Quadratic ρ-functional inequalities in banach spaces

In this paper, we solve the quadratic ρ-functional inequalities equation(0.1)   ‖f(x+y)+f(x−y)−2f(x)−2f(y)‖≤‖ρ(2f(x+y2)+2f(x−y2)−f(x)−f(y))‖,where ρ is a fixed complex number with |ρ| < 1, and equation(0.2)    ‖2f(x+y2)+2f(x−y2)−f(x)−f(y)‖≤‖ρ(f(x+y)+f(x−y)−2f(x)−2f(y))‖,where ρ is a fixed complex number with |ρ| < ½.Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces and prove the Hyers-Ulam stability of quadratic ρ-functional equations associated with the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces.