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  Volume 8, Issue 1, Article 14
"Non-strict" l'Hospital-Type Rules for Monotonicity: Intervals of Constancy

    Authors: Iosif Pinelis,  
    Keywords: L'Hospital-type rules for monotonicity, Intervals of constancy.  
    Date Received: 13/12/06  
    Date Accepted: 07/03/07  
    Subject Codes:

26A48; 26D10.

    Editors: Matti Vuorinen,  

Assuming that a ``derivative'' ratio $ $ rho:=f'/g'$ of the ratio $ r:=f/g$ of differentiable functions $ f$ and $ g$ is strictly monotonic (that is, $ $ rho$ is increasing or decreasing), it was shown in previous papers that then $ r$ can switch at most once, from decrease to increase or vice versa. In the present paper, it is shown that, if $ $ rho$ is non-strictly monotonic (that is, non-increasing or non-decreasing), then $ r$ can have at most one maximal interval of constancy (m.i.c.); on the other hand, any one m.i.c. of a given derivative ratio $ $ rho$ is the m.i.c. of an appropriately constructed original ratio $ r$.

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