Superstability of the $p$-power-radical functional equation related to sine function equation

In this paper, we find solutions and investigate the superstability bounded by a function (Gǎvruta sense) for the $p$-power-radical functional equation related to sine function equation:

$$f{\left(\sqrt[p]{\frac{x^p+y^p}{2}}\right)}^2-f{\left(\sqrt[p]{\frac{x^p-y^p}{2}}\right)}^2=f(x)f(y)$$

from an approximation of the $p$-power-radical functional equation:

$$\begin{array}{r}f{\left(\sqrt[p]{\frac{x^p+y^p}{2}}\right)}^2-f{\left(\sqrt[p]{\frac{x^p-y^p}{2}}\right)}^2=g(x)h(y),\end{array}$$

where $p$ is a positive odd integer, and $f,g$ and $h$ are complex valued functions on $\mathbb{R}$. Furthermore, the obtained results are extended to Banach algebras.