Quadratic ρ-functional inequalities in complex matrix normed spaces

Abstract

In this paper, we solve the following quadratic rho-functional inequalities

parallel to f(x + y) + f(x - y) - 2f(x) - 2f(y)parallel to <= parallel to rho(2f(x + y)/2) + 2f(x - y/2) - f(x) - f(y))parallel to,

where rho is a fixed complex number with |rho| < 1, and

parallel to 2f(x + y/2)+2f(x - y/2) - f(x) - f(y)parallel to <= parallel to rho(f(x + y) + f(x - y) - 2f(x) - 2f(y))parallel to,

where rho is a fixed complex number with |rho| < 1/2. By using the direct method, we prove the Hyers-Ulam stability of these inequalities in complex matrix normed spaces, and prove the Hyers-Ulam stability of quadratic rho-functional equations associated with these inequalities in complex matrix normed spaces.