A counterpart of the Taylor theorem and means

• Autorzy: Matkowski J.
• Języki publikacji: EN

Abstrakty

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

Słowa kluczowe

EN

Taylor theorem; mean; Taylor remainder mean; functional equation

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2014
• Tom: Nr 53
• Strony: 85--93
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Matkowski J.

• Faculty of Mathematics Computer Science and Econometrics University of Zielona Góra ul. Prof. Z. Szafrana 5a PL-65-516 Zielona Góra, Poland

Bibliografia

• [1] Berrone L.R., Moro J., Lagrangian means, Aequationes Math., (55)(5)(1998), 217-226.
• [2] Bullen P.S., Handbook of Means and Their Inequalities, Mathemat-ics and Its Applications, Vol. 560, Kluwer Academic Publishers, Dor-drecht/Boston/London, 2003.
• [3] Horwitz A., Means and Taylor polynomials, J. Math. Anal. Appl., 149(1990), 220-235.
• [4] Horwitz A., Means and averages of Taylor polynomials, J. Math. Anal. Appl., 176(1993), 404-412.
• [5] Matkowski J., Mean value property and associated functional equation, Aequationes Math., 58(1999), 46-59.
• [6] Matkowski J., On weighted extensions of Cauchy’s means, J. Math. Anal. Appl., 319(2006), 215-227.
• [7] Matkowski J., A mean-value theorem and its applications, J. Math. Anal. Appl., 373(2011), 227-234.
• [8] Matkowski J., Power means generated by some mean-value theorems, Proc. Amer. Math. Soc., 139(2011), 3601-3610.