Bi-additive s-functional inequalities and quasi-multipliers on Banach algebra

Abstract

In this paper, we solve the following bi-additive s-functional inequalities:

parallel to f( x + y, z - w ) + f( x - y, z + w ) - 2f( x, z )+2f ( y, w )parallel to <= parallel to s(2f(x+y/2, z-w) + 2f (x-y/2, z+w) - 2f(x, z) + 2f(y, w))parallel to,

parallel to 2f(x+y/2, z-w)+2f(x-y/2, z+w)-2f(x, z)+2f(y, w)parallel to <= parallel to s(f(x+y, z-w)+f(x-y, z+w)-2f(x, z)+2f(y, w))parallel to,

where s is a fixed nonzero complex number with vertical bar s vertical bar < 1. We also prove the Hyers-Ulam stability of quasi-multipliers on Banach algebras and unital C*-algebras associated with the bi-additive s-functional inequalities above.