Explore the vast range of titles available.

Here, we study the extension of p-trigonometric functions sinp and cosp family in complex domains and p-hyperbolic functions sinhp and the coshp family in hyperbolic complex domains. These functions satisfy analogous relations as their classical counterparts with some unknown properties. We show the relationship of these two classes of special functions viz. p-trigonometric and p-hyperbolic functions with imaginary arguments. We also show many properties and identities related to the analogy between these two groups of functions. Further, we extend the research bridging the concepts of hyperbolic and elliptical complex numbers to show the properties of logarithmic functions with complex arguments.

We introduce and study an approximate solution of the p-Laplace equation and a linearlization ℒ ϵ of a perturbed p-Laplace operator. By deriving an ℒ ϵ -type Bochner's formula and Kato type inequalities, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p-harmonic functions with finite q-energy on Riemannian manifolds. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type theorem for p-harmonic morphisms. 2010 Mathematics Subject Classification. Primary 53C21, 53C24; Secondary 58E20, 31C45. Key words and phrases. p-harmonic map, weakly p-harmonic function, perturbed p-Laplace operator, p-hyperbolic end, Liouville type properties.