Two applications of approximate differentiation formulae: An extremum problem for multiply monotone functions and the differentiation of asymptotic expansions

AbstractPresented in this report are two further applications of very elementary formulae of approximate differentiation. The first is a new derivation in a somewhat sharper form of the following theorem of V. M. Olovyanišnikov: Let Nn (n ⩾ 2) be the class of functions g(x) such that g(x), g′(x),…, g(n)(x) are ⩾ 0, bounded, and nondecreasing on the half-line −∞ < x ⩽ 0. A special element of Nn is g∗(x) = 0 if −∞ < x < −1, g∗(x) = (1 + x)n if −1 ⩽ x ⩽ 0. If g(x) ∈ Nn is such that g(0) ⩾ g∗(0) = 1, g(n)(0) ⩽ g∗(n)(0) = n!, then g(v)(0) ⩽ g∗(v)(0) for 1v = 1,…, n − 1. Moreover, if we have equality in (1) for some value of v, then we have there equality for all v, and this happens only if g(x) = g∗(x) in (−∞, 0].The second application gives sufficient conditions for the differentiability of asymptotic expansions (Theorem 4)