distribution and the true underlying distribution. If the minimum value of the probability mass function exceeds an $\alpha> 0$ (i.e. when the true underlying distribution is bounded sufficiently away from the boundary of the simplex), we prove an upper bound on the moment generating function of the centred relative entropy that matches (up to logarithmic factors in the alphabet size and $\alpha $) the optimal asymptotic rates, subsequently leading to a sharp concentration inequality for the centred relative entropy. As a corollary of this result we also obtain confidence intervals and moment bounds for the centred relative entropy that are sharp up to logarithmic factors in the alphabet size and $\alpha $.

Sharp multiplicative inequalities with BMO II

We find the best possible constant C in the inequality‖φ‖Lrpr⩽C‖φ‖Lppr‖φ‖BMO1−pr for all values of parameters p and r such that 1⩽p<r<+∞. To solve the problem, we define and explicitly compute the corresponding Bellman function. This function depends on three variables and has a rather complicated structure. We first solve the problem on an interval and then transfer our results to the circle and the line. We also obtain explicit estimates in several higher-dimensional settings.

Sharp Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on the Siegel domains and complex hyperbolic spaces

The aim of this paper is to establish higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on Siegel domains and complex hyperbolic spaces using the method of Helgason-Fourier analysis on the complex hyperbolic spaces. Firstly, we give a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller's operator, as well as the CR invariant differential operators on the Heisenberg group and CR sphere. (See Theorem 1.3.) Secondly, by using, among other things, the Kunze-Stein phenomenon on a closed linear group SU(1,n) and Helgason-Fourier analysis techniques on the complex hyperbolic spaces, we establish the Poincaré-Sobolev, Hardy-Sobolev-Maz'ya inequality on the Siegel domain Un and the unit ball BCn. (See Theorems 1.4, 1.5.) Finally, we establish the sharp Hardy-Adams inequalities and sharp Adams type inequalities on Sobolev spaces of any positive fractional order on the complex hyperbolic spaces. (See Theorems 1.9, 1.10 and 1.12.) The factorization theorem we prove on the complex hyperbolic spaces is considerably more difficult than in the case of real hyperbolic spaces and should be of its independent interest in the analysis on Heisenberg group and CR sphere and CR invariant differential operators therein.

Sobolev Inequalities in Manifolds with Nonnegative Curvature

AbstractWe prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael‐Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic volume ratio of the ambient manifold. © 2022 Wiley Periodicals LLC.

On a sharp inequality relating Yamabe invariants on a Poincare-Einstein manifold

For a Poincare-Einstein manifold under certain restrictions, X. Chen, M. Lai and F. Wang [Adv. Math. 343 (2019), pp. 16–35] proved a sharp inequality relating Yamabe invariants. We show that the inequality is true without any restriction.

Bounds for the Neuman–Sándor Mean in Terms of the Arithmetic and Contra-Harmonic Means

In this paper, the authors provide several sharp upper and lower bounds for the Neuman–Sándor mean in terms of the arithmetic and contra-harmonic means, and present some new sharp inequalities involving hyperbolic sine function and hyperbolic cosine function.

The Landau–Kolmogorov problem on a finite interval in the Taikov case

We solve the pointwise Landau–Kolmogorov problem on the interval I=[−1,1] on finding |f(k)(t)|→sup under constraints ‖f‖2⩽1 and ‖f(n)‖2⩽σ, where t∈I and σ>0 are fixed. For n=1 and n=2, we solve the uniform version of the Landau–Kolmogorov problem on the interval I in the Taikov case by proving the Karlin-type conjecture supt∈I|f(k)(t)|=|f(k)(1)| under above constraints. The proof relies on the analysis of the dependence of the norm of the solution to higher-order Sturm–Liouville equation (−1)nu(2n)+λu=−λf with boundary conditions u(s)(−1)=u(s)(1)=0, s=0,1,…,n−1, on non-negative parameter λ, where f is some piece-wise polynomial function. Furthermore, we find sharp inequality ‖f(k)‖∞⩽A‖f‖2+B‖f(n)‖2 with the smallest possible constant A>0 and the smallest possible constant B=B(A) for k∈{n−2,n−1}.

Isoperimetric inequality in noncompact 𝖬𝖢𝖯 spaces

We prove a sharp isoperimetric inequality for the class of metric measure spaces verifying the synthetic Ricci curvature lower bounds Measure Contraction property ( M C P ( 0 , N ) \mathsf {MCP}(0,N) ) and having Euclidean volume growth at infinity. We avoid the classical use of the Brunn-Minkowski inequality, not available for M C P ( 0 , N ) \mathsf {MCP}(0,N) , and of the PDE approach, not available in the singular setting. Our approach will be carried over by using a scaling limit of localization.

Willmore surfaces and Hopf tori in homogeneous 3-manifolds

Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space {mathbb {E}}^{3}(kappa ,tau ) with isometry group of dimension 4 is defined and its first variational formula is computed. Then, we characterize Clifford and Hopf tori as the only Willmore surfaces satisfying a sharp Simons-type integral inequality. On the other hand, we also obtain some integral inequalities for closed surfaces with constant extrinsic curvature in {mathbb {E}}^3(kappa ,tau ), becoming equalities if and only if the surface is a Hopf torus in a Berger sphere.

Sharp Hardy's type inequality for Laguerre expansions

A method of proving Hardy's type inequality for orthogonal expansions is presented in a rather general setting. Then, sharp multi-dimensional Hardy's inequality associated with the Laguerre functions of convolution type is proved for the type index $\alpha \in [-1/2, \infty)^{d}$. The case of the standard Laguerre functions is also investigated. Moreover, the sharp analogues of Hardy's type inequality involving $L^{1}$ norms in place of $H^{1}$ norms are obtained in both settings.

Nonparametric, Tuning-Free Estimation of S-Shaped Functions

AbstractWe consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimization problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a projection framework that demonstrates the consistency and robustness to misspecification of the estimator, our main theoretical results provide sharp oracle inequalities that yield worst-case and adaptive risk bounds for the estimation of the regression function, as well as a rate of convergence for the estimation of the inflection point. These results reveal not only that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations and a real data application to air pollution modelling also confirm the desirable finite-sample properties of the estimator, and our algorithm is implemented in the R package Sshaped.

A note on the stochastic version of the Gronwall lemma

We prove a stochastic version of the Gronwall lemma assuming that the underlying martingale has a terminal random value in Lp , where The proof of the present result is mainly based on a sharp martingale inequality of the Doob-type.

Maximization of the second Laplacian eigenvalue on the sphere

We prove a sharp isoperimetric inequality for the second nonzero eigenvalue of the Laplacian on S m S^m . For S 2 S^{2} , the second nonzero eigenvalue becomes maximal as the surface degenerates to two disjoint spheres, by a result of Nadirashvili for which Petrides later gave another proof. For higher dimensional spheres, the analogous upper bound was conjectured by Girouard, Nadirashvili and Polterovich. Our method to confirm the conjecture builds on Petrides’ work and recent developments on the hyperbolic center of mass and provides also a simpler proof for S 2 S^2 .

L1 isoperimetric inequality on the unit disk

In this paper, we obtain a sharp isoperimetric inequality on the unit disk for L1 functions and classify all extremal functions. The original isoperimetric inequality was obtained by Carleman for analytic functions [3].

A Minkowski Inequality for Horowitz–Myers Geon

We prove a sharp inequality for toroidal hypersurfaces in three- and four-dimensional Horowitz–Myers geon. This extend previous results on Minkowski inequality in the static spacetime to toroidal surfaces in asymptotically hyperbolic manifold with flat toroidal conformal infinity.

Triangular Ratio Metric Under Quasiconformal Mappings in Sector Domains

The hyperbolic metric and different hyperbolic type metrics are studied in open sector domains of the complex plane. Several sharp inequalities are proven for them. Our main result describes the behavior of the triangular ratio metric under quasiconformal maps from one sector onto another one.

On approximation of hypersingular integral operators by bounded ones

We solve the Stechkin problem about the approximation of generally speaking unbounded hypersingular integral operators by bounded ones. As a part of the proof, we also solve several related and interesting on their own problems. In particular, we obtain sharp Landau-Kolmogorov type inequalities in both additive and multiplicative forms for hypersingular integral operators and prove a sharp Ostrowski type inequality for multivariate Sobolev classes. We also give some applications of the obtained results, in particular, study the modulus of continuity of the hypersingular integral operators, and solve the problem of optimal recovery of the value of a hypersingular integral operator based on the argument known with an error.

On the sharp lower bound for duality of modulus

We establish a sharp reciprocity inequality for modulus in compact metric spaces X X with finite Hausdorff measure. In particular, when X X is also homeomorphic to a planar rectangle, our result answers a question of K. Rajala and M. Romney [Ann. Acad. Sci. Fenn. Math. 44 (2019), pp. 681-692]. More specifically, we obtain a sharp inequality between the modulus of the family of curves connecting two disjoint continua E E and F F in X X and the modulus of the family of surfaces of finite Hausdorff measure that separate E E and F F . The paper also develops approximation techniques, which may be of independent interest.