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

In cite{p}, Park introduced the quadratic $rho$-functional inequalities begin{eqnarray} && |f(x+y)+f(x-y)-2f(x)-2f(y)| && qquad le left|rholeft(2 fleft(frac{x+y}{2}right) + 2 flef (frac{x-y}{2}right)- f(x) - f(y)right)right|, nonumber end{eqnarray} where $rho$ is a fixed complex number with $|rho|and begin{eqnarray} && left|2 fleft(frac{x+y}{2}right) + 2 fleft(frac{x-y}{2}right)- f(x) - f(y)right| && qquad le |rho(f(x+y)+f(x-y)-2f(x)-2f(y))|, nonumber end{eqnarray} where $rho$ is a fixed complex number with $|rho| In this paper, we prove the Hyers-Ulam stability of the quadratic $rho$-functional inequalities (0.1) and (0.2) in $beta$-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of quadratic $rho$-functional equations associated with the quadratic $rho$-functional inequalities (0.1) and (0.2) in $beta$-homogeneous complex Banach spaces.