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  Volume 7, Issue 2, Article 40
On L'Hospital-Type Rules for Monotonicity

    Authors: Iosif Pinelis,  
    Keywords: L'Hospital-type rules, Monotonicity, Borwein-Borwein-Rooin ratio, Becker-Stark inequalities, Anderson-Vamanamurthy-Vuorinen inequalities, log-concavity, Maclaurin series, Hyperbolic geometry, Right-angled triangles.  
    Date Received: 18/05/05  
    Date Accepted: 14/11/05  
    Subject Codes:

26A48, 26A51, 26A82, 26D10, 50C10, 53A35

    Editors: Jonathan Borwein,  

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 $ rho:=f^{prime}/g^{prime}$ 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 $ infty$ 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 $ rho$ is monotonic, then $ tilderho:=r^{prime}cdot g^2/vert g^{prime}vert$ is so; hence, $ r^{prime}$ 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 $ rhonearrowsearrow$ on $ (a,b)$ - that is, for some $ cin(a,b)$, $ rhonearrow$ ($ rho$ is increasing) on $ (a,c)$ and $ rhosearrow$ on $ (c,b)$. Then $ rnearrow$ or $ nearrowsearrow$ on $ (a,b)$. Various applications and illustrations are given.

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