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

Abstract

Let $-\infty \le a<b\le \infty$. Let $f$ and $g$ be differentiable functions on $(a,b)$ and let $g^{\prime }\ne 0$ 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.

19 pages

References in corpus (1)

• New sharp Jordan type inequalities and their applications

Cited by in corpus (1)

• Convexity and monotonicity for the elliptic integrals of the first kind and applications