Results 1 
4 of
4
On l’Hospitaltype rules for monotonicity
 J. Inequal. Pure Appl. Math
"... ABSTRACT. Elsewhere we developed rules for the monotonicity pattern of the ratio r: = f/g of two differentiable functions on an interval (a, b) based on the monotonicity pattern of the ratio ρ: = f ′ /g ′ of the derivatives. Those rules are applicable even more broadly than l’Hospital’s rules for li ..."
Abstract

Cited by 12 (10 self)
 Add to MetaCart
(Show Context)
ABSTRACT. Elsewhere we developed rules for the monotonicity pattern of the ratio r: = f/g of two differentiable functions on an interval (a, b) based on the monotonicity pattern of the ratio ρ: = f ′ /g ′ of the derivatives. Those rules are applicable even more broadly than l’Hospital’s rules for limits, since in general we do not require that both f and g, or either of them, tend to 0 or ∞ at an endpoint or any other point of (a, b). Here new insight into the nature of the rules for monotonicity is provided by a key lemma, which implies that, if ρ is monotonic, then ˜ρ: = r ′ · g 2 /g ′  is so; hence, r ′ changes sign at most once. Based on the key lemma, a number of new rules are given. One of them is as follows: Suppose that f(a+) = g(a+) = 0; suppose also that ρ ↗ ↘ on (a, b) – that is, for some c ∈ (a, b), ρ ↗ (ρ is increasing) on (a, c) and ρ ↘ on (c, b). Then r ↗ or ↗ ↘ on (a, b). Various applications and illustrations are given.
“NONSTRICT” L’HOSPITALTYPE RULES FOR MONOTONICITY: INTERVALS OF CONSTANCY
, 2008
"... Let f and g be differentiable functions defined on the interval (a,b), where − ∞ � a < b � ∞, and let r: = f g and ρ:= It is assumed throughout that g and g ′ do not take on the zero value anywhere on (a,b). The function ρ may be referred to as a derivative ratio for the “original ” ratio r. In ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Let f and g be differentiable functions defined on the interval (a,b), where − ∞ � a < b � ∞, and let r: = f g and ρ:= It is assumed throughout that g and g ′ do not take on the zero value anywhere on (a,b). The function ρ may be referred to as a derivative ratio for the “original ” ratio r. In [11], general “rules ” for monotonicity patterns, resembling the usual l’Hospital rules for limits, were given. In particular, according to [11, Proposition 1.9 and Remark 1.14], one has the dependence of the monotonicity pattern of r ( on (a,b)) on that of ρ (and also on the sign of gg ′ ) as given by Table 1. The vertical double line in the table separates the conditions (on the left) from the corresponding conclusions (on the right). ρ gg ′ r> 0
L’HOSPITALTYPE RULES FOR MONOTONICITY, AND THE LAMBERT AND SACCHERI QUADRILATERALS IN HYPERBOLIC GEOMETRY
 JOURNAL OF INEQUALITIES IN PURE AND APPLIED MATHEMATICS
, 2005
"... ..."
(Show Context)
L'HospitalType Rules for Monotonicity: Include Them into Calculus Texts!
"... Ratios are ubiquitous. A recent Google search for \ratio " returned over 96 106 items. Among those are numerous and commonly used nancial ratios (visit, for example, ..."
Abstract
 Add to MetaCart
(Show Context)
Ratios are ubiquitous. A recent Google search for \ratio " returned over 96 106 items. Among those are numerous and commonly used nancial ratios (visit, for example,