Aequationes Mathematicae最新文献

Let A and B be (ntimes n) positive definite complex matrices, let (sigma ) be a matrix mean, and let (f: [0,infty )rightarrow [0,infty )) be a differentiable convex function with (f(0)=0). We prove that

where m represents the smallest eigenvalues of A and B and M represents the largest eigenvalues of A and B. If f is differentiable and concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under certain mild conditions. In particular, if f(x)/x is increasing, then

holds for all A and B with (Mle A+B). Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski’s determinant inequality.

We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps (f_{varvec{p}}) are parametrized by a quintuple (varvec{p}) of real numbers satisfying inequations. Viewing (f_{varvec{p}}) as a circle map, we show that it has a rotation number (rho (f_{varvec{p}})) and we compute (rho (f_{varvec{p}})) as a function of (varvec{p}) in terms of Hecke–Mahler series. As a corollary, we prove that (rho (f_{varvec{p}})) is a rational number when the components of (varvec{p}) are algebraic numbers.

Consider a real vector space denoted as X, and let cc(Y) represent the collection of all convex and compact subsets of a real Hausdorff topological vector space Y. This paper investigates set-valued solutions of the Pexiderized quadratic functional equation

for unknown functions (f_1,f_2,f_3,f_4:Xrightarrow cc(Y)). This functional equation incorporates many functional equations including the quadratic, Cauchy’s and Drygas’ equations. A characterization for set-valued solutions of this functional equation is presented in this paper.

Let ([a,b]subseteq mathbb {R}) be a non-empty and non singleton closed interval and (P={a=x_0<cdots <x_n=b}) is a partition of it. Then (f:Irightarrow mathbb {R}) is said to be a function of r-bounded variation, if the expression (sum nolimits ^{n}_{i=1}|f(x_i)-f(x_{i-1})|^{r}) is bounded for all possible partitions like P. One of the main results of the paper deals with the generalization of the classical Jordan decomposition theorem. We establish that for (rin ]0,1]), a function of r-bounded variation can be written as the difference of two monotone functions. While for (r>1), under minimal assumptions such a function can be treated as an approximately monotone function which can be closely approximated by a nondecreasing majorant. We also prove that for (0<r_1<r_2), the function class of (r_1)-bounded variation is contained in the class of functions satisfying (r_2)-bounded variations. We go through approximately monotone functions and present a possible decomposition for (f:I(subseteq mathbb {R}_+)rightarrow mathbb {R}) satisfying the functional inequality

A generalized structural study has also been done in that specific section. On the other hand, for (ell [a,b]ge d), a function satisfying the following monotonic condition under the given assumption will be termed as d-periodically increasing

We establish that in a compact interval any function satisfying d-bounded variation can be decomposed as the difference of a monotone and a d-periodically increasing function. The core details related to past results, motivation, structure of each and every section are thoroughly discussed below.

The paper covers the foundations of fractal calculus on fractal curves, defines different function classes, establishes vector spaces for (F^{alpha })-integrable functions, introduces local fractal integrable functions and fractal distribution functionals, defines the dual space of a fractal function space, proves completeness for (F^{alpha })-differentiable function spaces, defines Fractal Sobolev spaces, and introduces fractal gradian and fractal Laplace operators on fractal Hilbert spaces. It also presents criteria for the existence of unique solutions to fractal differential equations.

This paper characterizes curves where a regular polygon of either a variable side length or a constant side length is allowed to rotate during k full revolutions while having its vertices on the curve during the motion. A constructive method to generate these curves is given based on the curve described by the polygon centers (centroids) during the motion and some examples are shown. Moreover, if the regular polygon divides the curve perimeter into parts of equal length, it is proved that the curve is either a rotational symmetric curve in the case of a variable side length or a circle otherwise. Finally, in the case of a regular polygon of constant side length rotating along a curve, a simple relation between the algebraic areas of such a curve and the curve of polygon centers is revisited.

An arithmetical function f is multiplicative if (f(1)=1) and (f(mn)=f(m)f(n)) whenever m and n are coprime. We study connections between certain subclasses of multiplicative functions, such as strongly multiplicative functions, over-multiplicative functions and totients. It appears, among others, that the over-multiplicative functions are exactly same as the totients and the strongly multiplicative functions are exactly same as the so-called level totients. All these functions satisfy nice arithmetical identities which are recursive in character.