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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 $M_n^{[r]}(x_j,p_j)=\left(\sum_{j=1}^n p_jx_j^r\right)^{1/r}\quad(r\not= 0),\qquad M_n^{}(x_j,p_j)=\prod_{j=1}^n x_j^{p_j}.$
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 $\Gamma\left(M_n^{}(x_j,p_j)\right)\leq M_n^{}(\Gamma(x_j),p_j).$
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
##### Keywords:
gamma function; psi function; power means; complete monotonicity
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