In this paper we prove three power-exponential inequalities for positive real numbers. In particular, we conclude that this proofs give affirmatively answers to three, until now, open problems (conjectures~4.4, 2.1 and 2.2) posed by C{î}rtoaje in the following two works:
"{\it J. Inequal. Pure Appl. Math.} 10, Article 21, 2009" and "{\it J. Nonlinear Sci. Appl.} 4:2:130-137, 2011". Moreover, we present a new proof of the inequality $a^{ra}+b^{rb}\ge a^{rb}+b^{ra}$ for all positive real numbers $a$ and $b$ and $r\in [0,e]$. In addition, three new conjectures are presented.
We prove in this article the generalizations on the exponential Orlicz spaces Markov's - Bernstein's inequalities for algebraic polynomials and rational functions.
Nikolskii type inequalities for entire functions of exponential type on Rn for the Lorentz Zygmund spaces are obtained. Some new limiting cases are examined. Application to Besov type spaces of logarithmic smoothness is given.
The optimal constants in a class of exponential type inequalities for the Ornstein-Uhlenbeck operator in the Gauss space are detected. The existence of extremal functions in the relevant inequalities is also established. Our results disclose analogies and dissimilarities in comparison with Adams' inequality for the Laplace operator, a companion of our inequalities in the Euclidean space.
In this article, we present exponential-type inequalities for positive linear mappings and Hilbert space operators, by means of convexity and the Mond-Pe\v carić method. The obtained results refine and generalize some known results. As an application, we present extensions for operator-like geometric and harmonic means.
In this paper, we present a simple proof of a recent result of the second author which establishes that functional inverse-Santal{ó} inequalities follow from Entropy-Transport inequalities. Then, using transport arguments together with elementary correlation inequalities, we prove these sharp Entropy-Transport inequalities in dimension 1. We also revisit the proof of the functional inverse-Santal{ó} inequalities in the n dimensional unconditional case using these ideas.
In this paper, we derive a new proof on some sharp double integral inequalities of the Hermite-Hadamard type. Our approach is mainly based on well-known Taylor's theorem with the integral remainder.
The power $\frac{2m}{m+1}$ in the polynomial (and multilinear) Bohnenblust--Hille inequality is optimal. This result is well-known but its proof highly nontrivial. In this note we present a quite simple proof of this fact.
In a 2013 paper, the author showed that the convolution of a compactly supported measure on the real line with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). In a 2014 paper, the author gave bounds for the optimal constants in these LSIs. In this paper, we give a simpler, elementary proof of this result.
In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continous convex functions on a vector space ${\bf R}^m$ and vector-valued functions in a weakly compact subset of a Banach vector space generated by $m$ $L_μ^p$-spaces for $1\leq p<+\infty.$ Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by $m$ $L_μ^\infty$-spaces instead.
Let $d \ge 1$, $p \ge d$, and let $Ω$ be a smooth bounded open subset of $\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $Ω$ such that $$ \mathop{\int_Ω \int_Ω}_{|u(x) - u(y)| > δ} \frac{1}{|x-y|^{d+p}} \, dx \, dy < + \infty, $$ for some $δ> 0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincaré inequalities, and Hardy inequalities.
Decoupling inequalities disentangle complex dependence structures of random objects so that they can be analyzed by means of standard tools from the theory of independent random variables. We study decoupling inequalities for vector-valued homogeneous polynomials evaluated at random variables. We focus on providing geometric conditions ensuring decoupling inequalities with good constants depending only exponentially on the degree of the polynomial. Assuming the Banach space has finite cotype we achieve this for classical decoupling inequalities that compare the polynomials with their associate…
We present new proofs of two theorems of E.B. Davies and B. Simon about ultracontractivity property for of semigroups of operators and logarithmic Sobolev inequalities with parameter (LSIWP for short) satisfied by the generator of the semigroup. In our proof, we use neither the $L^p$ version of the $LSIWP$ nor Stein's interpolation. Our tool is Nash type inequality. We also present new results. In particular, we show that Nash type inequality and LSIWP are equivalent.
This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincaré inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequ…
In this paper we give alternate proofs of some well-known matrix inequalities. In particular, we show that under certain conditions the inequality holds \begin{align}\sum \limits_{λ_i\in \mathrm{Spec}(ab^{T})}\mathrm{min}\{\log |t-λ_i|\}_{[||a||,||b||]}&\leq \# \mathrm{Spec}(ab^T)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber \\&+\frac{1}{||b||-||a||}\sum \limits_{λ_i\in \mathrm{Spec}(ab^T)}\log \bigg(1-\frac{2λ_i}{||b||+||a||}\bigg).\nonumber \end{align}Also under the same condition, the inequality also holds\begin{align}\int \limits_{||a||}^{||b||}\log|\mathrm{det}(ab^{T}-tI)|dt&\leq \# \mat…
In this paper non-asymptotic exponential estimates are derived for the tail distribution of polynomial martingale differences in terms unconditional tails distributions of summands. Applications are considered in the theory of polynomials on independent random variables, to the theory of U-statistics, multiply martingale series and in the theory of weak compactness measures on the Banach spaces. Partially supported by the Israel Ministry of Absorbtion Mathematics Subject Classification (2000): 47A45, 47B10, 60F10, 60G42.
In this paper, we obtain a new type of inequalities for frames, which are parametrized by a parameter λ\in R . By suitable choices of λ, one obtains the previous results as special cases. Our new proof also makes the underlying mathematical structure that gives rise to these inequalities more transparent than previous approaches: Our proof shows that the main point is the splitting S = S1 + S2 of the positive denite frame operator S into the two positive semidenite operators S1 and S2 .
In this work, we deal with an iteration method for approximating a fixed point of a contraction mapping using the Mann's algorithm under functional random errors. We first show its almost complete convergence to the fixed point by mean of an exponential inequality and then we specify the induced rate of convergence. We finally build a confidence set for the fixed point.
By using Bernstein-type inequality we define analogs of spaces of entire functions of exponential type in $L_{p}(X), 1\leq p\leq \infty$, where $X$ is a symmetric space of non-compact. We give estimates of $L_{p}$-norms, $1\leq p\leq \infty$, of such functions (the Nikolskii-type inequalities) and also prove the $L_{p}$- Plancherel-Polya inequalities which imply that our functions of exponential type are uniquely determined by their inner products with certain countable sets of measures with compact supports and can be reconstructed from such sets of "measurements" in a stable way.
We give a direct proof of a functional Santalo inequality due to Fradelizi and Meyer. This provides a new proof of the Blaschke-Santalo inequality. The argument combines a logarithmic form of the Prekopa-Leindler inequality and a partition theorem of Yao and Yao.
We prove, using optimal transport tools, weighted Poincar'e inequalities for log-concave random vectors satisfying some centering conditions. We recover by this way similar results by Klartag and Barthe-Cordero-Erausquin for log-concave random vectors with symmetries. In addition, we prove that the variance conjecture is true for increments of log-concave martingales.
This note relies mainly on a refined version of the main results of the paper by F. Catrina and D. Costa (J. Differential Equations 2009). We provide very short and self-contained proofs. Our results are sharp and minimizers are obtained in suitable functional spaces. As main tools we use the so-called \textit{expand of squares} method to establish sharp weighted $L^{2}$-Caffarelli-Kohn-Nirenberg (CKN) inequalities and density arguments.
Frank and Lieb gave a new, rearrangement-free, proof of the sharp Hardy-Littlewood-Sobolev inequalities by exploiting their conformal covariance. Using this they gave new proofs of sharp Sobolev inequalities for the embeddings $W^{k,2}(\mathbb{R}^n)\hookrightarrow L^{\frac{2n}{n-2k}}(\mathbb{R}^n)$. We show that their argument gives a direct proof of the latter inequalities without passing through Hardy-Littlewood-Sobolev inequalities, and, moreover, a new proof of a sharp fully nonlinear Sobolev inequality involving the $σ_2$-curvature. Our argument relies on nice commutator identities deduce…
With direct and simple proofs, we establish Poincaré type inequalities (including Poincaré inequalities, weak Poincaré inequalities and super Poincaré inequalities), entropy inequalities and Beckner-type inequalities for non-local Dirichlet forms. The proofs are efficient for non-local Dirichlet forms with general jump kernel, and also work for $L^p (p>1)$ settings. Our results yield a new sufficient condition for fractional Poincaré inequalities, which were recently studied in \cite{MRS,Gre}. To our knowledge this is the first result providing entropy inequalities and Beckner-type inequalitie…
Our recent work on the Burchard-Choksi-Topaloglu flocking problem showed that in the large mass regime the ground state density profile is the characteristic function of some set. Here we show that this set is, in fact, a round ball. The essential mathematical structure needed in our proof is a strict rearrangement inequality with a quantitative error estimate, which we deduce from recent deep results of M. Christ.
In this paper we introduce the notion of sc-action functionals and their sc-gradient flow lines. Our approach is inspired by Floer's unregularized gradient flow. The main result of this paper is that under a Morse condition sc-gradient flow lines have uniform exponential decay towards critical points. The ultimate goal for the future is to construct a M-polyfold bundle over a M-polyfold such that the space of broken sc-gradient flow lines is the zero set of a appropriate sc-section. Here uniform exponential decay is essential. Of independent interest is that we derive exponential decay estimat…
In this paper, we first present simple proofs of Choi's results [4], then we give a short alternative proof for Fiedler and Markham's inequality [6]. We also obtain additional matrix inequalities related to partial determinants.
The use of Lyapunov conditions for proving functional inequalities was initiated in [5]. It was shown in [4, 30] that there is an equivalence between a Poincar{é} inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by $\bullet$ showing that strong functional inequalities are equivalent to Lyapunov type conditions; $\bullet$ showing that these Lyapunov conditions are characterized by the finiteness of generalized exponential momen…
In this paper, we offer a proof for a family of functional inequalities interpolating between the Poincar{é} and the logarithmic Sobolev (standard and weighted) inequalities. The proofs rely both on entropy flows and on a CD($ρ$, n) condition, either with $ρ$ = 0 and n > 0, or with $ρ$ > 0 and n $\in$ R. As such, results are valid in the case of a Riemannian manifold, which constitutes a generalization to what was proved in [BGS18, Ngu18].
We prove a local $L^p$-Poincaré inequality, $1\leq p < \infty$, on noncompact Lie groups endowed with a sub-Riemannian structure. We show that the constant involved grows at most exponentially with respect to the radius of the ball, and that if the group is nondoubling, then its growth is indeed, in general, exponential. We also prove a nonlocal $L^2$-Poincaré inequality with respect to suitable finite measures on the group.
