Tur\'an type inequalities for generalized inverse trigonometric functions

TL;DR

This paper investigates Turán type inequalities for generalized inverse trigonometric functions, focusing on the eigenfunction of the p-Laplace operator and extending results to related functions and Ramanujan's series.

Contribution

It introduces new Turán type inequalities for generalized inverse trigonometric functions and connects these to classical series studied by Ramanujan.

Findings

Established Turán inequalities for inverse trigonometric functions

Derived inequalities for hyperbolic counterparts

Linked inequalities to Ramanujan's series involving digamma function

Abstract

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.