LAUSR

Let $${\mathbb {C}}$$C be the set of complex numbers, X be a normed space and Y be a Banach space. We investigate the Hyers-Ulam stability theorem when $$f:{\mathbb {C}}\rightarrow Y$$f:C→Y satisfy the following $$\sigma -$$σ-quadratic inequality $$\begin{aligned} \Vert f(x+y)+f(x+\sigma (y))-2f(x)-2f(y)\Vert \le \epsilon \end{aligned}$$‖f(x+y)+f(x+σ(y))-2f(x)-2f(y)‖≤ϵin a set $$\Omega \subset {\mathbb {C}}^2$$Ω⊂C2 of Lebesgue measure $$m(\Omega )=0$$m(Ω)=0, where $$\sigma :X\rightarrow X$$σ:X→X is an involution.