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Abstract. 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.

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

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Abstract. We study the power mean inequality for the generalized trigonometric and hyperbolic functions with two parameters. The generalized p-trigonometric and (p, q)-trigonometric functions were introduced by Lindqvist and Takeuchi, respectively.