QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN BANACH SPACES

In this paper, we solve the 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, (0.1) where rho is a fixed complex number with vertical bar rho vertical bar < 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, (0.2) where rho is a fixed complex number with vertical bar rho vertical bar < 1/2. Using the direct method, we prove the Hyers-Ulam stability of the quadratic rho-functional inequalities (0.1) and (0.2) in 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 complex Banach spaces.