We discuss recent progress on issues surrounding the Brascamp–Lieb inequalities.

In this article, we formalize continuous differentiability of realvalued functions on n-dimensional real normed linear spaces. Next, we give a definition of the Ck space according to [23].

We continue our earlier investigations of radial subspaces of Besov and Lizorkin-Triebel spaces on ${ℝ}^d$. This time we study characterizations of these subspaces by differences.

The integral inequalities known for the Lebesgue integral are discussed in the framework of the Choquet integral. While the Jensen inequality was known to be valid for the Choquet integral without any additional constraints, this is not more true for the Cauchy, Minkowski, Hölder and other inequalities. For a fixed monotone measure, constraints on the involved functions sufficient to guarantee the validity of the discussed inequalities are given. Moreover, the comonotonicity of the considered functions...

Let be the cone of real univariate polynomials of degree ≤ 2n which are nonnegative on the real axis and have nonnegative coefficients. We describe the extremal rays of this convex cone and the class of linear operators, acting diagonally in the standard monomial basis, preserving this cone.

We consider the functional equation $f\left(xf\right(x\left)\right)=\varphi \left(f\right(x\left)\right)$ where $\varphi J\to J$ is a given increasing homeomorphism of an open interval $J\subset (0,\infty )$ and $f(0,\infty )\to J$ is an unknown continuous function. In a series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under $\varphi$ and which contains in its interior no fixed point except for $1$. They also provide a characterization of the class of monotone solutions and prove a necessary and sufficient condition for any solution...

Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if ${ʃ}_{\Omega }xyd\mu \le {\varphi }^{-1}\left({ʃ}_{\Omega }\varphi \circ xd\mu \right){\psi }^{-1}\left({ʃ}_{\Omega }\psi \circ xd\mu \right)$ for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then there exist...

We consider the functional equation $f\left(xf\right(x\left)\right)=\varphi \left(f\right(x\left)\right)$ where $\varphi J\to J$ is a given homeomorphism of an open interval $J\subset (0,\infty )$ and $f(0,\infty )\to J$ is an unknown continuous function. A characterization of the class $𝒮(J,\varphi )$ of continuous solutions $f$ is given in a series of papers by Kahlig and Smítal 1998–2002, and in a recent paper by Reich et al. 2004, in the case when $\varphi$ is increasing. In the present paper we solve the converse problem, for which continuous maps $f(0,\infty )\to J$, where $J$ is an interval, there is an increasing homeomorphism $\varphi$ of $J$ such that $f\in 𝒮(J,\varphi )$. We...

We show that the Covering Principle known for continuous maps of the real line also holds for functions whose graph is a connected $G_{\delta }$ subset of the plane. As an application we find an example of an approximately continuous (hence Darboux Baire 1) function f: [0,1] → [0,1] such that any closed subset of [0,1] can be translated so as to become an ω-limit set of f. This solves a problem posed by Bruckner, Ceder and Pearson [Real Anal. Exchange 15 (1989/90)].

In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval $\left[a,b\right]$ into a Banach space $X.$ It is shown that a Denjoy-Bochner integrable function on $\left[a,b\right]$ is Denjoy-Riemann integrable on $\left[a,b\right]$, that a Denjoy-Riemann integrable function on $\left[a,b\right]$ is Denjoy-McShane integrable on $\left[a,b\right]$ and that a Denjoy-McShane integrable function on $\left[a,b\right]$ is Denjoy-Pettis integrable on $\left[a,b\right].$ In addition, it is shown that for spaces that do not contain a copy of $c_0$, a measurable Denjoy-McShane integrable...