ON A FUNCTIONAL EQUATION ARISING FROM PROTH IDENTITY

Title & Authors
ON A FUNCTIONAL EQUATION ARISING FROM PROTH IDENTITY
Chung, Jaeyoung; Sahoo, Prasanna K.;

Abstract
We determine the general solutions $\small{f:\mathbb{R}^2{\rightarrow}\mathbb{R}}$ of the functional equation f(ux-vy, uy+v(x+y)) = f(x, y)f(u, v) for all x, y, u, $\small{v{\in}\mathbb{R}}$. We also investigate both bounded and unbounded solutions of the functional inequality $\small{{\mid}f(ux-vy,uy+v(x+y))-f(x,y)f(u,v){\mid}{\leq}{\phi}(u,v)}$ for all x, y, u, $\small{v{\in}\mathbb{R}}$, where $\small{{\ph}:\mathbb{R}^2{\rightarrow}\mathbb{R}_+}$ is a given function.
Keywords
exponential type functional equation;general solution;multiplicative function;Proth identity;stability;bounded solution;
Language
English
Cited by
References
1.
M. Albert and J. A. Baker, Bounded solutions of a functional inequality, Canad. Math. Bull. 25 (1982), no. 4, 491-495.

2.
R. Blecksmith and S. Broudno, Equal sums of three fourth powers or what Ramanujan could have said, Math. Magazine 79 (2006), 297-301.

3.
E. A. Chavez and P. K. Sahoo, On a functional equation arising from number theory, Appl. Math. Lett. 24 (2011), no. 3, 344-347.

4.
D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, 1998.

5.
P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, CRC Press, Taylor & Francis Group, Boca Raton, 2011.