On the Ulam-Hyers stability of the complex functional equation \(F(z)+F(2z)+\cdots +F(nz)=0 \) (Q2198307)

\textit{G. Mora} et al. [Comput. Math. Appl. 39, No. 9--10, 45--55 (2000; Zbl 0954.39016)] studied the general continuous solution of the functional equations \(f(x)+f(2x)=0\) and \(f(x)+f(2x)+f(3x)=0\), where \(f\) is a real function. In the present paper, the authors prove the generalized Hyers-Ulam stability of the following complex functional equation \[F(z) + F(2z) + \dots+ F(nz) = 0,\,\,\,n\geq 2,\,\,\,z\in\mathbb{C}- (-\infty, 0].\] In fact, it is a complex version of \[f(x)+f(2x)+ \dots+f(nx) = 0,\,\,\,n\geq 2,\,\,\,x>0\] that was studied in [\textit{G. Mora}, J. Math. Anal. Appl. 340, No. 1, 466--475 (2008; Zbl 1228.39020)].