Some special cases on Stolarsky’s means
DOI:
https://doi.org/10.23755/rm.v53i0.1637Abstract
In this paper we observe that one-parameter Stolarsky’s means (SM) are deduced from both the Mean Value Theorem for derivatives (MVTD) and the Mean Value Theorem for definite integrals (MVTI), and we study their elementary properties as the parameter varies. In the subfamily of SM having a natural number as parameter, we geometrically interpret one of them in particular as a real elliptic cone. We link SM having the integer power of a prime number as a parameter to classical means (i.e., harmonic mean, geometric mean, arithmetic mean, power mean). Finally, from an extension of Flett's Theorem (FT), we derive the expression of a new mean that is a upper bound of the arithmetic mean.References
H- Alzer and S. Ruscheweyh. On the Intersection of Two-Parameter Mean Value Families, : Proceedings of the American Mathematical Society, Vol. 129, No. 9, 2655-2662, Sep., 2001
Geo.D. Birkhoff and H.S. Vanvinder. On the Integral Divisors of an−bn, Annals of Mathematics Vol. 5 No. 4, 173-180, 1904.
P.Cerone and S.Dragomir. Stolarsky and Gini Divergence Measures in Information Theory. Research report collection, 7 (2), 2004
A.C. Chiang and K. Wainwright. Fundamental Methods of Mathematical Economics, 4th edition, McGraw Hill, 383-387, 2004.
R. Cisbani. Contributi alla teoria delle medie, Metron V. 13 N. 2, 23-34, 1938.
O. Dunkel. Generalized Geometric Means and Algebraic Equations, Annals of Mathematics, Second Series, Vol. 11, No. 1, 21-32, Oct., 1909
N. Elezovic and J. Pečarić, A note on Schur-convex functions, Rocky Mountain J. Math. 30, 853-856, 2000
H. Eves, Great Moments in Mathematics (Before 1650), Mathematical Association of America, 11-13, 1980.
T. M. Flett. A Mean Value Theorem The Mathematical Gazette , Feb., 1958, Vol. 42, No. 339, 38-39, Feb. 1958
L.Galvani. Dei limiti a cui tendono alcune medie, Bollettino dell’Unione Matematica Italiana, Serie 1, Vol. 6, n. 4, 173-179, 1927
H. W. Gould and m. E. Mays. Series Expansions of Means, Journal of Mathematical Analysis and Applications 101, 611-621, 1984 [13] M Heida, M Kantner, A Stephan. Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator, ESAIM: MathematicalModelling …, 2021 - esaim-m2an.org
Djordje Herceg, Dragoslav Herceg. Third-order modifications of Newton’s method based on Stolarsky and Gini means/Journal of Computational and Applied mathematics, volume 245 june 2013, 53-61
J.B Hiriart-Urruty, A note on the mean value theorem for convex functions, Bollettino UMI (5) 17-B , 765-775, 1980.
N. D.Kazarinoff. Analytic inequalities, Holt, Rinehart and Winston, 1960.
Kung Kuen Tse. A Note on the Mean Value Theorems. Advances in Pure Mathematics, Scientific Research Publishing, Vol. 11, No. 5, 395-399, 2021
E.B. Leach and M.C. Sholander. Extended mean values II, Journal of Mathematical Analysis and Applications 92, 207-223, 1983.
M.M. Mays. Function Which Parametrize Means, American Mathematical Monthly, Volume 90, Issue 10, 677-683, Dec., 1983.
Uta Merzbach and Carl Boyer. A History of Mathematics, Third Edition, 16-17
C. Palmisani. Un confronto tra i teoremi del valor medio, Periodico di Matematica (IV) Vol. V, 29-42, Marzo 2023
F. Qi. A note on Schur-convexity of extended mean values, Rocky Mountain J. Math. 35, no.5, 1787–1793, 2005
F.Qi. Logarithmic convexity of extended mean values, Proc. Amer. Math. Soc. 130, 1787-1796, 2002
P.K. Sahoo and T. Riedel. Mean value theorems an functional equations, World Scientific Publishing Company, Singapore, 1998.
J. Sandor. The Schur-convexity of Stolarsky and Gini means, Banach J. Math. Anal. 1, no. 2, 212–215, 2007
K. B. Stolarsky. Generalizations of the logarithmic mean, Mathematics Magazine , Mar., 1975, Vol. 48, No. 2 , 87-92, Mar. 1975.
K. B. Stolarsky. The Power and Generalized Logarithmic Means, The American Mathematical Monthly , Vol. 87, No. 7, 545-548, Aug. - Sep. 1980.
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