Research ArticleOpen AccessGoogle Scholar indexed
Solving Invariant Problem of Cauchy Means Based on Wronskian Determinant
School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang, China
School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang, China
- 1 School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang, China
- 2 School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang, China
Advances in Pure Mathematics·Volume 14 (2024)·Pages 515–522·Published 23 July 2024·DOI10.4236/apm.2024.147029
Copy link · social · email
Abstract
This paper studied the invariance of the Cauchy mean with respect to the arithmetic mean when the denominator functions satisfy certain conditions. The partial derivatives of Cauchy’s mean on the diagonal are obtained by using the method of Wronskian determinant in the process of solving. Then the invariant equation is solved by using the obtained partial derivatives. Finally, the solutions of invariant equations when the denominator functions satisfy the same simple harmonic oscillator equation or the denominator functions are power functions that have been obtained.
KeywordsCauchy MeanWronskian DeterminantArithmetic MeanInvariant Equation
- Cauchy, A.L. (1821) Cours d’analyse de l’École Royale Polytechnique, Ire partie: Analyse Algébrique. Imprimérie Royale.
- Daróczy, Z. and Páles, Z. (2002) Gauss-Composition of Means and the Solution of the Matkowski—Sutô Problem. Publicationes Mathematicae Debrecen , 61, 157-218. https://doi.org/10.5486/pmd.2002.2713
- Leach, E.B. and Sholander, M.C. (1984) Multi-Variable Extended Mean Values. Journal of Mathematical Analysis and Applications , 104, 390-407. https://doi.org/10.1016/0022-247x(84)90003-9
- Losonczi, L. (2000) Equality of Cauchy Mean Values. Publicationes Mathematicae Debrecen , 57, 217-230. https://doi.org/10.5486/pmd.2000.2338
- Losonczi, L. (2002) On the Comparison of Cauchy Mean Values. Journal of Inequ a lities and Applications , 2002, Article ID: 876016. https://doi.org/10.1155/s1025583402000024
- Losonczi, L. (2003) Equality of Two Variable Cauchy Mean Values. Aequationes Mathematicae , 65, 61-81. https://doi.org/10.1007/s000100300004
- Matkowski, J. (2004) Solution of a Regularity Problem in Equality of Cauchy Means. Publicationes Mathematicae Debrecen , 64, 391-400. https://doi.org/10.5486/pmd.2004.2939
- Berrone, L.R. and Moro, J. (2000) On Means Generated through the Cauchy Mean Value Theorem. Aequationes Mathematicae , 60, 1-14. https://doi.org/10.1007/s000100050131
- Berrone, L.R. (2015) Generalized Cauchy Means. Aequationes Mathematicae , 90, 307-328. https://doi.org/10.1007/s00010-015-0341-7
- Głazowska, D. (2011) Some Cauchy Mean-Type Mappings for Which the Geometric Mean Is Invariant. Journal of Mathematical Analysis and Applications , 375, 418-430. https://doi.org/10.1016/j.jmaa.2010.09.036
- Zhang, Q., Xu, B. and Han, M. (2020) Some Cauchy Mean-Type Mappings for Which the Arithmetic Mean Is Invariant. Aequationes Mathematicae , 95, 13-34. https://doi.org/10.1007/s00010-020-00743-0
- Zhang, Q. and Xu, B. (2011) An Invariance of Geometric Mean with Respect to Generalized Quasi-Arithmetic Means. Journal of Mathematical Analysis and Applic a tions , 379, 65-74. https://doi.org/10.1016/j.jmaa.2010.12.025
- Zhang, Q. and Xu, B. (2017) On Some Invariance of the Quotient Mean with Respect to Makó-Páles Means. Aequationes Mathematicae , 91, 1147-1156. https://doi.org/10.1007/s00010-017-0502-y
- Zhang, Q. (2024) Means and the Equations of Means. Advances in Mathematics ( China ), 53, 468-498.