Mean values for vector valued functions and corresponding functional equations

• Autorzy: Ger R. , Sablik M.

Identyfikatory

DOI

10.14708/cm.v53i2.796

• Języki publikacji: EN

Abstrakty

EN

Although, in general, a straightforward generalization of the Lagrange mean value theorem for vector valued mappings fails to hold we will look for what can be salvaged in that situation. In particular, we deal with Sanderson's and McLeod's type results of that kind (see [9] and [7], respectively). Moreover, we examine mappings with a prescribed intermediate point in the spirit of the celebrated Aczél's theorem characterizing polynomials of degree at most 2 (cf. [1]).

Słowa kluczowe

EN

mean value theorems; quasi-arithmetic means; Gauss-iteration; characterization of quadratic polynomials

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2013
• Tom: Vol. 53, No. 2
• Strony: 255--269
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Ger R.

• Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland

autor

Sablik M.

• Institute of Mathematics, Poznan University of Technology, Piotrowo 3A, 60-965 Poznan

Bibliografia

• [1] J. Aczél, A mean value property of the derivative of quadratic polynomials - without mean values and derivatives, Math. Magazine 58 (1985), 42-45.
• [2] Z. Daróczy and Zs. Páles, Gauss-composition of means and the solution of the Matkowski-Sutˆo problem, Publ. Math. Debrecen 61 (2002)157-218.
• [3] J. Ger, On Sahoo-Riedel equations on a real interval, Aequationes Math. 62 (2001), 1-12.
• [4] K. Heuvers, Another logarithmic functional equation, Aequationes Math. 58 (1999), 260-264.
• [5] M. Kuczma, On some functional equations with conic sections as solutions, Rocznik Naukowo-Dydaktyczny WSP w Krakowie 159 (1993), 197-213.
• [6] A. Lisak and M. Sablik, Trapezoidal rule revisited, Bull. Inst. Math. Acad. Sin. (N.S.) 6(3) (2011), 347-360.
• [7] R. McLeod, Mean value theorems for vector valued functions, Proc. Edin. Math. Soc. 14 (1965), 197-209.
• [8] P. Sahoo and T. Riedel, Mean value theorems and functional equations, World Scientific, Singapore-New Jersey-London-Hong Kong, 1998.
• [9] J.D.E. Sanderson, A versatile vector mean value theorem, Amer. Math. Monthly 79 (1972), 381-383.