On a functional equation arising from Proth identity

Bull. Korean Math. Soc. 0000 Vol. 31, No. 1, 131-138

Jaeyoung Chung and Prasanna K. Sahoo Kunsan National University, University of Louisville

Abstract : We determine the general solutions $f:\mathbb R^2 \to \mathbb R$ of the functional equation $ f(ux-vy, uy+v(x+y))=f(x, y)f(u, v) $ for all $x, y, u, v\in \mathbb R$. We also investigate both bounded and unbounded solutions of the functional inequality $ |f(ux-vy, uy+v(x+y))-f(x, y)f(u, v)|\le \phi(u, v) $ for all $x, y, u, v\in \mathbb R$, where $\phi:\mathbb R^2 \to \mathbb R_+$ is a given function.

Keywords : exponential type functional equation, general solution, multiplicative function, Proth identity, stability, bounded solution