A generalization of the Burkholder–Davis–Gundy inequalities

In this paper, by applying some improved inequalities, we extend the Burkholder–Davis–Gundy inequalities for α ∈ (0,1) to more general functions and submartingales. Moreover, a series of inequalities for a logarithmic function are also obtained correspondingly. Finally, we give an application to a stopped Brownian motion.     

Fefferman–Stein inequalities for the dyadic-like maximal operators

The paper contains a transference theorem which allows to extend a large class of unweighted inequalities for the dyadic maximal operator to their weighted Fefferman–Stein counterparts on general probability spaces.

     

On transversally symmetric pseudo-Einstein and Fefferman–Einstein spaces

We describe and construct pseudo-Hermitian structures θ without torsion (i.e. with transverse symmetry) whose Webster–Ricci curvature tensor is a constant multiple of the exterior differential dθ. We call these structures TS-pseudo-Einstein and our first result states that all these structures can locally be derived from Kähler–Einstein metrics. Then we discuss the corresponding Fefferman metrics of the TS-pseudo-Einstein structures. These are never Einstein. However, our second result states that they are locally always conformally Einstein.     

On the constants for some fractional Gagliardo–Nirenberg and Sobolev inequalities

We consider the inequalities of Gagliardo–Nirenberg and Sobolev in ${\mathbb{R}}^d$, formulated in terms of the Laplacian $\Delta$ and of the fractional powers $D^n?{\sqrt{?\Delta }}^n$ with real $n?0$; we review known facts and present novel, complementary results in this area. After illustrating the equivalence between these two inequalities and the relations between the corresponding sharp constants and maximizers, we focus the attention on the ${?}^2$ case where, for all sufficiently regular $f:{\mathbb{R}}^d\to \mathbb{C}$, the norm $6D^{j}f{6}_{{?}^r}$ is bounded in terms of $6f{6}_{{?}^2}$ and $6D^{n}f{6}_{{?}^2}$, for $1∕r=1∕2?(?n?j)∕d$, and suitable values of $j,n,?$ (with $j,n$ possibly noninteger). In the special cases $?=1$ and $?=j∕n+d∕2n$ (i.e., $r=+\infty$), related to previous results of Lieb and Ilyin, the sharp constants and the maximizers can be found explicitly; we point out that the maximizers can be expressed in terms of hypergeometric, Fox and Meijer functions. For the general ${?}^2$ case, we present two kinds of upper bounds on the sharp constants: the first kind is suggested by the literature, the second one is an alternative proposal of ours, often more precise than the first one. We also derive two kinds of lower bounds. Combining all the available upper and lower bounds, the sharp constants are confined to quite narrow intervals. Several examples are given, including the numerical values of the previously mentioned bounds.     

On Carlson–Levin inequalities

We present a new proof of Carlson–Levin inequality and some of its extensions, based on a dynamic programing-type approach.     

On Cowen–Pommerenke inequalities

C. Cowen and C. Pommerenke established inequalities for the derivatives of analytic self-mappings of the open unit disk at their fixed points. In this article, we show that these inequalities can be derived from Schwarz–Pick inequalities     

On some eigenvalue inequalities for Schatten–von Neumann operators

Michael Gil' recently obtained some bounds for eigenvalues in [J. Funct. Anal. 267 (2014) 3500–3506] and [Commun. Contemp. Math. 18 (2016) 1550022], which improve some classical results related to this aspect. We revisit these results by providing genuinely different arguments (e.g., using Aluthge transform, majorization). New results are derived along our discussions.     

The Fefferman–Szegő metric and applications

In this paper, we develop properties of the Szeg? kernel and Fefferman–Szeg? metric that were first introduced by D. Barrett and L. Lee. In particular, we produce a representative coordinate system related to the metric. We also explore the Poisson–Szeg? kernel. Additional analytic and geometric properties of the Szeg? kernel and Fefferman–Szeg? metric are developed.     

Brunn–Minkowski and Zhang inequalities for convolution bodies

A quantitative version of Minkowski sum, extending the definition of θ$\theta$-convolution of convex bodies, is studied to obtain extensions of the Brunn–Minkowski and Zhang inequalities, as well as, other interesting properties on Convex Geometry involving convolution bodies or polar projection bodies. The extension of this new version to more than two sets is also given.     

On fractional stochastic inequalities related to Hermite–Hadamard and Jensen types for convex stochastic processes

Stochastic convexity and its applications are very important in mathematics and probability (Aequationes Mathematicae 20:184–197, 1980). There are two well-known inequalities for convex stochastic processes: Jensen’s inequality and Hermite–Hadamard’s inequality. Recently, Hafiz (Stoch Anal Appl 22:507–523, 2004) has provided fractional calculus for some stochastic processes. The problem is how to formulate these inequalities for stochastic processes in the class of fractional calculus and that is what is done in this paper. Our results generalize the corresponding ones in the literature.     

On inequalities for conditional probabilities of unions of events and the conditional Borel–Cantelli lemma

New sharp upper and lower bounds for conditional (given a σ-algebra A) probabilities of unions of events and for a generalization of the conditional Borel–Cantelli lemma are obtained. Averaging the left- and right-hand sides of the corresponding inequalities yields bounds better than those obtained by directly estimating the probabilities of events. An example is given. New generalizations of the conditional Borel–Cantelli lemma are also obtained. Averaging yields new versions of this lemma under conditions different from the classical ones.     

On Stanley’s inequalities for character multiplicities

Let G be a group of automorphisms of a ranked poset \({{\mathcal Q}}\) and let N _k denote the number of orbits on the elements of rank k in \({{\mathcal Q}}\). What can be said about the N _k for standard posets, such as finite projective spaces or the Boolean lattice? We discuss the connection of this question to the representation theory of the group, and in particular to the inequalities of Livingstone-Wagner and Stanley. We show that these are special cases of more general inequalities which depend on the prime divisors of the group order. The new inequalities often yield stronger bounds depending on the order of the group.     

On Beckenbach–Radó type integral inequalities

Beckenbach and Radó characterized logarithmically subharmonic functions in the plane in terms of integral inequalities involving spherical averages. In this work we generalize this result to higher dimensions and thus answer to a question raised by Beckenbach and Radó. We also consider generalizations of integral inequalities suggested by Beckenbach and Radó and discuss connections to reverse Hölder inequalities and Muckenhoupt weights.     

On Young and Heinz inequalities for τ-measurable operators

The purpose of this article is to prove Young and Heinz inequalities for τ-measurable operators.     

Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo–Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb’s Translation Lemma and a Riesz energy version of the Brézis–Lieb lemma.     

On symmetrized stochastic convexity and the inequalities of Hermite–Hadamard type

Aequationes mathematicae - In this paper the concept of symmetrized convex stochastic processes is introduced. Some characterizations involving Hermite–Hadamard type inequalities and a...