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In this paper, the Schwarz type and Schwarz-Pick type inequalities for solutions of $\alpha$-harmonic equation for $\alpha>-1$ are investigated. By making use of the integral of trigonometric functions, we obtain the two types of…
In this paper, we study the sharp Poincar\'e inequality and the Sobolev inequalities in the higher order Lorentz--Sobolev spaces in the hyperbolic spaces. These results generalize the ones obtained in \cite{Nguyen2020a} to the higher order…
The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.
In this paper we consider anisotropic Lorentz-Karamata space $2\pi$ of periodic functions of $m$ variables and Nikol'skii--Besov's class . In this paper, we establish order-sharp estimates of the best approximation by trigonometric…
The well known table of Gradshteyn and Ryzhik contains indefinite and definite integrals of both elementary and special functions. We give proofs of several entries containing integrands with some combination of hyperbolic and trigonometric…
We find the optimal function norm on the left-hand side of the $m$th order Sobolev type inequality $\|u\|_{Y(\mathbb{H}^n)} \leq C \|\nabla_g^m u\|_{X(\mathbb{H}^n)}$ in the $n$-dimensional hyperbolic space $\mathbb{H}^n$, $1\leq m < n$.…
In this article, Bohr type inequalities for some complex valued harmonic functions defined on the unit disk are given. All the results are sharp.
In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, root-square means, etc. Some new means recently studied are also presented. Different kinds of refinement of inequalities among these means are…
The theory of Newton--Okounkov bodies provides direct relations and points out analogies between the theory of mixed volumes of convex bodies, on the one hand, and the intersection theories of Cartier divisors and of Shokurov $b$-divisors,…
In the article differential-difference (semi-discrete) lattices of hyperbolic type are investigated from the integrability viewpoint. More precisely we concentrate on a method for constructing generalized symmetries. This kind integrable…
We present a relationship between the generalized hyperharmonic numbers and the poly-Bernoulli polynomials, motivated from the connections between harmonic and Bernoulli numbers. This relationship yields numerous identities for the…
The asymptotic bound for a length-based attack on the Conjugacy Search Problem in relatively hyperbolic groups is cubic for hyperbolic elements and a "small" polynomial for parabolic elements, depending on the Conjugacy Search Problem for…
By using some tools of analysis, we establish some analytical properties such as monotonicity and inequalities involving the hyperbolic sine integral function. As applications of some of the established properties, we obtain some rational…
In this paper our aim is to present some monotonicity and convexity properties for the one dimensional regularization of the Coulomb potential, which has applications in the study of atoms in magnetic fields and which is in fact a…
This work is concerned with a P\'olya-Szeg\"o type inequality for anisotropic functionals of Sobolev functions. The relevant inequality entails a double-symmetrization involving both trial functions and functionals. A new approach that…
This paper is partially a review of the development of the Investigation Program announced by Stancho Dimiev at the Bedlevo Conference on Hypercomplex Analysis (2006). A new aspect related with hyperbolic complex numbers, their…
The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps.
Recent efforts in Analysis of Boolean Functions aim to extend core results to new spaces, including to the slice $\binom{[n]}{k}$, the hypergrid $[K]^n$, and noncommutative spaces (matrix algebras). We present here a new way to relate…
In this paper, we develop topological modules over the ring of bicomplex numbers. We discuss bicomplex convexivity, hyperbolic-valued seminorms and hyperbolic-valued Minkowski functionals in bicomplex modules. We also study the conditions…
Invariants of general linear system of two hyperbolic partial differential equations (PDEs) are derived under transformations of the dependent and independent variables by real infinitesimal method earlier. Here a subclass of the general…