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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]

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]

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]

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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]

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.

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]

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 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]

In a companion article we have introduced a notion of multiscale functional inequalities for functions X(A) of an ergodic stationary random field A on the ambient space ℝd. These inequalities are multiscale weighted versions of standard Poincaré, covariance, and logarithmic Sobolev inequalities. They hold for all the examples of fields A arising in the modelling of heterogeneous materials in the applied sciences whereas their standard versions are much more restrictive. In this contribution we first investigate the link between multiscale functional inequalities and more standard decorrelation or mixing properties of random fields. Next, we show that multiscale functional inequalities imply fine concentration properties for nonlinear functions X(A). This constitutes the main stochastic ingredient to the quenched large-scale regularity theory for random elliptic operators by the second author, Neukamm, and Otto, and to the corresponding quantitative stochastic homogenization results. [ABSTRACT FROM AUTHOR]

We study a class of isoperimetric problems on R + N where the densities of the weighted volume and weighted perimeter are given by two different non-radial functions of the type | x | k x N α. Our results imply some sharp functional inequalities, like for instance, Caffarelli-Kohn-Nirenberg type inequalities. [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]

This work is focused on the study of the nonlinear elliptic higher order equation ( − Δ ) m u = S k [ − u ] + λ f , x ∈ R N , where the k -Hessian S k [ u ] is the k th elementary symmetric polynomial of eigenvalues of the Hessian matrix of the solution and the datum f belongs to a suitable functional space. This problem is posed in R N and we prove the existence of at least one solution by means of topological fixed point methods for suitable values of m ∈ N . Questions related to the regularity of the solutions and extensions of these results to the nonlocal setting are also addressed. On the way to construct these proofs, some technical results such as a fixed point theorem and a refinement of the critical Sobolev embedding, which could be of independent interest, are introduced. [ABSTRACT FROM AUTHOR]

For displacement convex functionals in the probability space equipped with the Monge–Kantorovich metric we prove the equivalence between the gradient and functional type Łojasiewicz inequalities. We also discuss the more general case of λ -convex functions and we provide a general convergence theorem for the corresponding gradient dynamics. Specialising our results to the Boltzmann entropy, we recover Otto–Villani's theorem asserting the equivalence between logarithmic Sobolev and Talagrand's inequalities. The choice of power-type entropies shows a new equivalence between Gagliardo–Nirenberg inequality and a nonlinear Talagrand inequality. Some nonconvex results and other types of equivalences are discussed. [ABSTRACT FROM AUTHOR]

We consider d × d tensors A ( x ) that are symmetric, positive semi-definite, and whose row-divergence vanishes identically. We establish sharp inequalities for the integral of ( det ⁡ A ) 1 d − 1 . We apply them to models of compressible inviscid fluids: Euler equations, Euler–Fourier, relativistic Euler, Boltzman, BGK, etc. We deduce an a priori estimate for a new quantity, namely the space–time integral of ρ 1 n p , where ρ is the mass density, p the pressure and n the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional. [ABSTRACT FROM AUTHOR]

We study functional inequalities for Markov chains on discrete spaces with entropic Ricci curvature bounded from below. Our main results are that when curvature is non-negative, but not necessarily positive, the spectral gap, the Cheeger isoperimetric constant and the modified logarithmic Sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space, with respect to a suitable metric. These estimates are discrete analogues of classical results of Riemannian geometry obtained by Li and Yau, Buser and Wang. [ABSTRACT FROM AUTHOR]

In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad's angular cut-off assumption. This is done by an adaptation of the famous entropy method and its variants, resulting in explicit algebraic, or even stretched exponential, rates of convergence to equilibrium under appropriate assumptions. The novelty in our approach is that it involves functional inequalities relating the entropy to its production rate, which have independent applications to equations with mixed linear and non-linear terms. We also briefly discuss some properties of the equation in the non-cut-off case and conjecture what we believe to be the right rate of convergence in that case. [ABSTRACT FROM AUTHOR]

In this paper we deal with free functional inequalities on the circle. There are some interesting changes from their classical counterparts. For example, the free Poincaré inequality has a slight change which seems to account for the lack of invariance under rotations of the base measure. Another instance is the modified Wasserstein distance on the circle which provides the tools for analyzing transportation, Log-Sobolev, and HWI inequalities. These new phenomena also indicate that they have classical counterparts, which does not seem to have been investigated thus far. [ABSTRACT FROM AUTHOR]

We present various inequalities for the complete elliptic integral of the first kind, K(r) = ∫π/20 1/√1-r2sin2(t)dt (0

The authors present the power series expansions of the function R ( a ) − B ( a ) at a = 0 and at a = 1 / 2 , show the monotonicity and convexity properties of certain familiar combinations defined in terms of polynomials and the difference between the so-called Ramanujan constant R ( a ) and the beta function B ( a ) ≡ B ( a , 1 − a ) , and obtain asymptotically sharp lower and upper bounds for R ( a ) in terms of B ( a ) and polynomials. In addition, some properties of the Riemann zeta function ζ ( n ) , n ∈ N , and its related sums are derived. [ABSTRACT FROM AUTHOR]

Let \(( f^t )_{t \in \mathbb{R}}\) be a measurable iteration group on an open interval \(I\). Under some conditions, we prove that the inequalies \(g\circ f^a \leq f^a \circ g\) and \(g\circ f^b \leq f^b\circ g\) for some \(a,b \in \mathbb{R}\) imply that \(g\) must belong to the iteration group. Some weak conditions under which two iteration groups have to consist of the same elements are given. An extension theorem of a local solution of a simultaneous system of iterative linear functional equations is presented and applied to prove that, under some conditions, if a function \(g\) commutes in a neighbourhood of \(f\) with two suitably chosen elements \(f^a\) and \(f^b\) of an iteration group of \(f\) then, in this neighbourhood, \(g\) coincides with an element of the iteration group. Some weak conditions ensuring equality of iteration groups are considered.

Continuous solutions of iterative linear inequalities of the first and second order are considered, belonging to a class \(\mathcal{F}_T\) of functions behaving at the origin as a prescribed function \(T\).

In this work, a new inequality by sharpening the well-known Hölder inequality by means of a theorem based on abstract convexity is derived. [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]

We prove the strict concavity of Dirichlet's eta function η ( s ) = ∑ j = 1 ∞ ( − 1 ) j − 1 j s on ( 0 , ∞ ) . This extends a result of Wang, who proved in 1998 that η is strictly logarithmically concave on ( 0 , ∞ ) . Several new inequalities satisfied by η are also presented. Among them is the double-inequality log ⁡ 2 < η ( x ) 1 / x η ( y ) 1 / y η ( x y ) 1 / x y < 1 , for all x , y ∈ ( 1 , ∞ ) . Both bounds are sharp. [ABSTRACT FROM AUTHOR]

We analyse the asymptotic behaviour of solutions to the one dimensional fractional version of the porous medium equation introduced by Caffarelli and Vázquez [1,2] , where the pressure is obtained as a Riesz potential associated with the density. We take advantage of the displacement convexity of the Riesz potential in one dimension to show a functional inequality involving the entropy, entropy dissipation, and the Euclidean transport distance. An argument by approximation shows that this functional inequality is enough to deduce the exponential convergence of solutions in self-similar variables to the unique steady states. [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]

Abstract: Various miscellaneous functional inequalities are deduced for the so-called generalized inverse trigonometric and hyperbolic functions. For instance, functional inequalities for sums, difference and quotient of generalized inverse trigonometric and hyperbolic functions are given, as well as some Grünbaum inequalities with the aid of the classical Bernoulli inequality. Moreover, by means of certain already derived bounds, bilateral bounding inequalities are obtained for the generalized hypergeometric Clausen function. [Copyright &y& Elsevier]

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]