Derivation-homomorphism functional inequalities

Abstract

In this paper, we introduce and solve the following additive-additive (s,t)-functional inequality

parallel to g(x +y) - g(x) - g(y)parallel to + parallel to h(x + y) + h(x - y) - 2h(x)parallel to <= parallel to s(2g(x+y/2) - g(x) - g(y)) parallel to + parallel to t (2h(x+y/2) + 2h(x-y/2) - 2h(x)parallel to, (1)

when s and t arc fixed nonzero complex numbers with vertical bar s vertical bar < 1 and vertical bar t vertical bar < 1. Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of derivation-homomorphisms in complex Banach algebras, associated to the additive-additive (s,t)-functional inequality (1) and the following functional inequality

parallel to g(xy) - g(x)y - xg(y)parallel to + parallel to h(xy) - h(x)h(y)parallel to <= phi(x,y). (2)