ADDITIVE rho-FUNCTIONAL INEQUALITIES AND EQUATIONS

Abstract

In this paper, we investigate the additive rho-functional inequalities parallel to f(Sigma(k)(j=1)xj) - Sigma(k)(j=1)f(xj)parallel to <=parallel to rho(kf(Sigma(k)(j=1)xj/k) - Sigma(k)(j=1)f(xj)parallel to (0.1) and parallel to kf(Sigma(k)(j=1)xj/k - Sigma(k)(j=1)f(xj)parallel to <=parallel to rho(f(Sigma(k)(j=1)xj) - Sigma(k)(j=1)f(xj)parallel to, (0,2) where rho is a fixed complex number with vertical bar rho vertical bar < 1. Furthermore, we prove the Hyers-Ulam stability of the additive rho-functional inequalities (0.1) and (0.2) in complex Banach spaces and prove the Hyers-Ulam stability of additive rho-functional equations associated with the additive rho-functional inequalities (0.1) and (0.2) in complex Banach spaces.