LAUSR

In this paper, we solve the additive s -functional inequalities ‖ f (x+ y− z)− f (x)− f (y)+ f (z)‖ ‖s( f (x− y)+ f (y− z)− f (x− z))‖, (0.1) where s is a fixed nonzero complex number with |s| < 1 , and ‖ f (x− y)+ f (y− z)− f (x− z)‖ ‖s( f (x+ y− z)− f (x)− f (y)+ f (z))‖, (0.2) where s is a fixed nonzero complex number with |s| < 1 . Furthermore, we prove the Hyers-Ulam stability of the additive s -functional inequalities (0.1) and (0.2) in complex Banach spaces. This is applied to investigate partial multipliers in Banach ∗ -algebras and unital C∗ -algebras, associated with the additive s -functional inequalities (0.1) and (0.2). Mathematics subject classification (2010): 39B52, 46L05, 39B62, 43A22.