Extremal values of multiple gamma and sine functions
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.
scientific article; zbMATH DE number 5712548