Cyclic mean-value inequalities for the gamma function
Volume 132 / 2013
Abstract
We present two cyclic inequalities involving the classical $\varGamma$-function of Euler and the (unweighted) power mean $$ M_t(a,b)=\left(\frac{a^t+b^t}{2}\right)^{1/t} \quad (t\neq 0), \quad\ M_0(a,b)=\sqrt{ab} \quad (a,b>0). $$
(I) Let $2\leq n\in\mathbb{N}$ and $r\in\mathbb{R}$. The inequality $$ \prod_{j=1}^n \varGamma\left(\frac{1}{1+M_r(x_j,x_{j+1})}\right) \leq \prod_{j=1}^n \varGamma\left(\frac{1}{1+x_j}\right) \quad\ (x_{n+1}=x_1) $$ holds for all $x_j>0$ $(j=1,\ldots ,n)$ if and only if $r\leq 0$. (II) Let $2\leq n \in\mathbb{N}$ and $s\in \mathbb{R}$. The inequality $$ \prod_{j=1}^n \varGamma\left(\frac{1}{1+x_j}\right) \leq \prod_{j=1}^n \varGamma\left(\frac{1}{1+M_s(x_j,x_{j+1})} \right) \quad\ (x_{n+1}=x_1) $$ is valid for all $x_j>0$ $(j=1,\ldots,n)$ if and only if $$ s\geq \max_{0 < x < 1} P(x)=1.0309\ldots . $$ Here, $$ P(x)=2x-1+x(x-1)\frac{\psi'(x)}{\psi(x)} \quad\mbox{and} \quad{\psi=\varGamma'/\varGamma}. $$
