Functional inequalities and pairs of hom-derivations and homomorphisms

Abstract

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

2g

x + y 2

− g(x) − g(y)

+

2h

x + y 2

+ 2h

x − y 2

− 2h(x)

(1) ≤ ∥s (g (x + y) − g(x) − g(y))∥ + ∥t(h(x + y) + h(x − y) − 2h(x))∥ , where s and t are fixed nonzero complex numbers with |s|+|t| < 1. We define a pair of hom-derivation and homomorphism in complex Banach algebras, and using the direct method and the fixed point method, we prove the Hyers-Ulam stability of pairs of hom-derivations and homomorphisms in complex Banach algebras, associated to the additive-additive (s, t)-functional inequality (1) and the following functional inequality (2) ∥g(xy) − g(x)h(y) − h(x)g(y)∥ + ∥h(xy) − h(x)h(y)∥ ≤ φ(x, y)