An extension of Rademacher’s theorem is proved for Lipschitz mappings between Banach spaces without the Radon–Nikodým property.

Answering a problem posed by John Michael Rassias, we study the functional inequality

$$\begin{eqnarray*}f(x+ y+ xy)\leq f(x)+ f(y)+ f(xy),\end{eqnarray*}$$

$f$

We study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.

We investigate island systems with continuous height functions and strongly laminar systems which are laminar systems containing sets with disjoint boundaries. In the discrete case, we show that for a maximal rectangular system of islands $ \mathcal{H} $ on an $m$ by $n$ rectangular grid we have $\lceil \min (m, n)/ 4\rceil \leq \vert \mathcal{H} \vert \leq \lceil m/ 2\rceil \lceil n/ 2\rceil $. In the continuous case we show that under some conditions maximal strongly laminar systems $ \mathcal{H} $ have cardinality ${\aleph }_{0} $ or ${2}^{{\aleph }_{0} } $ and present examples with $\vert \mathcal{H} \vert = {\aleph }_{0} $.

Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function ${K}_{ir} (x)$ of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of ${K}_{ir} (x)$ and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of $r$ . Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of ${K}_{ir} (x)$ .

Necessary and sufficient conditions are presented for a function involving the divided difference of the psi function to be completely monotonic and for a function involving the ratio of two gamma functions to be logarithmically completely monotonic. From these, some double inequalities are derived for bounding polygamma functions, divided differences of polygamma functions, and the ratio of two gamma functions.

In this paper, we establish various inequalities for some differentiable mappings that are linked with the illustrious Hermite–Hadamard integral inequality for mappings whose derivatives are $s$-$(\alpha , m)$-convex. The generalised integral inequalities contribute better estimates than some already presented. The inequalities are then applied to numerical integration and some special means.

We obtain the approximate functional equation for the Rankin–Selberg zeta function in the critical strip and, in particular, on the critical line $\operatorname {Re} s= \frac {1}{2}$.

We study the question whether a Riemann–Stieltjes integral of a positive continuous function with respect to a nonnegative function of bounded variation is positive.

We present a family of radical convolution Banach algebras on intervals (0,a] which are of Sobolev type; that is, they are defined in terms of derivatives. Among other properties, it is shown that all epimorphisms and derivations of such algebras are bounded. Also, we give examples of nontrivial concrete derivations.

We characterise solutions f,g:ℝ→ℝ of the functional equation f(x+g(x)y)=f(x)f(y) under the assumption that f is continuous. Our considerations refer mainly to a paper by Chudziak [‘Semigroup-valued solutions of the Goła̧b–Schinzel functional equation’, Abh. Math. Semin. Univ. Hambg. 76, (2006), 91–98], in which the author studied the same equation assuming that g is continuous.

We consider the convex hull ℬk of the symmetric moment curve Uk(t)=(cos t,sin t,cos 3t,sin 3t,…,cos (2k−1)t,sin (2k−1)t) in ℝ2k, where t ranges over the unit circle 𝕊=ℝ/2πℤ. The curve Uk(t) is locally neighborly: as long as t1,…,tk lie in an open arc of 𝕊 of a certain length ϕk>0 , the convex hull of the points Uk (t1),…,Uk (tk)is a face of ℬk. We characterize the maximum possible length ϕk, proving, in particular, that ϕk >π/2for all k and that the limit of ϕk is π/2as k grows. This allows us to construct centrally symmetric polytopes with a record number of faces.

The main objective of this paper is a study of some new refinements and converses of multidimensional Hilbert-type inequalities with nonconjugate exponents. Such extensions are deduced with the help of some remarkable improvements of the well-known Hölder inequality. First, we obtain refinements and converses of the general multidimensional Hilbert-type inequality in both quotient and difference form. We then apply the results to homogeneous kernels with negative degree of homogeneity. Finally, we consider some particular settings with homogeneous kernels and weighted functions, and compare our results with those in the literature.

We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise processes with jumps. We illustrate the result in some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.

The purpose of this paper is to study the existence of periodic solutions and the topological structure of the solution set of first-order differential equations involving the distributional Henstock–Kurzweil integral. The distributional Henstock–Kurzweil integral is a general integral, which includes the Lebesgue and Henstock–Kurzweil integrals. The main results extend some previously known results in the literature.

In this paper, using the Schauder Fixed Point Theorem and the Vidossich Theorem, we study the existence of solutions and the structure of the set of solutions of the Darboux problem involving the distributional Henstock–Kurzweil integral. The two theorems presented in this paper are extensions of the previous results of Deblasi and Myjak and of Bugajewski and Szufla.

In this paper we extend some estimates of the right-hand side of a Hermite–Hadamard type inequality for functions whose derivatives’ absolute values are P-convex. Applications to the trapezoidal formula and special means are introduced.

General three-point quadrature formulas for the approximate evaluation of an integral of a function f over [0,1], through the values f(x), f(1/2), f(1−x), f′(0) and f′(1), are derived via the extended Euler formula. Such quadratures are sometimes called “corrected” or “quadratures with end corrections” and have a higher accuracy than the adjoint classical formulas, which only include the values f(x), f(1/2)and f(1−x) . The Gauss three-point, corrected Simpson, corrected dual Simpson, corrected Maclaurin and corrected Gauss two-point formulas are recaptured as special cases. Finally, sharp estimates of error are given for this type of quadrature formula.

A generalisation of Descartes’ rule of signs to other functions is derived and a bound for the number of positive zeros of a class of integral transforms is deduced from that. A more precise rule of signs is also discussed in the light of these results.

Some inequalities of Jensen type for Q-class functions are proved. More precisely, a refinement of the inequality f((1/P)∑ ni=1pixi)≤P∑ ni=1(f(xi)/pi) is given in which p1,…,pn are positive numbers, P=∑ ni=1pi and f is a Q-class function. The notion of the jointly Q-class function is introduced and some Jensen type inequalities for these functions are proved. Some Ostrowski and Hermite–Hadamard type inequalities related to Q-class functions are presented as well.