Quadratic \rho-functional inequalities in \beta-homogeneous normed spaces

In [۱۲], Park introduced the quadratic \rho-functional inequalities\begin{eqnarray}\label{E۰۱}&& \|f(x+y)+f(x-y)-۲f(x)-۲f(y)\| \\ && \qquad \le  \left\|\rho\left(۲ f\left(\frac{x+y}{۲}\right) + ۲ f\left(\frac{x-y}{۲}\right)- f(x) -  f(y)\right)\right\|,  \nonumber\end{eqnarray}where \rho is a fixed complex number with |\rho|<۱,and\begin{eqnarray}\label{E۰۲}&& \left\|۲ f\left(\frac{x+y}{۲}\right) + ۲ f\left(\frac{x-y}{۲}\right)- f(x) -  f(y)\right\| \\ && \qquad \le  \|\rho(f(x+y)+f(x-y)-۲f(x)-۲f(y))\|   , \nonumber\end{eqnarray}where \rho is a fixed complex number with |\rho|<\frac{۱}{۲}.In this paper, we prove the Hyers-Ulam stability of the quadratic \rho-functional inequalities (۰.۱) and (۰.۲)  in \beta-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of quadratic \rho-functional equations associated with  the quadratic \rho-functional inequalities(۰.۱) and (۰.۲) in \beta-homogeneous complex Banach spaces.