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In this paper, we prove a logarithmic Sobolev inequality for closed submanifolds with constant length of mean curvature vector in a manifold with nonnegative sectional curvature. [ABSTRACT FROM AUTHOR]

We establish a framework for the existence and uniqueness of solutions to stochastic nonlinear (possibly multi-valued) diffusion equations driven by multiplicative noise, with the drift operator L being the generator of a transient Dirichlet form on a finite measure space (E , B , μ) and the initial value in F e ∗ , which is the dual space of an extended transient Dirichlet space. L and F e ∗ replace the Laplace operator Δ and H - 1 , respectively, in the classical case. This framework includes stochastic fast diffusion equations, stochastic fractional fast diffusion equations, the Zhang model, and applies to cases with E being a manifold, a fractal, or a graph. In addition, our results apply to operators - f (- L) , where f is a Bernstein function, e.g., f (λ) = λ α or f (λ) = (λ + 1) α - 1 , 0 < α < 1 . [ABSTRACT FROM AUTHOR]

Motivated by the notion of the Hyers–Ulam stability, we will first study the stability problem of a certain class of functional inequalities associated with Jordan–von Neumann type additive functional equation in quasi-Banach spaces. The second part of this research is devoted to investigate the approximation of generalized derivations satisfying some specific functional inequalities in quasi-Banach algebras. [ABSTRACT FROM AUTHOR]

In the present note we are interested in proving the counterpart of the (right-hand side of the) celebrated Hermite–Hadamard inequality for φ -convex functions. In particular, we prove that the only φ -convex function for which the Hermite–Hadamard inequality holds with the Lagrangian mean on the right-hand side is (up to an affine transformation) the log -convex function. [ABSTRACT FROM AUTHOR]

We define the quantum p-divergence and introduce Beckner’s inequalities for primitive quantum Markov semigroups on a finite-dimensional matrix algebra satisfying the detailed balance condition. Such inequalities quantify the convergence rate of the quantum dynamics in the noncommutative L p -norm. We obtain a number of implications between Beckner’s inequalities and other quantum functional inequalities, as well as the hypercontractivity. In particular, we show that quantum Beckner’s inequalities interpolate between Sobolev-type inequalities and Poincaré inequality in a sharp way. We provide a uniform lower bound for the Beckner constant α p in terms of the spectral gap and establish the stability of α p with respect to the invariant state. As applications, we compute the Beckner constant for the depolarizing semigroup and discuss the mixing time. For symmetric quantum Markov semigroups, we derive the moment estimate, which further implies a concentration inequality. We introduce a new class of quantum transport distances W 2 , p interpolating the quantum 2-Wasserstein distance by Carlen and Maas (J Funct Anal 273(5):1810–1869, 2017) and a noncommutative H ˙ - 1 Sobolev distance. We show that the quantum Markov semigroup with σ -GNS detailed balance is the gradient flow of a quantum p-divergence with respect to the metric W 2 , p . We prove that the set of quantum states equipped with W 2 , p is a complete geodesic space. We then consider the associated entropic Ricci curvature lower bound via the geodesic convexity of p-divergence, and obtain an HWI-type interpolation inequality. This enables us to prove that the positive Ricci curvature implies the quantum Beckner’s inequality, from which a transport cost and Poincaré inequalities can follow. [ABSTRACT FROM AUTHOR]

We consider generalised Mehler semigroups and, assuming the existence of an associated invariant measure σ , we prove functional integral inequalities with respect to σ , such as logarithmic Sobolev and Poincaré type. Consequently, some integrability properties of exponential functions with respect to σ are deduced. [ABSTRACT FROM AUTHOR]

We show that certain inequalities involving differences of the Bernstein basis polynomials and values of a function f ∈ C [ 0 , 1 ] , which is twice differentiable in [0, 1], imply that the function is q-monotone. This provides a partial answer to an open problem of the authors in a recent paper. [ABSTRACT FROM AUTHOR]

We study the Dyson–Ornstein–Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, relative entropy, and transportation cost. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes abruptly at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein–Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions. [ABSTRACT FROM AUTHOR]

A correction to article "An Extension of Raşa's Conjecture to q-Monotone Functions" published in a previous issue of the periodical is presented.

We show that certain inequalities involving differences of the Bernstein basis polynomials and values of a function f ∈ C [ 0 , 1 ] , imply that the function is q-monotone. In view of previous results of the authors (see the list of references), the current results provide, among others, a characterization of q-monotone functions in C[0, 1]. [ABSTRACT FROM AUTHOR]

Three functional inequalities are shown to uniquely characterize the exponential function. Each of the three inequalities is indispensable in the sense that no two of the three suffice. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R n , n ≥ 2 , be a bounded, connected, open set with Lipschitz boundary. Let F be a suitable norm in R n and let Δ F u = div F ξ (∇ u) F (∇ u) be the so-called Finsler Laplacian, with u ∈ H 1 (Ω) . In this paper, we prove two inequalities for λ F (β , Ω) , the first eigenvalue of Δ F with Robin boundary conditions involving a positive function β (x) . As a consequence of our result, we obtain the asymptotic behavior of λ F (β , Ω) when β is a positive constant which goes to zero [ABSTRACT FROM AUTHOR]

In recent years, it has been shown that some classical inequalities follow from a local stochastic dominance for naturally associated random polytopes. We strengthen planar isoperimetric inequalities by attaching a stochastic model to some classical inequalities, such as Mahler's Theorem, and a reverse Lutwak–Zhang inequality, the polar for L p centroid bodies. In particular, we obtain the dual counterpart to a result of Bisztriczky–Böröczky. [ABSTRACT FROM AUTHOR]

In this paper, we have established some new integral identities connected with the left-hand side of Hermite–Hadamard inequality. By using this identity, we have obtained some new bounds for functions whose derivatives in absolute values are s-convex. [ABSTRACT FROM AUTHOR]

In this paper, we prove multilevel concentration inequalities for bounded functionals f = f (X 1 , ... , X n) of random variables X 1 , ... , X n that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of k-tensors of higher order differences of f. We provide applications for both dependent and independent random variables. This includes deviation inequalities for empirical processes f (X) = sup g ∈ F | g (X) | and suprema of homogeneous chaos in bounded random variables in the Banach space case f (X) = sup t ‖ ∑ i 1 ≠ ... ≠ i d t i 1 ... i d X i 1 ⋯ X i d ‖ B . The latter application is comparable to earlier results of Boucheron, Bousquet, Lugosi, and Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for U-statistics with bounded kernels h and for the number of triangles in an exponential random graph model. [ABSTRACT FROM AUTHOR]

We derive sufficient conditions for non-emptiness of the efficient sets for stochastic dominance relations, usually employed in economics and finance. We do so via the concept of stochastic spanning and its characterization by a saddle-type property. Under the appropriate framework, sufficiency takes the form of semicontinuity of a related functional. In some cases, this boils down to weak continuity of the parameterization of the underlying set of probability distributions. [ABSTRACT FROM AUTHOR]

A new proof is given for the fact that centered Gaussian functions saturate the Euclidean forward–reverse Brascamp–Lieb inequalities, extending the Brascamp–Lieb and Barthe theorems. A duality principle for best constants is also developed, which generalizes the fact that the best constants in the Brascamp–Lieb and Barthe inequalities are equal. Finally, as the title hints, the main results concerning finiteness, structure, and Gaussian-extremizability for the Brascamp–Lieb inequality due to Bennett, Carbery, Christ, and Tao are generalized to the setting of the forward–reverse Brascamp–Lieb inequality. [ABSTRACT FROM AUTHOR]

We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry–Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein–Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than N - 3 . For the purely harmonic chain the order of the rate is in [ N - 3 , N - 1 ] . This shows that, in this set up, the spectral gap decays at most polynomially with N. [ABSTRACT FROM AUTHOR]

This note is concerned with an extension, at second order, of an inequality on the discrete cube C n = { - 1 , 1 } n (equipped with the uniform measure) due to Talagrand (Ann Probab 22:1576–1587, 1994). As an application, the main result of this note is a theorem in the spirit of a famous result from Kahn et al. (cf. Proceedings of 29th Annual Symposium on Foundations of Computer Science, vol 62. Computer Society Press, Washington, pp 68–80, 1988) concerning the influence of Boolean functions. The notion of the influence of a couple of coordinates (i , j) ∈ { 1 , ... , n } 2 is introduced in Sect. 2, and the following alternative is obtained: For any Boolean function f : C n → { 0 , 1 } , either there exists a coordinate with influence at least of order (1 / n) 1 / (1 + η) , with 0 < η < 1 (independent of f and n), or there exists a couple of coordinates (i , j) ∈ { 1 , ... , n } 2 with i ≠ j , with influence at least of order (log n / n) 2 . In Sect. 4, it is shown that this extension of Talagrand inequality can also be obtained, with minor modifications, for the standard Gaussian measure γ n on R n ; the obtained inequality can be of independent interest. The arguments rely on interpolation methods by semigroup together with hypercontractive estimates. At the end of the article, some related open questions are presented. [ABSTRACT FROM AUTHOR]

In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan R-function) R(a), by showing some monotonicity, concavity and convexity properties of certain combinations defined in terms of R(a), sin (π a) and polynomials. Some properties of the Riemann zeta function and its related special sums are presented, too. [ABSTRACT FROM AUTHOR]

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional C ∗ -algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates. [ABSTRACT FROM AUTHOR]

Let Fα,β(x) = βEβ(xβ) - αEα(xα), where Eα denotes the Mittag-Lefler function. We prove that if α, β ∈ (0, 1], then Fα,β is completely monotonic on (0,∞) if and only if α ≤ β. This extends a result of T. Simon, who proved in 2015 that Fα,1 is completely monotonic on (0,∞) if α ∈ (0, 1]. Moreover, we apply our monotonicity theorem to obtain some functional inequalities involving Fα,β. [ABSTRACT FROM AUTHOR]

We will prove a reverse Rogers-Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume estimates for polars of ℓp-diferences of convex bodies under the condition that their polar bodies have opposite barycenters. [ABSTRACT FROM AUTHOR]

The generalized modulus has numerous applications in geometric function theory and analytic number theory. In this paper, we study the monotonicity and convexity of the generalized modulus and obtain sharp functional inequalities for this function. Our results are the extensions of some well-known inequalities of the modulus of the Grötzsch ring. [ABSTRACT FROM AUTHOR]

For $x\in R$, the error function $\operatorname{erf}(x)$ is defined as In this paper, we answer the question: what are the greatest value p and the least value q, such that the double inequality $\operatorname {erf}(M_{p}(x,y;\lambda))\leq G(\operatorname{erf}(x),\operatorname {erf}(y);\lambda)\leq\operatorname{erf}(M_{q}(x,y;\lambda))$ holds for all $x,y\geq1$ (or $0< x,y<1$) and $\lambda\in(0,1)$? Here, $M_{r}(x,y;\lambda)=(\lambda x^{r}+(1-\lambda)y^{r})^{1/r}$ ( $r\neq0$), $M_{0}(x,y;\lambda)=x^{\lambda}y^{1-\lambda}$ and $G(x,y;\lambda )=x^{\lambda}y^{1-\lambda}$ are the weighted power and the weighted geometric mean, respectively. [ABSTRACT FROM AUTHOR]

In this paper, some mathematical inequalities of Hölder type are established. Applications for some operator inequalities as well as for functional inequalities in convex analysis are provided as well. [ABSTRACT FROM AUTHOR]

In this paper, some functional versions for operator entropies as well as for their related operator inequalities are investigated. The theoretical results obtained by our functional approach immediately imply those of operator version in a simple and nice way. [ABSTRACT FROM AUTHOR]

Let $$V\in C^2(\mathbb{R }^d)$$ such that $$\mu _V(\text{ d }x):= \text{ e }^{-V(x)}\,\text{ d }x$$ is a probability measure, and let $$\alpha \in (0,2)$$ . Explicit criteria are presented for the $$\alpha $$ -stable-like Dirichlet form to satisfy Poincaré-type (i.e., Poincaré, weak Poincaré and super Poincaré) inequalities. As applications, sharp functional inequalities are derived for the Dirichlet form with $$V$$ having some typical growths. Finally, the main result of [] on the Poincaré inequality is strengthened. [ABSTRACT FROM AUTHOR]

In this paper, we find the least value r and the greatest value p such that the double inequality erf(Mp(x, y;λ)) ≤ H(erf(x), erf(y);λ) ≤ erf(Mr(x, y;λ)) holds for all x, y ≥ 1 (or 0 < x, y < 1) with 0 < λ < 1, where erf(x) = 2/√π ∫x0 e-t2 dt, and Mp(x, y;λ) = (λxp + (1 – λ)yp)1/p (p ≠ 0) and M0(x, y;λ) = xλy1-λ are, respectively, the error function, and weighted power mean. [ABSTRACT FROM AUTHOR]

We propose variants of non-asymptotic dual transcriptions for the functional inequality of the form $$ f + g + k\circ H \ge h$$ . The main tool we used consists in purely algebraic formulas on the epigraph of the Legendre-Fenchel transform of the function $$ f + g + k\circ H$$ that are satisfied in various favorable circumstances. The results are then applied to the contexts of alternative type theorems involving composite and DC functions. The results cover several Farkas-type results for convex or DC systems and are general enough to face with unpublished situations. As applications of these results, nonconvex optimization problems with composite functions, convex composite problems with conic constraints are examined at the end of the paper. There, strong duality, stable strong duality results for these classes of problems are established. Farkas-type results and stable form of these results for the corresponding systems involving composite functions are derived as well. [ABSTRACT FROM AUTHOR]

In this paper we consider a sum of modified Bessel functions of the first kind of which particular case is used in the study of Kanter's sharp modified Bessel function bound for concentrations of some sums of independent symmetric random vectors. We present some monotonicity and convexity properties for that sum of modified Bessel functions of the first kind, as well as some Turán type inequalities, lower and upper bounds. Moreover, we point out an error in Kanter's paper (J Multivariate Anal 6:222-236, ). [ABSTRACT FROM AUTHOR]

In Avantaggiati and Loreti [Ric. Mat. 57(2):171-202, ] we studied the Cauchy problem for a class of Hamilton-Jacobi equations with initial data verifying the Lipschitz condition. In this paper we extend those results to the case in which the initial data are lower semicontinuous [in the following lsc], and are lower bounded and semiconvex. Here we prove hypercontractivity results and new Logarithm Sobolev Inequalities (shortly, LSI). [ABSTRACT FROM AUTHOR]

We study approximations of evolving probability measures by an interacting particle system. The particle system dynamics is a combination of independent Markov chain moves and importance sampling/resampling steps. Under global regularity conditions, we derive non-asymptotic error bounds for the particle system approximation. In a few simple examples, including high dimensional product measures, bounds with explicit constants of feasible size are obtained. Our main motivation are applications to sequential MCMC methods for Monte Carlo integral estimation. [ABSTRACT FROM AUTHOR]

In this paper, we study a subelliptic heat kernel on the Lie group SL(2, ℝ) and on its universal covering $\widetilde{\mathbf{SL}(2,\mathbb{R})}$. The subelliptic structure on SL(2,ℝ) comes from the fibration SO(2)→ SL(2,ℝ) → H and it can be lifted to $\widetilde{\mathbf{SL}(2,\mathbb{R})}$. First, we derive an integral representation for these heat kernels. These expressions allow us to obtain some asymptotics in small times of the heat kernels and give us a way to compute the subriemannian distance. Then, we establish some gradient estimates and some functional inequalities like a Li-Yau type estimate and a reverse Poincaré inequality that are valid for both heat kernels. [ABSTRACT FROM AUTHOR]

We present various inequalities for the error function. One of our theorems states: Let α ≥ 1. For all x, y > 0 we have with the best possible bounds [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of a Borell-Brascamp-Lieb inequality for metric measure spaces ( M,, m) denoted by ( K, N) for two numbers K, N ∈ ℝ with N ≥ 1. In the first part we prove that ( K, N) holds true on metric measure spaces satisfying a curvature-dimension condition ( K, N) developed and studied by Lott and Villani in (Ann Math 169:903–991, ) as well as by Sturm in (Acta Math 196(1):133–177, ). The aim of the second part is to show that ( K, N) is stable under convergence of metric measure spaces with respect to the 2-transportation distance. [ABSTRACT FROM AUTHOR]

Summary. Let erf be the error function. We determine the best possible constants $ \alpha_n $ and $ \beta_n $ such that the functional inequalities $ \alpha_n \, \text{\rm erf}\Bigl(\sum_{i=1}^{n}x_i\Bigr) \leq{\sum_{i=1}^{n}\text{\rm erf}(x_i) -\prod_{i=1}^{n}\text{\rm erf}(x_i)}\leq{\beta_n \, \text{\rm erf}\Bigl(\sum_{i=1}^{n}x_i\Bigr)} $ hold for all $ x{_i} \geq 0 (i = 1,..., n) $. Furthermore, we determine the sharp constants $ \alpha_n $ and $ \beta_n $ under the assumption that $ x{_i} \leq 0 (i = 1,..., n) $. Our results extend and complement an inequality due to Mitrinović-Esseen. [ABSTRACT FROM AUTHOR]

For a group (G,·) and a real or complex inner product space (E, 〈·,·〉) with norm ... . ... we consider the functional inequality f : G → E, ...2f(x) + 2f(y) - f(xy-1)... ≤ ...f(xy)... (∀x, y ∈ G) (I) and describe situations in which (I) implies the Jordan-von Neumann parallelogram equation f : G → E, 2f(x) + 2f(y) = f(xy) + f(xy-1) (∀x, y ∈ G). (JvN) [ABSTRACT FROM AUTHOR]

Functional version for the so-called Furuta parametric relative operator entropy is here investigated. Some related functional inequalities are also discussed. The theoretical results obtained by our functional approach immediately imply those of operator versions in a simple, fast, and nice way. [ABSTRACT FROM AUTHOR]