LAUSR

Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of biderivations and bihomomorphisms in Banach algebras and unital $$C^*$$ C ∗ -algebras, associated with the bi-additive functional inequality: 1 $$\begin{aligned}&\Vert f(x+y, z+w) + f(x+y, z-w) + f(x-y, z+w) \nonumber \\&\quad + f(x-y, z-w) -4f(x,z)\Vert \nonumber \\&\quad \le \left\| s \left( 2f\left( x+y, z-w\right) + 2f\left( x-y, z+w\right) - 4f(x,z )+ 4 f(y, w)\right) \right\| , \end{aligned}$$ ‖ f ( x + y , z + w ) + f ( x + y , z - w ) + f ( x - y , z + w ) + f ( x - y , z - w ) - 4 f ( x , z ) ‖ ≤ s 2 f x + y , z - w + 2 f x - y , z + w - 4 f ( x , z ) + 4 f ( y , w ) , where s is a fixed nonzero complex number with $$|s |< 1$$ | s | < 1 .