Quadratic ρ-functional inequalities in complex matrix normed spaces

In this paper, we solve the following quadratic ρ-functional inequalities ║f (x + y)+f (x − y) − 2f (x) − 2f (y)║ ≤ ║ρ(2f (x+y/2) +2f (x-y/2) − f(x) − f(y) )║ , where ρ is a fixed complex number with |ρ| 1, and ∥2f(x+y/2)+2f(x−y/2)−f(x)−f(y)∥≤∥ρ(f(x+y)+f(x−y)−2f(x)−2f(y)∥, where ρ is a fixed complex number with | ρ | 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 ρ -functional equations associated with these inequalities in complex matrix normed spaces.