We establish some inequalities of Schwarz–Pick type for harmonic and hyperbolic functions on the unit ball $${\mathbb {R}^n}$$ we disprove a recent conjecture Liu (Int Math Res Not, 2021).

we shall give a lower estimate of harmonic‎ ‎functions of order greater than one in a half space‎, ‎which‎ ‎generalize the result obtained by b‎. ‎ya‎. ‎levin in a half plane‎.

We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(−1) spaces are strongly hyperbolic. On the way, we determine the best ...

‎Einstein‎, ‎M"{o}bius‎, ‎and Proper Velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper Lorentz group in the real Minkowski space-time $bkR^{n,1}.$ Using the gyrolanguage we study their gyroharmonic analysis‎. ‎Although there is an algebraic gyroisomorphism between the three models we show that there are some differences between them‎. ‎Our...

Abstract We consider a magnetic Laplacian −∆A = (id + A)?(id + A) on a noncompact hyperbolic surface M with finite area. A is a real one-form and the magnetic field dA is constant in each cusp. When the harmonic component of A satifies some quantified condition, the spectrum of −∆A is discrete. In this case we prove that the counting function of the eigenvalues of −∆A satisfies the classical We...