Alternative proofs of Shafer's inequality for inverse hyperbolic tangent

TL;DR

This paper identifies and corrects a mistake in a previous proof of Shafer's inequality for inverse hyperbolic tangent and provides alternative proofs to strengthen the mathematical understanding.

Contribution

It corrects an error in a prior proof and offers new alternative proofs of Shafer's inequality for inverse hyperbolic tangent.

Findings

Corrected a mistake in the original proof of Theorem 2

Provided alternative proofs for Shafer's inequality

Enhanced mathematical rigor in the proof of the theorem

Abstract

We point out that a concise proof of Theorem 2 in the article, 'On a quadratic estimate of Shafer' by L. Zhu contains a small mistake. Correcting this mistake and giving alternative proofs of Theorem 2 is the main aim of this note.