Quadratic ρ-functional inequalities in β-homogeneous normed spaces

Abstract

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 (4f (x + y/2) + f (x - y) - 2f(x) - 2f(y)parallel to,

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

parallel to 4f (x + y/2) + f (x - y) - 2f(x) - 2f(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 vertical bar rho vertical bar < 1.

Using the direct method, we prove the Hyers-Ulam stability of the quadratic beta-functional inequalities (1) and (2) in fi homogeneous complex Banach spaces.