حمید خدائی

‎We consider the following quadratic and quartic functional equations‎ ‎\begin{align*}‎ ‎f\left(\sqrt{xx^*+yy^*}\hspace{.05cm}\right)=f(x)+f(y),‎ ‎\end{align*}‎ ‎\begin{align*}‎ ‎f\left(\sqrt[4]{(xx^*+yy^*)(xx^*+yy^*)^*}\right)+f\left(\sqrt[4]{(xx^*-yy^*)(xx^*-yy^*)^*}\right)=2f(x)+2f(y)‎ ‎\end{align*}‎ ‎in $C^*$-algebras‎. ‎We also prove the stability of these functional‎ ‎equations in $\beta$-normed spaces by using the Banach fixed point‎ ‎theorem and a direct method.‎