Remarks on the functional equation $$f(x+1)=g(x)f(x)$$ and a uniqueness theorem for the gamma function

The topic of gamma type functions and related functional equation \(f(x+1)=g(x)f(x)\) has been seriously studied from the first half twentieth century till now. Regarding unique solutions asymptotic condition \(\displaystyle {\lim _{x \rightarrow \infty }}\frac{g(x+w)}{g(x)}=1\), for each \(w>0\), is considered in many serial papers including R. Webster’s article (1997). Motivated by limit summability real (introduced author 2001), we show that paper can be reduced to convergence sequence \(\frac{g(n+1)}{g(n)}\) 1, then removing initial \(f(1)=1\) generalize it. On other hand, Rassias Trif proved if g satisfies another appropriate condition, admits at most one solution f, which eventually \(\log \)-convex second order. We also sow without changing this case, a uniqueness theorem similar result valid f This implies new function states \)-convexity Bohr-Mollerup Theorem replaced \)-concavity order two. At last, some important questions about them will asked.