A new way to prove L'Hospital Monotone Rules with applications

TL;DR

This paper introduces a novel, simplified proof technique for L'Hospital Monotone Rules using an auxiliary function, leading to new inequalities for special functions and means.

Contribution

It presents a new, concise proof method for L'Hospital Monotone Rules and applies it to derive sharp inequalities for hyperbolic, trigonometric functions, and bivariate means.

Findings

Established three new sharp inequalities for special functions and means.

Provided a natural, easier-to-understand proof of L'Hospital Monotone Rules.

Extended the application of these rules to various mathematical inequalities.

Abstract

Let $-\infty \leq a<b\leq \infty $. Let $f$ and $g$ be differentiable functions on $(a,b)$ and let $g^{\prime }\neq 0$ on $(a,b)$. By introducing an auxiliary function $H_{f,g}:=\left( f^{\prime }/g^{\prime }\right) g-f$, we easily prove L'Hoipital rules for monotonicity. This offer a natural and concise way so that those rules are easier to be understood. Using our L'Hospital Piecewise Monotone Rules (for short, LPMR), we establish three new sharp inequalities for hyperbolic and trigonometric functions as well as bivariate means, which supplement certain known results.