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A counterpart of the Taylor theorem and means

Matkowski J.

Fasciculi Mathematici

2014

Nr 53

85--93

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, …[wzór]. The connection between Tnƒ ana Mnƒ is given. A functional equation related to M2 ƒ is derived and an open problem is posed.