Additive-quadratic rho-functional inequalities in Banach spaces

Abstract

LetM(1)f(x,y): = 3/4 f(x + y) - 1/4 f(-x -y)+1/4f(x - y) + 1/4f (y - x) - f (x) - f(y),M(2)f(x, y) := 2f (x+ y/2) + f(x-y/2)+f(y-x/2) f(x) - f(y).We solve the additive-quadratic rho-functional inequalitiesparallel to M(1)f(x,y)parallel to <= parallel to rho M(2)f(x, y)parallel to, (0.1)where rho is a fixed complex number withvertical bar rho vertical bar < 1/2 and parallel to M(2)f(x,y)parallel to <= parallel to rho M(1)f(x,y)parallel to, (0.2)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 additive -quadratic rho-functional inequalities (0.1) and (0.2) in complex Banach spaces.