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

In this paper, we solve the following additive-quadratic ρ-functional inequalities ‖f(x + y) + f(x - y) - 2f(x) - f(y) - f(-y)‖ (0.1) ⩽ ǁρ 2f ( x + y 2 ) + 2f ( x - y 2 ) - 3/2 f(x) + 1 2 f(-x) - 1/2 f(y) - 1/2 f(-y)ǁ ; where ρ is a fixed complex number with ǀρǀ < 1, and ǁ2f ( x + y 2 ) + 2f (-x - y 2) - 3/2 f(x) + 1 2 f(-x) - 12 f(y) - 1/2 f(-y)ǁ ⩽‖ρ(f(x + y) + f(x - y) - 2f(x) - f(y) - f(-y)ǁk; (0.2) where ρ is a fixed complex number with ǀρǀ < 1 2 , and prove the Hyers-Ulam stability of the additivequadratic ρ-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of additive-quadratic ρ-functional equations associated with the additive-quadratic β-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banach spaces. [ABSTRACT FROM AUTHOR]