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A power mean inequality for the gamma function. (English) Zbl 0964.33002
In this interesting paper, the author extends a result due to L. G. Lucht [Aequationes Math. 39, No. 2/3, 204-209 (1990; Zbl 0705.39002)] on convexity-like inequalities for Euler’s gamma function, involving the geometric mean.
Let \(x_j>0,\;p_j>0\;(1\leq j\leq n),\;p_1+\cdots +p_n=1\) and \(n\geq 2\). Then the old result can be stated as: \[ \Gamma\left(x_1^{p_1}\cdots x_n^{p_n}\right)\leq \left(\Gamma\left(x_1^{p_1}\right)\right) \cdots \left(\Gamma\left(x_n^{p_n}\right)\right) \] for all \(x_j\geq a\) or \(\leq a\), with \(a\) the unique positive zero of \(\psi(x)+x\psi'(x)\). Equality holds if and only if \(x_1=\cdots =x_n\).
In the paper the power mean for the numbers \(\{x_j\}\), weights \(\{p_j\}\) and a real parameter \(r\) is defined as \[ M_n^{[r]}(x_j,p_j)=\left(\sum_{j=1}^n p_jx_j^r\right)^{1/r}\quad(r\not= 0),\qquad M_n^{[0]}(x_j,p_j)=\prod_{j=1}^n x_j^{p_j}. \]
As the gamma function is strictly convex for \(x>0\), we have the well known inequality \[ \Gamma(p_1x_1+\cdots p_nx_n)\leq p_1\Gamma(x_1)+\cdots p_n\Gamma(x_n), \] which can now be seen as \[ \Gamma\left(M_n^{[1]}(x_j,p_j)\right)\leq M_n^{[1]}(\Gamma(x_j),p_j). \]
The author now introduces the following real numbers: \(r_0=0.21609\ldots\) : the unique positive zero of \(\psi(x)+x\psi'(x)\), \(r_1=1.46163\ldots\) : the unique positive zero of \(\psi(x)\), \(r_2=2.08907\ldots\) : the unique positive zero of \(x\psi(x)-1\), \[ \begin{alignedat}{2} \alpha&=0.01317\ldots&&= \sup_{0<x<r_0} \left\{\psi(x)+x\psi'(x)\right\}/\left\{\psi(x)-x(\psi(x))^2 \right\}, \\ \beta&=11.29416\ldots &&= \inf_{r_1<x<r_2} \left\{\psi(x)+x\psi'(x)\right\}/\left\{\psi(x)-x(\psi(x)) ^2\right\}. \end{alignedat} \] The main result of the paper is: \[ \Gamma\left(M_n^{[r]}(x_j,p_j)\right)\leq M_n^{[r]}(\Gamma(x_j),p_j) \] holds for all positive \(n\)-tuples \(x_j\) and all positive weights \(p_j\) with sum \(1\), if and only if \(\alpha\leq r\leq \beta\).
Furthermore some corollaries are given, one of which concerns the case \(r=-1\): the classical harmonic mean. For this mean the author shows \[ \text{all }x_j\in [a,b]\Rightarrow\Gamma\left(M_n^{[-1]}(x_j,p_j)\right)\leq M_n^{[-1]}(\Gamma(x_j),p_j), \]
\[ \text{either all }x_j\in (0,a]\text{ or all }x_j\in [b,\infty) \Rightarrow\Gamma\left(M_n^{[-1]}(x_j,p_j)\right)\geq M_n^{[-1]}(\Gamma(x_j),p_j), \] where \(0<a<b\) are the only positive zeros of \(2\psi(x)-x(\psi(x))^2+x\psi'(x)\). Again equality holds if and only if \(x_1=\cdots =x_n\).

MSC:
33B15 Gamma, beta and polygamma functions
26D15 Inequalities for sums, series and integrals
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