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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]

Motivated by the recent investigations of several authors, the main aim of this article is to derive several functional inequalities for a class of functions related to the incomplete Fox-Wright functions that were introduced and studied recently. Moreover, new functional bounds for functions related to the Fox-Wright function are deduced. Furthermore, a class of completely monotonic functions related to the Fox-Wright function is given. The main mathematical tools used to obtain some of the main results are the monotonicity patterns and the Mellin transform for certain functions related to the two-parameter Mittag-Leffler function. Several potential applications for this incomplete special function are mentioned.

Motivated by the recent investigations of several authors, the main aim of this article is to derive several functional inequalities for a class of functions related to the incomplete Fox-Wright functions that were introduced and studied recently. Moreover, new functional bounds for functions related to the Fox-Wright function are deduced. Furthermore, a class of completely monotonic functions related to the Fox-Wright function is given. The main mathematical tools used to obtain some of the main results are the monotonicity patterns and the Mellin transform for certain functions related to the two-parameter Mittag-Leffler function. Several potential applications for this incomplete special function are mentioned. [ABSTRACT FROM AUTHOR]

Coarse-graining techniques play a central role in reducing the complexity of stochastic models and are typically characterized by a mapping which projects the full state of the system onto a smaller set of variables which captures the essential features of the system. Starting with a continuous-time Markov chain, in this work we propose and analyze an effective dynamics, which approximates the dynamical information in the coarse-grained chain. Without assuming explicit scale-separation, we provide sufficient conditions under which this effective dynamics stays close to the original system and provide quantitative bounds on the approximation error. We also compare the effective dynamics and corresponding error bounds to the averaging literature on Markov chains which involve explicit scale-separation. We demonstrate our findings on an illustrative test example. [ABSTRACT FROM AUTHOR]

We prove two new functional inequalities of the forms JG \varphi(\psi -- \psi) < a JG \psi ln(ψ-ψ} Jg 1/a and fG \ψ < ψ/ψ fG llikp)² f°r anY finitely connected, bounded C2-domain G C R2, a constant ßo > 0, any a > 0, and sufficiently regular functions p, ß>. We then illustrate their usefulness by proving long time stabilization and eventual smoothness properties for certain generalized solutions to the chemotaxis-Navier--Stokes system {nt + u ΔVn = Δn -- V • (nS'(æ,n,c)Vc);Δt + u Δc = Δc -- nf(c);ut + (« • Δ)u = Δu + ΔP + ΔΔo. V • u = 0} on a smooth, bounded, convex domain Q C R² with no-flux boundary conditions for n and c as well as a Dirichlet boundary condition for u. We further allow for a general chemotactic sensitivity S attaining values in R2x2 as opposed to a scalar one. [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 study the convergence rate of a continuous-time simulated annealing process $ (X_t; \, t \ge 0) $ for approximating the global optimum of a given function $ f $. We prove that the tail probability $ \mathbb{P}(f(X_t) > \min f +\delta) $ decays polynomial in time with an appropriately chosen cooling schedule of temperature, and provide an explicit convergence rate through a non-asymptotic bound. Our argument applies recent development of the Eyring-Kramers law on functional inequalities for the Gibbs measure at low temperatures. [ABSTRACT FROM AUTHOR]

Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell et al. and the Schramm–Steif inequality for the Fourier‐Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm–Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions—expressed in terms of randomized stopping sets—yielding sharp phase transitions, quantitative noise sensitivity, exceptional times and bounds on critical windows for monotonic Boolean Poisson functions. Our analysis is based on a new general definition of "stopping set", not requiring any topological property for the underlying measurable space, as well as on the new concept of a "continuous‐time decision tree", for which we establish several fundamental properties. We apply our findings to the k$$ k $$‐percolation of the Poisson Boolean model and to the Poisson‐based confetti percolation with bounded random grains. In these two models, we reduce the proof of sharp phase transitions for percolation, and of noise sensitivity for crossing events, to the construction of suitable randomized stopping sets and the computation of one‐arm probabilities at criticality. This enables us to settle an open problem suggested by Ahlberg et al. (a special case of which was conjectured earlier by Ahlberg et al. on noise sensitivity of crossing events for the planar Poisson Boolean model with random balls whose radius distribution has finite (2+α)$$ \left(2+\alpha \right) $$‐moments and also show the same for planar confetti percolation model with bounded random balls. We also prove that critical probability is 1/2$$ 1/2 $$ for the planar confetti percolation model with bounded, π/2$$ \pi /2 $$‐rotation invariant and reflection invariant random grains. Such a result was conjectured by Benjamini and Schramm in the case of fixed balls and proved by Müller, Hirsch and Ghosh and Roy in the case of balls, boxes and random boxes, respectively; our results contain all previous findings as special cases. [ABSTRACT FROM AUTHOR]

We are interested in sharp functional inequalities for the coherent state transform related to the Wehrl conjecture and its generalizations. This conjecture was settled by Lieb in the case of the Heisenberg group, Lieb and Solovej for SU(2), and Kulikov for SU(1, 1) and the affine group. In this article, we give alternative proofs and characterize, for the first time, the optimizers in the general case. We also extend the recent Faber-Krahn-type inequality for Heisenberg coherent states, due to Nicola and Tilli, to the SU(2) and SU(1, 1) cases. Finally, we prove a family of reverse Hölder inequalities for polynomials, conjectured by Bodmann.

Functional inequalities involving special functions are very useful in mathematical analysis, and several interesting results have been obtained in this topic. Several methods have been used by many authors in order to derive upper or lower bounds of certain special functions. In this paper, we establish some general integral inequalities involving strictly monotone functions. Next, some special cases are discussed. In particular, several estimates of trigonometric and hyperbolic functions are deduced. For instance, we show that Mitrinović-Adamović inequality, Lazarevic inequality, and Cusa-Huygens inequality are special cases of our obtained results. Moreover, an application to integral equations is provided. [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]

In this study, the generalized power mean inequalities with a negative parameter are refined using an optimality theorem on the generator function. The optimality theorem requires the study of different cases for the exponents and yields a refinement of the inequality in a neighbourhood of the vectors for which the equality occurs. Then, these local inequalities are generalized to all positive vectors by an appropriate selection of parameters. Also, some of the results are exemplified by numerical calculations. [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]

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In this paper we extend complex uniform convexity estimates for C to Rn and determine best constants. Furthermore, we provide the link to log-Sobolev inequalities and hypercontractivity estimates for ultraspherical measures. [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]

The purpose of this paper is to introduce the logarithmic mean of two convex functionals that extends the logarithmic mean of two positive operators. Some inequalities involving this functional mean are discussed as well. The operator versions of the functional theoretical results obtained here are immediately deduced without referring to the theory of operator means.

Some differentiability results from the paper of D.Ş. Marinescu & M. Monea [7] on delta-convex mappings, obtained for real functions, are extended for mappings with values in a normed linear space. In this way, we are nearing the completion of studies established in papers [2], [5] and [7].

Functional inequalities involving special functions are very useful in mathematical analysis, and several interesting results have been obtained in this topic. Several methods have been used by many authors in order to derive upper or lower bounds of certain special functions. In this paper, we establish some general integral inequalities involving strictly monotone functions. Next, some special cases are discussed. In particular, several estimates of trigonometric and hyperbolic functions are deduced. For instance, we show that Mitrinović-Adamović inequality, Lazarevic inequality, and Cusa-Huygens inequality are special cases of our obtained results. Moreover, an application to integral equations is provided.

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]

We revisit the variational characterization of conservative diffusion as entropic gradient flow and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for diffusions of Langevin--Smoluchowski type, the Fokker--Planck probability density flow maximizes the rate of relative entropy dissipation, as measured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in the backwards direction of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the maximal rate of entropy dissipation along the Fokker--Planck flow and measure exactly the deviation from this maximum that corresponds to any given perturbation. A bonus of our trajectorial approach is that it derives the HWI inequality relating relative entropy (H), Wasserstein distance (W), and relative Fisher information (I). [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]

This paper generalizes and unifies different recent results as well as provides a new methodology concerning vector optimization problems involving composite mappings in locally convex Hausdorff topological vector spaces. The Lagrangian and weak Lagrangian dual problems are proposed. Characterizations of strong duality results are proved at the same time with characterizations of Farkas lemmas for composite vector mappings in a general setting (i.e. without any assumptions on convexity or continuity of the mappings involved). Corresponding results in the convex setting are also proposed by establishing as main tools some variants of representations of epigraphs of conjugate mappings in our composite vector framework. As by-products, several Farkas-type results for composite vector functions are proposed, which extend and cover several known ones recently appeared in the literature. Lastly, the results are applied to get duality results for a class of convex semi-vector bilevel optimization problems. [ABSTRACT FROM AUTHOR]

Report from the conference.

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 revisit the Wyner-Ahlswede-Körner network, focusing especially on the converse part of the dispersion analysis, which is known to be challenging. Using the functional-entropic duality and the reverse hypercontractivity of the transposition semigroup, we lower bound the error probability for each joint type. Then by averaging the error probability over types, we lower bound the c-dispersion (which characterizes the second-order behavior of the weighted sum of the rates of the two compressors when a nonvanishing error probability is small) as the variance of the gradient of $\inf _{ {P}_{ {U}| {X}}}\{{ { cH}}({Y}| {U})+ {I}({U}; {X})\}$ with respect to $ {Q}_{{{ XY}}}$ , the per-letter side information and source distribution. In comparison, using standard achievability arguments based on the method of types, we upper-bound the c-dispersion as the variance of $ {c}\imath _{ {Y}| {U}}({Y}| {U})+\imath _{ {U}; {X}}({U}; {X})$ , which improves the existing upper bounds but has a gap to the aforementioned lower bound. Our converse analysis should be immediately extendable to other distributed source-type problems, such as distributed source coding, common randomness generation, and hypothesis testing with communication constraints. We further present improved bounds for the general image-size problem via our semigroup technique. [ABSTRACT FROM AUTHOR]

Abstract 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.

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]

We provide an estimation of the dissipation of the Wasserstein 2 distance between the law of some interacting -particle system, and the times tensorized product of the solution to the corresponding limit nonlinear conservation law. It then enables to recover classical propagation of chaos results [20] in the case of Lipschitz coefficients, uniform in time propagation of chaos in [17] in the case of strictly convex coefficients. And some recent results [7] as the case of particle in a double well potential. [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 the behavior of various Lyapunov functionals (relative entropies) along the solutions of a family of nonlinear drift-diffusion-reaction equations coming from statistical mechanics and population dynamics. These equations can be viewed as gradient flows over the space of Radon measures equipped with the Hellinger-Kantorovich distance. The driving functionals of the gradient flows are not assumed to be geodesically convex or semi-convex. We prove new isoperimetric-type functional inequalities, allowing us to control the relative entropies by their productions, which yields the exponential decay of the relative entropies. [ABSTRACT FROM AUTHOR]