On a Functional Equation Related to the Determinant of Symmetric Two-By-Two Matrices

Authors

  • Kelly B. Houston Department of Mathematics, University of Louisville, Louisville, KY, USA
  • P. K. Sahoo Department of Mathematics, University of Louisville, Louisville, KY, USA

DOI:

https://doi.org/10.5644/SJM.03.1.07

Keywords:

Determinant, functional equation, logarithmic function, multiplicative function

Abstract

The present work aims to find the solutions $f,g,h, \ell , m : \mathbb{R}^2 \to \mathbb{R}$ of the functional equation $f( ux - vy , \, uy - vx ) = g(x, y ) + h(u, v) + \ell (x , y ) \, m(u , v)$ for all $x, y, u, v \in \mathbb{R}$ without any regularity assumptions on the unknown functions. This equation is a generalization of a functional equation which arises from the characterization of the determinant of symmetric matrices.

 

2000 Mathematics Subject Classification. Primary 39B22

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Author Biography

Kelly B. Houston, Department of Mathematics, University of Louisville, Louisville, KY, USA

Department of Mathematics, University of Louisville, Louisville, KY, USA

References

J. Acz´el and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989.

J. K. Chung and P. K. Sahoo, General solution of some functional equations related to the determinant of symmetric matrices, Demonstr. Math., 35 (2002), 539-544.

S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, Singapore 2002.

K. B. Houston and P. K. Sahoo, On two functional equations and their solutions, submitted to Aequationes Math., 2006.

M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality, Prace Nauk. Uniw. Sla 489, Polish Scientific Publishers, Warsaw-Cracow-Katowice, 1985.

P. K. Sahoo, Solved and Unsolved Problems, Problem 2, News Letter of the European Mathematical Society, 58 (2005), 43-44.

P. K. Sahoo and T. R. Riedel, Mean Value Theorems and Functional Equations, World Scientific Publishing Co., NJ, 1998.

J. Smital, On Functions and Functional Equations, Adam Hilger, Bristol-Philadelphia, 1988.

L. Sz´ekelyhidi, Convolution Type Functional Equations on Topological Abelian Groups. World Scientific, Singapore 1991.

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Published

12.06.2024

How to Cite

Houston, K. B., & Sahoo, P. K. (2024). On a Functional Equation Related to the Determinant of Symmetric Two-By-Two Matrices. Sarajevo Journal of Mathematics, 3(1), 61–71. https://doi.org/10.5644/SJM.03.1.07

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Articles