CUBIC ρ-FUNCTIONAL INEQUALITY AND QUARTIC ρ-FUNCTIONAL INEQUALITY

Abstract

In this paper, we solve the following cubic rho-functional inequality parallel to (2x + y) + f (2x - y) - 2f (x + y) - 2f (x - y) - 12f (x)parallel to <= parallel to rho (4f + (x + y/2) + 4f f(x-y/2) - f (x + y) - 6f(x))parallel to, (0.1)

where rho is a fixed complex number with broken vertical bar p broken vertical bar < 2, and the quartic rho-functional inequality <= parallel to rho (8f (x + y/2) + 8f (x -y/2) -4f (x - y) -24f (x) + 6f(y)parallel to (0.2) <= parallel to rho(8f (x + y/2) + 8f (x-y/2) -2f (x + y) -2f(x -y) -12f(x) + 3f(y) parallel to where rho is a fixed complex number with broken vertical bar rho broken vertical bar < 2.

Using the direct method, we prove the Hyers-Ulam stability of the cubic p -functional inequality (0.1) and the quartic rho-functional inequality (0.2) in complex Banach spaces.