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Two families of approximations for the gamma function
In this paper, we establish two families of approximations for the gamma function: $$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{...
Estimating gamma function by digamma function
The aim of this paper is to refine some recent results stated by Alzer and Grinshpan [H. Alzer, A.Z. Grinshpan, Inequalities for the gamma and q-gamma functions J. Approx. Theory 144 (2007) 67-83] and Batir [N. Batir, An interesting double inequality ...
Some Mean Value Inequalities for the Gamma Function
We determine the infimum of the harmonic mean of $\Gamma (x_1 ),\Gamma (x_2 ), \cdots ,\Gamma (x_n )$ under the constraints $\Pi _{k = 1}^n x_k = 1$, all $x_k > 0$. We present numerical evidence for this infimum to be equal to $\Gamma (1) = 1$ if $n \...
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