A counterpart of the Taylor theorem and means
2014
scientific article
english
- Taylor theorem
- mean
- Taylor remainder mean
- functional equation
EN For an n-times differentiable real function ƒ defined in an a real interval I, some properties of the Taylor remainder means Tn[ƒ] are considered. It is proved that Tn[ƒ] is symmetric iff n – 1, and a conjecture concerning the equality Tn[g]- Tn[ƒ] is formulated. The main result says that if ƒ(n) is one-to-one, there exists a unique mean Mn[ƒ] : ƒ(n) (I) x ƒ(n) (I) → ƒ(n) (I) such that, for all x, y ϵ I, ∑k=0n – 1 (f(k) (x)/k!)(y — x)k + (Mn[f](f(n)(x), f(n)(y))/n!)(y – x)n. The connection between Tn[ƒ] and Mn[ƒ] is given. A functional equation related to M2[ƒ] is derived and an open problem is posed.
85 - 93
CC BY-NC-ND (attribution - noncommercial - no derivatives)
public
10