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STABILITY OF AN EXPONENTIAL-MONOMIAL FUNCTIONAL EQUATION
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 3 / June 2018
- Published online by Cambridge University Press:
- 28 March 2018, pp. 471-479
- Print publication:
- June 2018
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- Article
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- You have access
- Export citation
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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 for all
$$\begin{eqnarray}\displaystyle & \displaystyle \big|f\big(\!\sqrt[N]{x^{N}+y^{N}}\big)-f(x)f(y)\big|\leq \unicode[STIX]{x1D719}(x), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \big|f\big(\!\sqrt[N]{x^{N}+y^{N}}\big)-f(x)f(y)\big|\leq \unicode[STIX]{x1D713}(x,y) & \displaystyle \nonumber\end{eqnarray}$$
$x,y\in \mathbb{R}$, 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.
