In this talk we present a Tur\'an type inequality for the inverse of the eigenfunction $\sin_p$ of the one-dimensional $p$-Laplace operator and. We deduce similar inequalities 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. Moreover, we present various miscellaneous functional inequalities for the so-called generalized inverse trigonometric and hyperbolic functions. For instance, functional inequalities for sums, difference and quotient of generalized inverse trigonometric and hyperbolic functions are given, as well as some Gr\"unbaum inequalities with the aid of the classical Bernoulli inequality. Moreover, by means of certain already derived bounds, bilateral bounding inequalities are obtained for the generalized hypergeometric ${}_3F_2$ Clausen function. The talk is based on the papers:

1. \'A. Baricz, B.A. Bhayo, M. Vuorinen: Tur\'an type inequalities for generalized inverse trigonometric functions. http://arxiv.org/abs/1305.0938

2. \'A. Baricz, B.A. Bhayo, T.K. Pog\'any: Functional inequalities for generalized inverse trigonometric and hyperbolic functions. http://arxiv.org/abs/1401.4863