  In this paper we study the inverse of the eigenfunction $\sin_p$ of the
one-dimensional $p$-Laplace operator and its dependence on the parameter $p$,
and we present a Tur\'an type inequality for this function. Similar
inequalities are given also for other generalized inverse trigonometric and
hyperbolic functions. In particular, we deduce a Tur\'an type inequality for a
series considered by Ramanujan, involving the digamma function.
