Extremal values of multiple gamma and sine functions (Q973929)

Barnes's multiple gamma function \(\Gamma_{r}(x)\) is defined by \[ \Gamma_{r}(x)=\exp\Big(\frac{\partial}{\partial s}\zeta_{r}(s,\,x)\Big|_{s=0}\Big) \;\;\;(x>0) \] where \(\zeta_{r}(s,\,x)\) is the multiple Hurwitz zeta function defined as \[ \zeta_{r}(s,\,x)=\sum_{n_{1},\dots,n_{r}=0}^{\infty}\frac{1}{(n_{1}+\dots + n_{r}+x)^{s}}\;\;\;\;(\text{Re} (s)>r). \] The multiple sine function \(S_{r}(x)\) is defined by \[ S_{r}(x)=\Gamma_{r}(x)^{-1}\,\Gamma_{r}(r-x)^{(-1)^{r}}\;\;\;\;(0<x<r). \] The author studies the extremal values of the functions \(\Gamma_{r}(x)\) and \(S_{r}(x)\) in the fundamental intervals, shows the number and locations of the extremal points and also proves that all the local maximum and minimum values are greater and less than one, respectively.