Communications of the
Korean Mathematical Society CKMS

In this paper, we obtain the generalization of the Hyers-Ulam-Rassias stability in the sense of G{\v a}vruta and Ger of the generalized $G$-type functional equations of the form $f(\varphi(x))=\Gamma(x)f(x)$. As a consequence in the cases $\varphi(x):=x+p:=x+1$, we obtain the stability theorem of $G$-functional equation : the reciprocal functional equation of the double gamma function.

Keywords: Functional equation, Hyers-Ulam stability, Hyers-Ulam-Rassias stability, G-function, double gamma function

MSC numbers: 39B52, 39B72, 39B82