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

Abstract

Let −∞ ≤ a < b ≤ ∞. Let f and g be differentiable functions on (a, b) and let g ′ = 0 on (a, b). By introducing an auxiliary function H f,g := (f ′ /g ′ ) 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.

Sharp Power Mean Bounds for Two Seiffert-like Means

“…It follows from Proposition 1 that U sinh /V is increasing on (0, 1). This together with (37) and (38)…”

“…To introduce the second tool, we introduce an important auxiliary function H f ,g , which appeared in [38] and was called Yang's H-function in [39]. For −∞ ≤ a < b ≤ ∞, let f and g be differentiable on (a, b) and g = 0 on (a, b).…”

“…The following proposition was proved in [38] and is called the L'Hospital piece monotonic rule (LPMR). Proposition 2.…”

Convexity of ratios of the modified Bessel functions of the first kind with applications

“…To prove the monotonicity of F θ,ν (x), we need three tools. The first tool is the so-called H -function H f ,g and two identities, which appeared in [20]…”