Inequalities for eigenfunctions of the $p$-Laplacian

Motivated by the work of P. Lindqvist, we study eigenfunctions of the one-dimensional $p$-Laplace operator, the $\sin_{p}$ functions, and prove several inequalities for these and $p$-analogues of other trigonometric functions and their inverse functions. Similar inequalities are given also for the $p$-analogues of the hyperbolic functions and their inverses.