论文标题
$ q- $ gamma和$ q- $ digamma功能的谐波平均不平等
A harmonic mean inequality for the $q-$gamma and $q-$digamma functions
论文作者
论文摘要
我们证明了其他结果，结果表明，$γ_Q（x）$和$γ_Q（1/x）$的谐波平均值大于或等于$ 1 $，nutionary $ x> 0 $和$ q \ in J $ in J $，其中$ j $是$ [0，+\ \ infty）$的子集。另外，我们证明在（1,9/2）$中有$ p_0 \，因此，对于$ q \ in（0，p_0）$，$ψ_q（1）$是$ψ_q（x）$和$ c（1/x）$的谐音平均值的最低含量，$ x> $ x> $ q y;最大。我们的结果概括了由于Alzer和Gautschi引起的一些已知不平等现象。
We prove amongs others results that the harmonic mean of $Γ_q(x)$ and $Γ_q(1/x)$ is greater than or equal to $1$ for arbitrary $x > 0$ and $q\in J$ where $J$ is a subset of $[0,+\infty)$. Also, we prove that for there is $p_0\in(1,9/2)$, such that for $q\in(0,p_0)$, $ψ_q(1)$ is the minimum of the harmonic mean of $ψ_q(x)$ and $ψ_q(1/x)$ for $x > 0$ and for $q\in(p_0,+\infty)$, $ψ_q(1)$ is the maximum. Our results generalize some known inequalities due to Alzer and Gautschi.
