Symmetric biderivations on Banach algebras

Abstract

In [25], Park introduced the following bi-additive s-functional inequalities (1) $$\matrix{{{\rm{||}}f(x + y, z - w) + f(x - y, z + w) - 2f(x, z) + 2f(y, w){\rm{||}}} \hfill \cr {\;\;\;\; \le \left\| {s\left({2f\left({{{x + y} \over 2}, z - w} \right) + 2f\left({{{x - y} \over 2}, z + w} \right) - 2f(x, z) + 2f(y, w)} \right)} \right\|} \hfill \cr}$$ and (2) $$\matrix{{\left\| {2f\left({{{x + y} \over 2}, z - w} \right) + 2f\left({{{x - y} \over 2}, z + w} \right) - 2f(x, z) + 2f(y, w)} \right\|} \hfill \cr {\;\; \le {\rm{||}}s(f(x + y, z - w) + f(x - y, z + w) - 2f(x, z) + 2f(y, w)){\rm{||,}}} \hfill \cr} $$ where s is a fixed nonzero complex number with |s| < 1. In this paper, we prove the Hyers-Ulam stability of symmetric biderivations and skew-symmetric biderivation on Banach algebras and unital C* -algebras, associated with the bi-additive s-functional inequalities (1) and (2).