Mean-value theorems and some symmetric means

• Autorzy: Matkowski J.
• Języki publikacji: EN

Abstrakty

EN

Some variants of the Lagrange and Cauchy mean-value theorems lead to the conclusion that means, in general, are not symmetric. They are symmetric iff they coincide (respectively) with the Lagrange and Cauchy means. Under some regularity assumptions, we determine the form of all the relevant symmetric means.

Słowa kluczowe

EN

mean value theorem; mean; symmetric mean; Lagrange mean; Cauchy mean; translated quasi-geometric mean; differential equation

PL

twierdzenie o wartości średniej; średnia; średnia symetryczna; twierdzenie Lagrange'a; równanie różniczkowe

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2013
• Tom: Vol. 46, nr 3
• Strony: 461--474
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Matkowski J.

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, PL-65246 Zielona Góra, Poland

Bibliografia

• [1] P. S. Bullen, D. S. Mitrinović, P. M. Vasić, Means and their Inequalities, D. Reidel Publishing Company, Dordrecht–Boston–Lancaster–Tokyo, 1988.
• [2] P. S. Bullen, Handbook of Means and their Inequalities, Mathemtics and Its Applications, Vol. 560, Kluwer Academic Publishers, Dordrecht–Boston–London, 2003.
• [3] D. Głazowska, J. Matkowski, An invariance of geometric mean with respect to Lagrangian means, J. Math. Anal. Appl. 331(2) (2007), 1187–1199.
• [4] J. Matkowski, Mean value property and associated functional equations, Aequationes Math. 58(1-2) (1999), 46–59.
• [5] J. Matkowski, Lagrangian mean-type mappings for which the arithmetic mean is invariant, J. Math. Anal. Appl. 309(1) (2005), 15–24.
• [6] J. Matkowski, On iterations of means and functional equations. Iteration theory (ECIT'04), Grazer Math. Ber., 350, 2006, 184–201.
• [7] J. Matkowski, On weighted extensions of Cauchy’s means, J. Math. Anal. Appl. 319(1) (2006), 215–227.
• [8] J. Matkowski, A mean-value theorem and its applications, J. Math. Anal. Appl. 373 (2011), 227–234.
• [9] J. Matkowski, Power means generated by some mean-value theorems, Proc. Amer. Math. Soc. 139 (2011), 3601–3610.
• [10] J. Matkowski, I. Pawlikowska, Homogeneous means generated by a mean-value therem, J. Math. Ineq. 4 (2010), 467–479.