Cyclic mean-value inequalities for the gamma function

Horst Alzer¹

¹ Morsbacher Str. 10 D-51545 Waldbröl, Germany

Colloquium Mathematicum, Tome 132 (2013) no. 1, pp. 27-34

Résumé

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, $$ P(x)=2x-1+x(x-1)\frac{\psi'(x)}{\psi(x)} \quad\mbox{and} \quad{\psi=\varGamma'/\varGamma}. $$

Détail

DOI : 10.4064/cm132-1-3

Keywords: present cyclic inequalities involving classical vargamma function euler unweighted power mean b frac t right quad neq quad sqrt quad leq mathbb mathbb inequality prod vargamma frac j right leq prod vargamma frac right quad holds ldots only leq leq mathbb mathbb inequality prod vargamma frac right leq prod vargamma frac j right quad valid ldots only geq max ldots here x x frac psi psi quad mbox quad psi vargamma vargamma