Remark on the stability of the gamma functional equation (Q1297017)

Let \(\delta>0\) and let \(f:(0,+\infty)\rightarrow (0,+\infty)\) satisfy \(| f(x+1)-xf(x)| \leq\delta\) for \(x>0\). It is proved that there exists a unique \(F:(0,+\infty)\rightarrow (0,+\infty)\) such that \(F(x+1)=xF(x)\) and (1) \(| f(x)-F(x)| \leq e\delta /x\) for \(x>0\), where \(e\) denotes Euler's number. Moreover, it is shown that \(e\) is the best possible constant in (1). This brief note generalizes a result obtained by \textit{S.-M. Jung} [Result. Math. 33, No. 3-4 , 306-309 (1998; Zbl 0907.39027)].