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Power product inequalities for the \(\Gamma_k \) function. (English) Zbl 1207.33001

Summary: We give an upper and a lower power product estimate for the \(k\)-gamma function
\[ \begin{split} \frac{k^nn!}{(x)_{n,k}}\cdot \bigg(\frac{x_kn}{k+kn}\bigg)^{kn}\cdot e^{[\frac1k H(n)-1](x-k)}\leq \Gamma_k(x)\leq \frac{k^{n+1}(n+1)!}{(x)_{n+1,k}}\cdot \bigg(\frac{x_kn}{k+kn}\bigg)^{kn}\cdot e^{[\frac1k H(n+1)-1](x-k)}\\ (x>0,\;k>0,\;n=1,2,\dots) \end{split} \]
hold, where \(H(n)=1+\frac12+\cdots+\frac1n\).

MSC:

33B15 Gamma, beta and polygamma functions
26D07 Inequalities involving other types of functions
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