STABILITY OF AN EXPONENTIAL-MONOMIAL FUNCTIONAL EQUATION

Abstract

Let $N$ be a fixed positive integer and $f:\mathbb{R}\rightarrow \mathbb{C}$. As a generalisation of the superstability of the exponential functional equation we consider the functional inequalities $$\begin{eqnarray}\displaystyle & \displaystyle \big|f\big(\!\sqrt[N]{x^{N}+y^{N}}\big)-f(x)f(y)\big|\leq \unicode[STIX]{x1D719}(x), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \big|f\big(\!\sqrt[N]{x^{N}+y^{N}}\big)-f(x)f(y)\big|\leq \unicode[STIX]{x1D713}(x,y) & \displaystyle \nonumber\end{eqnarray}$$ for all $x,y\in \mathbb{R}$, where $\unicode[STIX]{x1D719}:\mathbb{R}\rightarrow \mathbb{R}^{+}$ is an arbitrary function and $\unicode[STIX]{x1D713}:\mathbb{R}^{2}\rightarrow \mathbb{R}^{+}$ satisfies a certain condition.