CUBIC ρ-FUNCTIONAL INEQUALITY AND QUARTIC ρ-FUNCTIONAL INEQUALITY.

In this paper, we solve the following cubic ρ-functional inequality ‖f(2x + y) + f(2x - y) - 2f(x + y) - 2f(x - y) - 12f(x)‖ ⩽ ǁ ρ 4f (x + y/2 )+ 4f (x - y/2) - f(x + y) - f(x - y) - 6f(x))ǁ ; (0.1) where ρ is a ρ xed complex number with ǀρǀ < 2, and the quartic ρ-functional inequality ‖f(2x + y) + f(2x - y) - 4f(x + y) - 4f(x - y) - 24f(x) + 6f(y)‖ (0.2) ⩽ ǁ ρ(8f x + y/2( + 8f (x - y/2) - 2f(x + y) - 2f(x - y) - 12f(x) + 3f(y) ǁ ; where ρ is a fixed complex number with ǀρǀ <2. Using the direct method, we prove the Hyers-Ulam stability of the cubic ρ-functional in- equality (0.1) and the quartic ρ-functional inequality (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]