ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN β-HOMOGENEOUS NORMED SPACES

Abstract

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

parallel to f(x + y) + f(x - y) - 2f(x) - f(y) - f(-y)parallel to (0.1)

<= parallel to rho(2f (x + y/2) +2f (x - y/2) -3/2f(x) + 1/2f(x) -1/2f(y) - 1/2f(-y))parallel to,

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) - 3/2f(x) + 1/2f(-x) - 1/2f(y) - 1/2f(-y)parallel to

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

where rho is a fixed complex number with vertical bar rho vertical bar < 1/2-, and prove the Hyers-Ulam stability of the additive quadratic rho-functional inequalities (0.1) and (0.2) in beta-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of additive-quadratic rho-functional equations associated with the additive-quadratic rho-functional inequalities (0.1) and (0.2) in beta-homogeneous complex Banach spaces.