Quadratic rho- functional inequalities in beta-homogeneous normed spaces

In this paper, we solve the quadratic rho functional inequalities \[\|f(x+y)+f(xy)2f(x)2f(y)\|\leq\|\rho(4f(\frac{x+y}{2})+f(xy)2f(x)2f(y))\|,\] where rho is a fixed complex number with \(|\rho| lt;1 , and\[\|4f(\frac{x+y}{2})+f(xy)2f(x)2f(y)\|\leq\|\rho(f(x+y)+f(xy)2f(x)2f(y))\|,\] where rho is a fixed complex number with \(|\rho| lt;1 . Using the direct method, we prove the HyersUlam stability of the quadratic rho functional inequalities (1) and (2) in beta homogeneous complex Banach spaces.