26A48,26D10,26A51,26D15,60E15,62P10

An algorithmic description of the dependence of the oscillation pattern of the ratio  f / g  of two functions f and g on the oscillation pattern of the ratio  f' / g'  of their derivatives is given. This tool is then used in order to refine and extend the Yao-Iyer inequality, arising in bioequivalence studies. The convexity conjecture by Topsøe concerning information inequalities is addressed in the context of a general convexity problem. This paper continues the series of results begun by the l'Hospital type rule for monotonicity. Other applications of this rule are given elsewhere: to certain information inequalities, to monotonicity of the relative error of a Padé approximation for the complementary error function, and to probability inequalities for sums of bounded random variables.;